物理科学现代艺术图片欣赏用形象的艺术语言表达深奥的物理科学
Caustic I
Three-dimensional caustics formed on a flat sea bottom by light passing through two consecutive wavy surfaces.
Caustics are places where things accumulate; in this case light is accumulating,We often think of focal points as places where light gathers after passing through a lens,but more generally,for "random" lenses,there are many more interesting patterns to examine,Caustics are seen abundantly in the two-dimensional electron flow Transport series; there,we are looking at the flow as it moves along in two dimensions,Here,we are seeing the flow of light in three dimensions as it is interrupted by the sea bottom surface,This pattern is not possible for a true sea bottom,because the light has passed through seven consecutive surfaces,being refracted twice,(Caustic I uses two such wavy surfaces to refract the light)
Caustic II
Three-dimensional caustics formed on a flat sea bottom by light passing through seven consecutive wavy surfaces.
Caustics are places where things accumulate; in this case light is accumulating,We often think of focal points as places where light gathers after passing through a lens,but more generally,for "random" lenses,there are many more interesting patterns to examine,Caustics are seen abundantly in the two-dimensional electron flow Transport series; there,we are looking at the flow as it moves along in two dimensions,Here,we are seeing the flow of light in three dimensions as it is interrupted by the sea bottom surface,This pattern is not possible for a true sea bottom,because the light has passed through seven consecutive surfaces,being refracted twice,(Caustic I uses two such wavy surfaces to refract the light,Caustic II uses seven).
Caustic IV
Caustic IV displays folds,cusps,and swallowtails,which are typical caustic structures,here formed by looking through a ruled,transparent colored three-dimensional curved sheet.
Caustic IV displays folds,cusps,and swallowtails,which are typical caustic structures,here formed by looking through a ruled,transparent colored three-dimensional curved sheet,The sheet itself is smooth (but not flat); when we project it onto a plane (by looking through it from a certain angle) we see accumulation regions where material builds up along the line of sight,One of the most common caustics is called a cusp,Cusps result when a flat part of a sheet develops a fold somewhere along the sheet,At a definite point,we can see two new edges or caustics where before none existed,Several of these can be seen in Caustic IV,Caustic IV emphasizes the appearance of caustics in projections of higher dimensional objects onto lower dimension,a property also present in Torus III and Torus IV,A caustic is a region where the higher dimensional surface lies tangent to the projection,thus it’s shadow,piles up” along a caustic,Color hue and value in this image are determined in part by color subtraction of overlapping parts of the sheet.
Interpenetrating Surfaces
We peer through several interpenetrating approximately ellipically shaped surfaces.
Electrons in molecules normally move a lot faster than the nuclei,This leads to the concept of effective potential energy surfaces that govern the motion of nuclei,These surfaces interpenetrate,and intersect in various ways,Their intersection actually causes a breakdown in the assumption that the motion of the nuclei can take place on the surfaces in the first place,The breakdown leads to interesting experimental consequences,Here we peer through several interpenetrating approximately ellipically shaped surfaces,transparent enough to see through into the next surface,Caustics develop as the viewing direction goes parallel to the surfaces,The breakdown of the so-called "adiabatic" approximation is suggested by the broken nature and color chaos of the surfaces.
2DEG
Variation and elaboration of an illustration for an article on two dimensional electron gases in Physics Today.
This image began as an illustration for an article on two dimensional electron gases in Physics Today,Depicting various layers in the micron sized heterostructures,and especially the electron donating atoms together with the effective potential energy landscape which induce on the electrons living in the interfacial layer between Gallium Arsinide and Aluminum Gallium Arsinide semiconductor crystals,the original image invited variation and elaboration.
Analyzed Collision
Collisions between polyatomic and diatomic molecules,with acceleration vectors included.
Two different collisions are shown,one behind the other,Each collision is performed in time steps,and at each step the atoms making up the molecules are drawn,For example,the twisting red and green track on the lower left is a diatomic molecule,vibrating and rotating as it moved toward the edge of the image,Each of the two collisions actually took place in two dimensions,i.e,the plane of the image,Therefore,when the track of one atom is hiding another,it is because that atom appeared there after the other had passed by,The overall effect is three dimensional,but the third dimension is really time,not depth! Knowing this,it is possible to reconstruct much of the history of the collisions from the image,The acceleration arrows show how much and in what direction each atom was accelerating at each step,The collision in the foreground proceeded from top to bottom,the molecules entering the scene from the top and upper left,and after colliding in the middle,exit on the bottom and lower right.
Collision II
Stroboscopic record of a collision between four diatomic molecules,a tetra-atomic molecule,and an atom.
This is the stroboscopic record of a collision between four diatomic molecules,a tetra-atomic molecule,and an atom,The vibrations and rotations of the molecules before and after collision are seen in the peculiar intertwining paths executed by the atoms in the molecule,This is a classical picture of the collision showing chaos in the very complicated and sensitive dependence on exactly where the atoms are as they begin to collide,The collision proceeded from top to bottom,the molecules entering the scene from the top and upper left,and after colliding in the middle,exit on the bottom and lower right,although two diatomic molecules started within the scene.
Rotating Rotator I
Tracks of three different four segmented rotators are seen in the foreground,as they proceeded from lower part of the image towards the upper.
The simplest rotator consists of two rigid bars pivoted together end to end,The bars are free to rotate around the pivot like the segments of an old-fashioned carpenter's ruler,only without the friction,If you throw such a rotator into the air,the segments will pivot around each other in interesting ways,while the object as a whole flies smoothly through the air,If there are three or more segments,the pivoting is chaotic,First one segment may spin wildly,then all three segments may rotate as a unit,then perhaps the two end segments spin in opposite directions,etc,No matter how many segments there are,the wild rotations of the individual segments,together with the overall rotation as a whole,proceed independently of the smooth motion of what is called the "center of mass" of the object,The reason for this is that the rotator is acting only on itself,The forces,which cause the segments to rotate at different rates,are exerted by the rotator and not by some outside agent,If there is no outside agent,the center of mass moves uniformly,according to Newton's laws,
In Rotating Rotator I,the tracks of three different four segmented rotators are seen in the foreground,as they proceeded from lower part of the image towards the upper,In the background,more rotator paths are shown,The difference is that in the background three sets of rotators actually collided near the middle of the picture,leading to changes in the pattern of rotation and changes in the directions of the center of mass of each of the rotators.
Rotating Rotator II
Three sets of four-segmented rotators were set spinning and traveling from left to right.
The simplest rotator consists of two rigid bars pivoted together end to end,The bars are free to rotate around the pivot like the segments of an old-fashioned carpenter's ruler,only without the friction,If you throw such a rotator into the air,the segments will pivot around each other in interesting ways,while the object as a whole flies smoothly through the air,If there are three or more segments,the pivoting is chaotic,First one segment may spin wildly,then all three segments may rotate as a unit,then perhaps the two end segments spin in opposite directions,etc,No matter how many segments there are,the wild rotations of the individual segments,together with the overall rotation as a whole,proceed independently of the smooth motion of what is called the "center of mass" of the object,The reason for this is that the rotator is acting only on itself,The forces that cause the segments to rotate at different rates are exerted by the rotator and not by some outside agent,If there is no outside agent,the center of mass moves uniformly,according to Newton's laws,
In Rotating Rotator II,three sets of four-segmented rotators were set spinning and traveling from left to right,The images are stroboscopic; meaning that after very short intervals a new picture of what the rotator is doing is taken and added to existing pictures,The rule is that the current image of the rotator overwrites and overlays all prior images,In this way,the history of the rotation and the progress of the rotator can be deduced from the image,
The tracks of three different four segmented rotators are seen in the foreground,as they proceeded from lower part of the image towards the upper,In the background,more rotator paths are shown,The difference is that in the background three sets of rotators actually collided near the middle of the picture,leading to changes in the pattern of rotation and changes in the directions of the center of mass of each of the rotators,
Banyan
Electrons are injected at the lower left and ride over a bumpy landscape,with the elevation slowly rising toward the top of the image.
Electrons are launched from the lower left over a bumpy landscape simulating donor atoms near an electron gas,There is also a potential gradient in this case,slowing the electrons down as the approach the top of the image,and turning them back so that they head toward the bottom again,As they slow down,the bumps become too high to surmount,and several hilltops are visible as excluded holes in the upper third of the picture
Dendrite
Electrons injected at the top of the image rain down in the branched flow pattern seen,as they ride over a bumpy landscape.
These images render electron flow paths in a "two dimensional electron gas",Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,The electron flow images are excellent examples of the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories looks like looking through translucent kelp! Or,like ridges on a mountain.
End of Coherence
Electron flow launched from the upper center in a weakly random potential,showing quantum phase as color,and the decay of the coherence of the phase to the lower right.
Electron flow launched from the upper left in a weakly random potential is colored according to quantum phase,Trajectories were written by color addition,so that color saturation represents coherence of the trajectories,they get farther from the source,they loose coherence with each other,The color banding diminishes,A data subtraction was done using saturation as the marker,leaving the least coherent region in the lower right transparent and showing the artificial color gradient beneath,
Exponential
Electrons launched from the upper right fan out and then form branch,as indirect effects of travelling over bumps.
These images render electron flow paths in a "two dimensional electron gas." Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,These two images are excellent examples of the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories is like looking through translucent kelp,or like ridges on a mountain.
Nanowire
Electron paths in a nanowire,including imperfections in the wire.
As components of electronic devices get ever smaller,wires connecting those components must also shrink in proportion,At the micron to nanometer scale,where devices are now being built,the wave nature of matter is becoming critical,This may be an advantage or it may be a problem,Making and understanding nanowires is certainly a challenge,Real nanowires have imperfections,The image Nanowire grew out of a study of electron flow in a wire riddled with random imperfections,It shows electrons injected at one point contact,the,sun,” flowing out from there to all regions of the wire,The disturbance of the electron tracks by the imperfections is shown in their somewhat unruly paths,The quantum aspect of the electrons is shown in color,we can follow the wave nature of the electrons by assigning yellow to the crest of the wave,blue to a trough,continuously around the color circle,The creative process leading to Naonwire is typical of my artwork,a synthesis of research and artistic creation,each one enhancing the other,Experiments conducted by M,Topinka,B,LeRoy and B,Westervelt measuring electron transport in semiconductor microstructures led to scientific illustrations of electrons riding over bumpy landscape potentials,Experimentation with various methods of recording individual electron tracks (overwrite,transparency,color combination) led to a variety of effects and expanded the horizon of the medium,The resulting Transport series is the first of large format high resolution electron flow images using branched flow physics,These images revealed the caustics formed when electrons flow from a particular point over a hilly landscape.
Transport II
Electrons launched from the center in all directions fan and then form branches,as indirect effects of travelling over bumps.
These images render electron flow paths in a "two dimensional electron gas",Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,These two images are excellent examples of the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories looks like looking through translucent kelp,or like ridges on a mountain.
Transport III
Electron flow with folds and caustics,resulting from injection at the lower right followed by branching due to the rough landscape over which the electrons are travelling.
Loosely speaking,caustics are edges,lines along which one object or space ends and another begins,But edges are usually much more,In a drawing,caustics determine where a line should fall,and where it should begin and end,If the object being rendered is a smooth,three dimensional form,light will usually collect or diminish rapidly at an edge,and detail will accumulate there,This is because the caustic of a curved surface is where we look tangent to (i.e,along) the surface,If we imagine the surface as a thin shell of smoky plastic in front of a uniform gray sky,then the edges (caustics) will be very dark,because there light must pass through much more material to get through than at a typical place,This points to the more general concept of caustics,places where accumulation occurs,
Whether by training or by instinct,we associate a line in a simple drawing with a caustic in the real world,But caustics are not always found at the obvious places,For example,look at the "proper" way to draw a torus (donut) from a certain perspective,This simple drawing is much more convincing than the way that might seem more obvious,The reason is that the top edge persists up to a certain point and then just vanishes; where it vanishes we stop the line,There is no reason for the top edge to stop where the bottom one meets it,as in the incorrect drawing,Caustics are found everywhere,and nowhere are they as beautiful as when looking through a thin folded translucent sheet,such as translucent kelp,One of the caustics we are bound to see is called a cusp It happens when a flat part of the kelp develops a fold as we follow along a blade,At a definite point,we start to see two new edges or caustics arise where before there were none,Light coming through folded kelp is an example of projection,in this case from three dimensions into two (think of taking a picture of the kelp,it is three dimensional,but its image will have to live on the two dimensional surface of the film).
Transport IV
Electrons launched from the bottom fan out and then form branch,as indirect effects of travelling over bumps.
These images render electron flow paths in a "two dimensional electron gas",Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw of my group,and me,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,These two images are excellent examples of 1) the degree to which I manipulate the raw data to get an image,and 2) more important still,the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories looks like looking through translucent kelp! Or,like ridges on a mountain.
Transport IX
Paths of electrons followed for a short time,representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live,
In Transport IX we see the paths of electrons followed for a short time,representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live,The distinct and overlapping patterns resulted from the particular hills and valleys encountered from a new location,It is seen that the branches emanating from different initial launch points cross and seem independent,confirming that they are not due to any fixe features in the potential landscape but rather are due to the history of encounters with hills and valleys "upstream".
Transport VI
Electrons launched from the upper left fan out and then form branches,as indirect effects of travelling over bumps.
Transport VII
Electron flow over a two dimensional hilly terrain.
Electron flow over a two dimensional bumpy surface; the electrons were injected from the top in a uniform sheet,all initially heading straight down,The focussing and defocussing effects of the hilly terrain are clearly seen.
Transport XI
Electron tracks in nano device launched at various places,The bunching or branching of electron tracks depends on where the electrons are launched.
Electron flow started at several points on a single random potential surface,some moved and copied Small-scale electronic devices,the size of a bacterium or even a hundred times smaller,inevitably have minute imperfections in them which cause electrons to scatter and spread out as they progress through the device,We recently discovered that the electrons tend to bunch up and form branches,as is seen in many of the Transport images,This image shows that the branching is not a matter only of the landscape over which electrons are traveling,but also depends on where the electrons begin their journey,In this image,the electrons are launched at different places,over a very small range of initial angles,represented by the narrow "Stems",Smaller initial differences in angle grow quickly,as evidenced by the fanning out of electron paths,This is the beginning of the eventual chaotic motion of these electrons,However,the branching is also evident,Note that the branches cross,This means that the branches are not following specific valleys in the landscape,but are rather indirect effects caused by focusing as electrons travel over bumps and hill
Transport XII
Electrons launched from a particular point ride over a bumpy landscape,forming cusps and branches as they move away from launch point.
Electrons launched from a particular point ride over a bumpy landscape,forming cusps and branches as they move away from launch point,The color of the electron tracks in this image is keyed to the initial angle of launch of the electrons,This was done to see whether the branches which form are all coming from a small range of angles,or are collected from several different angles,Orange and predominately yellow branches show that these branches are coming from a very small range of initial angles of launch,Shortly after the electrons are launched,bombs and dips in the landscaped plaza to focus into V shaped regions called cusps,The edges of these regions often become branches,as is clearly seen in this image.
Transport XIII
This image shows two kinds of chaos,a random quantum wave on the surface of a sphere,and chaotic electron paths launched over a range of angles from a particular point,
This image shows two kinds of chaos,a random quantum wave on the surface of a sphere,and chaotic electron paths launched over a range of angles from a particular point,Transport 13 juxtaposes two kinds of chaos,One is quantum mechanical,and one is classical,The quantum mechanical chaos is represented by the random wave confined to the surface of a sphere,This random wave corresponds to a classical particle moving wildly on the surface of the sphere,The foreground we see the tracks electrons launched from a particular point with a range of angles,The different angles were assigned colors,so that one could see more readily the mechanisms by which the electrons' spread out,The increasingly random paths of the electrons as they travel farther from the launching point are the classical analog of the chaos seen on the sphere.
Transport XVI
Electron flow in a two-dimensional electron gas
Electrons flow in a sheet from a line on the top of the image; quantum phase is shown as color,The electrons are experiencing a random potential that defects them and causes the "curtains" of caustics to form.
Transport XVII
Electron flow launched from the upper center in a weakly random potential.
Electron flow launched from the upper center in a weakly random potential,Trajectories launched evenly over 180 degrees,using strict overwrite of trajectories,This gives a hidden surface effect to the caustics (cusps) which has much the same topology as a true erosion landscape,Color assgned from direction of trajectory.
Transport XVIII
Electron flow launched from the upper center in a weakly random potential.
Electron flow launched from the upper center in a weakly random potential,Trajectories launched evenly over 180 degrees,using strict over write of trajectories,This gives a hidden surface effect to the caustics (cusps) that has much the same topology as a true erosion landscape,Color assigned from direction of trajectory.
Dissippation
Classical dissipoative map with whorls.
Dissipation is an image of a mathematical map,The term "map" is used differently here than the usual meaning of the word,Rather than showing a representation of geographic features,a mathematical map assigns a new point to go on a region to from every point in the region shown on the map,Starting in one place,or point,the map specifies the next point to go to,But that point has another that it goes to,and so on,This is repeated many times,yielding figures of fantastic complexity,
We can imagine mapping every point inside some region,say a circle,which will result in a new region with a new distorted shape and more than likely centered somewhere else,If the area of the mapped circle remains the same,and just the shape changes,we have an area preserving map,If the area contracts,it might after many steps settle on a region of lower dimensionality than the one it started in; this is the case for Dissipation,
In Dissipation,the mapped region is the surface of a sphere,several circles of different colors on a black field were mapped 16 times,There are attracting vortices (whirlpools),where area is destroyed,Some places become so stretched and thinned out that they are transparent,revealing the surface of a "sun" below,The net effect is not unlike the convective cells seen on the surface of star
Homoclinic
Early evolution of a homoclinic tangle ina chaotic map.
Map I
Homoclinic oscillations from iteration of a chaotic map,with color chosen to represent quantum phase.
This image results from a chaotic "map",We start with the yellow region at the upper left,which maps onto the yellow region to its immediate right,and so on as we would read a book,This map has been pixelized so that if any point in a pixel is accessed the whole pixel lights up,The colors are given by the deBroglie wavelength,using a so-called semiclassical approximation to what the quantum map would be.
Map II
An iterated classical map with color chosen by quantum considerations.
An iterated classical map with color chosen by quantum considerations,In the top half,an entire region has been mapped and we are looking at one iteration,in the lower half,a single point has been mapped many times and we have recorded everywhere it went.
Nodal V
Rapid variation of a quantum wave function (middle) gives way to slow variation in the "forbidden" regime.
Tunneling is one of the starkest manifestations of the strange quantum realm,Particles can get through energy barriers that cannot penetrate classically,It is theoretically possible for a baseball to pass through a window or a wall without breaking anything,although the chances of this are so astronomically tiny that it may as well be impossible,However,smaller,lighter particles,such as carbon 60,have been seen to tunnel through barriers,
Nodal V is inspired by this tunneling phenomenon,It is an abstracted version of a quantum wave function confined to an infinitely deep pit,The bottom of the pit is the cent of the images,and the energy of the particle is such that classically it could not progress beyond the radius of the red flame region,The wave is sometimes positive,sometimes negative,and when it changes sign from plus to minus there is always an abrupt color shift,The lines along which the sign change takes place are called nodal lines,Beyond the red zone,the particle is in a "classically forbidden" realm,and the nodal lines change qualitatively,This images shows that some nodal lines persist to infinity,and it indicates haw many such lines do so,The wave function seen in the center is an example of a random quantum wave,and now we know how this randomness is manifested in the tunneling region.
Suris
A point-to-point map of the plane onto itself iterated many times with 500 initial point of various colors initiated at many locations.
This image results from a mathematical "map",which is like each house in a development having a unique "sister" house,There is a funny address book,which just says for each house what its sister house is that it must send letters to,Each house receives letters from just one other house and sends them to just one house,Either house can be itself,which is very rare,or the sending and receiving houses could be the same house,which is also rare,Typically the letter goes on a long journey,which either closes on itself in a simple way or fills a whole region of houses,Suppose a letter is sent from a house to its sister house,The sister sends it on to its sister,etc,All sister houses are somewhere in the same development,Now,it is not obvious (and not true in general) that the letter will reach all the houses in the development,Eventually however it must come back to the house where it originated,even if it has to go to every other house first,Even the same adress book,or map,can differ wildly as to how many houses are reached from a given house,If there are 1000 houses in the development,some houses would reach only a few others before the letter came back,and others could reach 100 or 500,What you see here is the result of sending about 500 "letters" in a development of about 5 million houses,The letter were colored differently to see where they went,There aren't that many strongly distinguishable colors,so sometimes its hard to tell,For example,in the "sun" region,the letters all went around in little ovals,but all were colored only slightly differently in that region,A blue "letter" went to about a million houses,as you can see,These maps show structures which are ubiquitous in nature,The islands are resonances,and the filled in regions (e.g,"sky") are regions of chaos where the letter covers every house in certain areas,but doesn't visit other areas.
Crystal I
Looking corner-on at a small cubic sample of a perfect crystal consisting of a periodic array of three different atoms.
In Crystal I we are looking corner-on at a small cubic sample of a perfect crystal consisting of a periodic array of three different atoms,The top layer (red) and the other two surfaces visible of Crystal I shows the orchard effect,with a much more complicated pattern looking through the bulk of the crystal,which is a three dimensional orchard,In spite of the complexity seen peering through the crystal,Crystal I is a perfectly regular array of three different atoms,This reminds us that even perfectly ordered systems may be quite complex,Note the different atoms that appear on different faces,only red atoms on the top,red and green on the right face,and all three on the left face.
Crystal III
Looking top down into a three-dimensional crystal,quasi-crystalline elements are seen in projection.
Looking top down into a three-dimensional perfect crystal,both crystalline and quasi-crystalline elements are seen,For,if we look through a perfect 3D crystal at irrational angles,we see a quasi-crystalline (non-repeating but non-random space filling pattern) structure,The various "diophantine" solutions of number theory are seen as clear paths through the crystal.
Diophantine
Rectangular ordered array rendered with panoramic "camera".
This image records several interesting phenomena,The crystalline images (Orchard,Crystal I) are the antithesis of chaos,although both images illustrate how perfectly "simple" periodic arrays can have their own intricacies and complexities,Orchard is truly simple,a perfectly rectangular array of dots,This array was "photographed" at close range by a panoramic camera,giving the apparent distortion and distinctive shape,Everyone has had the pleasure of looking down rows in an orchard or field with a repetitive array of objects,The more perfect the array and the thinner the objects,the more we can see through the orchard at various angles,Seeing through the orchard is a matter of looking in directions where a straight line never touches one of the objects in the orchard,Imagine standing near the orchard,and taking a wide angle photograph of it,You (and the camera) see straight paths,lines leading from your eye or the lens through the orchard,The camera records such paths also as straight lines,as you can see in Orchard,Imagine you have your camera set on a road which runs along the orchard,On one side of the road is the first row of trees,and the road stretches to your right and to your left,Now,even though the road and the first row are also a straight line,it is not a line that leads from the camera lens in one direction,and the camera is under no obligation to record it as straight! Thus,the distortions caused by the panoramic camera are very specific,and in spite of the serious camera distortion the paths through the orchard are rendered perfectly straight,
The possible paths through the orchard without hitting a tree are limited by the size of the trees,All of the directions possible through the orchard are given by two integers,n and m,which specify the number of rows n "over" and the number of rows m "up" that we travel as we proceed along the path,So,every direction is specified by a rational fraction n/m,because any common factor of m and n doesn't matter,(For example,going over 1 and up 2 is the same as going over 3 and up 6),For a given size tree,there are a finite number of paths,because if n or m gets too large you hit a tree.
Pyramid
A perfectly regular and predictable crystalline array displays surprising complexity when viewed at a typical angle
Quantize I
Color Quantization
If we can directly construct the wave behavior from the rays,then we understand the relation of classical chaos and wave,or quantum chaos,An example of such a construction in the case of non-chaotic ("regular") motion is given in Quantize I and Rosetta Stone,The separate dots represent the location of a classical particle at equally spaced times,The particle is oscillating in a somewhat asymmetrical bowl,The colors are taken to correspond to the phase of the wave,Phase corresponds to whether you are at a crest or a trough or somewhere in between,A simple sine wave for example oscillates from crest to trough and back to crest again,We represent the phases as colors,so that red corresponds to a crest,and cyan to a trough,When the trajectory (the track the particle or marble takes) overlaps itself,we do not just overwrite what was there before,but rather we add to it,so that we get the effect of interference of waves,In Rosetta Stone we see the wave pattern (see "Chladni") which is constructed out of the trajectory,right before our eyes.
Torus I
Track left by resonant energy transfer between two different types of motion
This image is a three dimensional image (plotted in two dimension) of a four dimensional object,When classical motion of particles is not chaotic,we say it is integrable,it can be confined to the surface of donut-shaped objects or "tori" which live in four or more dimensions,We cannot accurately represent such objects on the two-dimensional surface of an image,But we can try,The twisting surface seen here follows a torus wrapped around a simpler,larger torus,This corresponds to a classical resonance,in which one kind of motion efficiently exchanges energy back and forth with another.
Torus II
Track left by resonant energy transfer between two different types of motion
This image is a three dimensional image (plotted in two dimension) of a four dimensional object,When classical motion of particles is not chaotic,we say it is integrable; it can be confined to the surface of donut-shaped objects or "tori" which live in four or more dimensions,We cannot accurately represent such objects on the two dimensional surface of an image,But we can try,The torus appears to intersect itself,but this is because we are pretending it exists in three dimensions,In the four dimension space,it does not intersect,The surface of the torus was made partial transparent to reveal the structure within.
Torus III
Apparently self-intersecting tori (Torus III and IV).
The dynamics of a system of two degrees of freedom involves two positions and their rate of change,i.e,two velocities; altogether four coordinates,Thus a system of two degrees of freedom,lives” in four mathematical dimensions,If it is non-chaotic,it lives on the surface of two-dimensional torus,embedded in the four dimensional space,In projecting this motion into the plane of the image we have only a,shadow” of the full four dimensional space onto two chosen dimensions,The shadow appears to self-intersect (Torus III) or to turn itself inside out (Torus IV),but in the full four-dimensional space it does no such thing,
TorusIV
A four-dimensional tranparent torus is projected onto two dimensions,showing characteristic caustic structures,caused by viewing tangent to surfaces at some place
Bessel 21
A superposition of 21 plane waves,travelling in 21 evenly space directions around the compass.
A superposition of 21 plane waves,traveling in 21 evenly space directions around the compass,results in and approximation to a Bessel function near the center of the "comet",and increasingly disordered structure away from there,The resulting image,if displayed over a much wider area,would reveal its quasi-crystalline structure,The approximate Bessel function breaks down outside a radius inside the sun region,The region of breakdown is most of the image,The slice shown in lighter colors is one repeating unit as if copied 20 times would regenerate the whole image,This image is a quasi-crystal as well,except that the approximate repetitions of the sun region are very far away,outside the image,All additions of more than three plane waves are quasi-crystals,but the near-repetition distance moves out rapidly as the number of plane waves used increases.
Quasicrystal II
A quasi-crystal generated as an addition of 5 plane waves equally spaced in angle.
This image is a quasi-crystal,showing some aspects of crystalline order,but missing crucial long range order we expect of a crystal,That is,with an ordinary crystal we can always move around by multiples of the repeating distances ("lattice constants") and come to an repeated atom or structure; not so in this quasi-crystal,It looks like a crystal at first glance,but then stubbornly refuses to yield to our human tendency to search for a pattern,There is no repeating pattern,not ever!
It is easy to "tile" a floor with simple triangles,squares,or hexagons (three,four or six sided objects) with no gaps,but it is a maddening fact that you can't tile a floor with pentagons without gaps,Adding two sets of plane waves at right angles gives a square array,just like a square tiling of a floor,Similarly,three wave sets at 120 degrees from each other gives a perfect equilateral triangular,crystalline array,So the idea occurs naturally to "force" a pentagonal tiling with FIVE wavesets equally spaced in angle,that is,at 72 degrees from one another,What we get is not a regular tiling at all,but a quasi-crystal,It is frustrating to look too long at this image! Inspection shows that again and again similar structures appear,some with five-fold symmetry,but no two are exactly alike,and the similar structures do not occur on a regular lattice of positions,The penta-quasicrystal is the beginning of an infinite variety of quasi-crystals,since no regular n-sided tile with n greater than 6 can tile the plane without gaps,As we make n bigger,taking more and more wave sets equally spaced in angle from each other,the 2n-sided "flowers" which are formed make a quasicrystalline array,The flowers grow much farther apart as n increases so that we see but one good flower in Comet and Quasicrystal III.
Quasicrystal III
In addition of 9 plane waves equally spaced in angle "pivoted" about the center of the "sun".
An addition of 9 plane waves equally spaced in angle,The plane waves were "pivoted" about the center of the "sun",giving an approximation to a Bessel function centered there,which breaks down outside a radius which is already inside the sun region,The region of breakdown is most of the image,The slice shown in lighter colors is one repeating unit which if copied 8 times would regenerate the whole image,This image is a quasicrystal as well,except that the approximate repetitions of the sun region are very farther away from each other than in Quasicrystal II,which has 5 waves added together,but not so far away as in Comet,which has 21 waves,All additions of more than three plane waves are quasicrystals,but the near-repetition distance moves out rapidly as the number of plane waves used increases,What you call a "near-repetition" is somewhat arbitrary,The more exact one makes the repetition requirement,the further away the repetitions become,There are no really good repetitions in the image,although there are many poor ones.
Chladni
Hand colored version of one of Chladni's original drawing of nodal lines.
The diagrams of Ernst Chladni (1756-1827) are the scientific,artistic,and even the sociological birthplace of the modern field of wave physics and quantum chaos,Educated in Law at the University of Leipzig,and an amateur musician,Chladni soon followed his love of science and wrote one of the first treatises on acoustics,"Discovery of the Theory of Pitch",Chladni had an inspired idea,to make waves in a solid material visible,This he did by getting metal plates to vibrate,stroking them with a violin bow,Sand or a similar substance spread on the surface of the plate naturally settles to the places where the metal vibrates the least,making such places visible,These places are the so-called nodes,which are wavy lines on the surface,The plates vibrate at pure,audible pitches,and each pitch has a unique nodal pattern,Chladni took the trouble to carefully diagram the patterns,which helped to popularize his work,Then he hit the lecture circuit,fascinating audiences in Europe with live demonstrations,This culminated with a command performance for Napoleon,who was so impressed that he offered a prize to anyone who could explain the patterns,More than that,according to Chladni himself,Napoleon remarked that irregularly shaped plate would be much harder to understand! While this was surely also known to Chladni,it is remarkable that Napoleon had this insight,Chladni received a sum of 6000 francs from Napoleon,who also offered 3000 francs to anyone who could explain the patterns,The mathematician Sophie Germain took he prize in 1816,although her solutions were not completed until the work of Kirchoff thirty years later,Even so,the patterns for irregular shapes remained (and to some extent remains) unexplained,Government funding of waves research goes back a long way! (Chladni was also the first to maintain that meteorites were extraterrestrial; before that,the popular theory was that they were of volcanic origin.) One of his diagrams is the basis for image,which is a playfully colored version of Chaldni's original line drawing,Chladni's original work on waves confined to a region was followed by equally remarkable progress a few years later.
Mixed Chamber
Standing wave in a lemon shaped cavity
Imagine splitting a circle in two equal halves,If we move the halves apart and connect the gaps with straight lines,we have a "stadium" shape,which is perhaps the most famous shape in all modern billiard history,The stadium is also called the Bunimovich stadium,after its inventor,who showed that all of its trajectories are chaotic,except a subset of 0% of them which are periodic,If instead of moving the half circles apart we push them together,the enclosed shape is a "lemon",seen here,The lemon billiard is a "typical" dynamical system,which combines with ordered and chaotic behavior,depending on how the system is set into motion,As seen in Correspondence,there are four classes of trajectories,three of which are ordered and one chaotic,(This changes if we make fatter or thinner lemon shapes by moving the circle halves in or out),This particular standing wave corresponds to the most complicated of the three ordered trajectories,billiard wave functions.
Modes I
A series of standing waves in a lemon shaped cavity.
This image shows a series of standing waves in a lemon shaped cavity,Each standing wave is a different resonant frequency of the chamber,The modes closely follow the patterns of classical ray paths bouncing within the chamber,
Linear Ramp
Random wave on a linear ramp potential in two dimensions
Chaos and unpredictability are also universal,Put waves and chaos together and you get random waves,There is something immediately familiar about them; perhaps we know intuitively that this is the stuff of the universe,
In the atomic domain we must grapple with the fact that particles often act like waves instead of little points of mass,Imagine waves coming at you in a long "set",The energy of the wave and the waves themselves are traveling toward you,but the crests of the waves stretch out in long lines to your right and left,A particle traveling somewhere actually corresponds to the energy in the wave,and in some sense the particle is really a wave,not a particle at all,At the particle bounces around,its associated wave goes with it,and we are led to consider superpositions of many wave "sets" or as we say,wave trains,The rule is the wave crests and troughs are perpendicular to the path of the particle,and the faster the particle goes,the shorter the distance from crest to crest,When we add waves together,well,we just add them add each point,so that when two crests are in the same place we get an even higher crest,but if a crest meets a trough of equal and opposite magnitude,they cancel out and we are on a nodal point,where the wave is neither above or below the mean height,All this "waveology" applies to sound waves,water waves,and almost as simply to light waves,But it absolutely mysterious for "matter waves",or the waves belonging to particles,The mystery has to do with the fact that we have both wavelike and particle like properties in all matter,and we still don't know how to resolve this,because those properties are such opposites,
Linear Ramp shows a random quantum wave,made of waves heading in all directions as described above,The analogous particles find themselves on a ramp,and like marbles on a sheet of plywood they reach a point (actually a line) where they can go no further,Along this line the wavelength is the longest,where the wavelengths are the shortest,In effect,the waves are lapping up against a shoreline,The shoreline is the pink band at the top of the blue area.
Monolith
Approaching chaos,perspective of random wave in two dimensions.
Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy that corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986.
Random Sphere I
Random and repeated superposition of waves on the surface of a sphere,simulating extreme quantum chaos
Sphere and Random Sphere show random super positions of waves on the surface of a sphere,Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy which corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986.
Random Sphere II
Random superposition of waves on the surface of a sphere,simulating extreme quantum chaos
Blue moon and Random Sphere show random super positions of waves on the surface of a sphere,Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy that corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986.
Random Sphere II
Random superposition of waves on the surface of a sphere,simulating extreme quantum chaos
Random Sphere I and Random Sphere II show random superpositions of waves on the surface of a sphere,Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy which corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986,
There is an eerie familiarity with the strange and yet strangely familiar structures you see in random waves,Knowing that chaos is the norm in nature,especially at the level of molecular motion,and knowing that everything is really quantum and that random waves are the quantum manifestation of chaos,it is perhaps not too melodramatic to say that we are made of the stuff you see in Random Wave II.
Random Synthesis
Simulation of quantum chaos,Blurred and colored sets of parallel lines ruled on a plane,overlapping in the central region.
By blurring and coloring sets of parallel lines ruled on a plane,we simulate the addition of plane waves,Here the wave sets converged in the center of the "flower",each wave having a limited width (the rule lines were each only about 1/4 the length of the image; but each set of ruled lines extends across the whole "flower",The sets all overlapped in the flower region,and were randomly chosen in direction and offset ("phase"),By blurring and coloring the lines we see an excellent random wave produced in the central region,Addition of such sets of lines was once a favorite motif of Sol Le Witt,Sometimes he used parallel lines as seen here,sometimes concentric circles,It turns out not to matter that is used as far as the qualitative result goes,This is in agreement with the mathematical fact that random addition of plane waves sets gives the exact same result as random addition of Bessel functions,using different places in the plane of the image (randomly) to locate the centers of the Bessel functions.
Trichaotic
Trichaotic is a composite three manifestations of chaos; two classical,one quantum.
Trichaotic is a composite three manifestations of chaos; two classical,one quantum,At the top of the image,the sky,is a random wave,corresponding to the quantum manifestation of classical chaos,In the middle distance is a stroboscopic accumulation of the motion of a chaotic rotator,as in Rotating Rotator I,In the foreground the motion of electrons in a nanowire is shown,The wire has some roughness in its shape (attributable to the method by which the wire is produced) which causes the trajectories to behave randomly over time.
Resonator I
A quantum wave builds up in a resonant cavity between the straight and curved walls.
A quantum wave builds up in a resonant cavity between the straight and curved walls,when waves are arriving from below,Most of the wave energy is reflected back,but a surprisingly large fraction of it gets through the tiny hole if the wavelength is just right to make the cavity resonant,Westervelt Resonator-In this image,a quantum wave builds up in a resonant cavity between the straight and curved walls,when waves are arriving from below,Most of the wave energy is reflected back,but a surprisingly large fraction of it gets through the tiny hole if the wavelength is just right to make the cavity resonant,Prof,Robert Westervelt and his research group invented the "Westervelt resonator" around 1995 at Harvard University,for the purpose of investigating electron waves,In this picture you see various aspects of waves all acting together,reflection,diffraction,and resonance,The whole device is just a few microns across,or smaller than a bacterium,
The idea is to place a semicircular reflecting mirror facing a wall punctured by a little hole,Electrons are aimed at the wall,and most hit it and bounce back,But there are exceptions to this,Some electrons make it through the little hole,where they emerge and then get reflected back to the wall by the mirror,Most of these reflect back to the mirror,and so forth,Since not many electrons can get through the hole,you would think it would be dark (or quiet) in the "cavity" between the wall and the mirror (think of light,or sound,which are also waves),But as the speed of the electrons is increased,their wavelength gets shorter,according to the deBroglie formula l = h/(m v),where m is the mass,v is the velocity,and l is the wavelength,Some speeds are just right to fit a round number of wavelengths in the cavity,and a big buildup of waves takes place in there,the "resonance",If this were sound coming through a small hole in a wall,it would still be very loud in "room" at that one wavelength,or frequency of sound,The resonance is rare and usually it is pretty quiet in the cavity,Finally,even though the electrons would be trapped in the cavity (except for going back through the hole),they manage to leak out the sides,This is diffraction; waves do this,but particles cannot.
Resonance II
Westervelt Resonator
In this image,a quantum wave builds up in a resonant cavity between the straight and curved walls,when waves are arriving from below,Most of the wave energy is reflected back,but a surprisingly large fraction of it gets through the tiny hole if the wavelength is just right to make the cavity resonant,Prof,Robert Westervelt and his research group invented the "Westervelt resonator" around 1995 at Harvard University,for the purpose of investigating electron waves,In this picture you see various aspects of waves all acting together,reflection,diffraction,and resonance,The whole device is just a few microns across,or smaller than a bacterium,
The idea is to place a semicircular reflecting mirror facing a wall punctured by a little hole,Electrons are aimed at the wall,and most hit it and bounce back,But there are exceptions to this,Some electrons just miss the wall and wind up bending their paths AFTER they pass the wall,to fill in the shadow behind the wall,There would be a hard shadow behind the wall if we send particles at it from the same direction,Some electrons make it through the little hole,where they emerge and then get reflected back to the wall by the mirror,Most of these reflect back to the mirror,and so forth,Since not many electrons can get through the hole,you would think it would be dark (or quiet) in the "cavity" between the wall and the mirror (think of light,or sound,which are also waves),But as the speed of the electrons is increased,their wavelength gets shorter,according to the deBroglie formula l = h/(m v),where m is the mass,v is the velocity,and l is the wavelength,Some speeds are just right to fit a round number of wavelengths in the cavity,and a big buildup of waves takes place in there,the "resonance",If this were sound coming through a small hole in a wall,it would still be very loud in "room" at that one wavelength,or frequency of sound,The resonance is rare and usually it is pretty quiet in the cavity,Finally,even though the electrons would be trapped in the cavity (except for going back through the hole),they manage to leak out the sides.
Correspondence
A second "Rosetta Stone",showing the close correspondence between the quantum waves and classical motion
This is another "Rosetta Stone",showing the close correspondence between the standing waves in black (a standing wave keeps the same shape at time evolves) and classical trajectories seen in red,There are four types of classical motion for this lemon-shaped billiard,including a chaotic motion seen in the lower right,The motion depends only on the initial "launch" the trajectory is given,Other trajectories can differ in appearance somewhat but they always fall into one of the four classes,As seen in black,the standing waves (quantum wave functions; or alternately the sound level in a lemon shaped room at certain "resonant" frequencies) also fall into these four classes,Here,no attempt has been made to color the trajectories to reproduce the wave structure of the standing waves,
Double Diamond
Scarred quantum wave function in a stadium shaped enclosure.
Tradition had it years ago that the standing waves in chaotic cavities would be just like random waves,i.e,made of of wave sets added randomly from all directions,The reason is that the analogous classical trajectories bounce around wildly inside the stadium and suggest we add corresponding wave sets in all directions,The wave trapped in this cavity,shown here in perspective with the wave intensity as small hills ina very jumbled landscape,does share many features with random waves,However,there is an important difference,the concentration of strongest intensity lies along classical periodic orbits,here given by an orbit that makes a double diamond shape,The simple periodic orbits have a strong influence on the waves,which was not expected,We say the wave is "scarred" by the periodic orbit.
Rosetta Stone
A direct link or translation between classical mechanics (the trajectory weaving its path,like a marble rolling in a bowl) and quantum mechanics (the patchwork quilt effect seen at the lower right,exhibiting wavelike behavior)
The Rosetta stone was found in Egypt in 1799,and gave the same text in Greek,hieroglyphics,and ancient Egyptian,In much the same way,this image translates between classical mechanics of the 19th century and quantum mechanics of the 20th century,The classical mechanics (the trajectory weaving its path,like a marble rolling in a bowl) and quantum mechanics (the patchwork quilt effect seen at the lower right,exhibiting wavelike behavior) are connected by deBroglie's relation p l = h,where l is the wavelength of the particle and p is its momentum,h is called Planck's constant.
Scar
Scarred quantum wave function in a stadium shaped enclosure.
Tradition had it years ago that the standing waves in chaotic cavities would be just like random waves,i.e,made of of wave sets added randomly from all directions,The reason is that the analogous classical trajectories bounce around wildly inside the stadium and suggest we add corresponding wave sets in all directions,The wave trapped in this cavity does share many features with random waves,but there is an important difference,the concentration of strong waves along classical periodic orbits,here given by an orbit that makes an "X" in the center and bounces horizontally at the top and bottom of the stadium,The simple periodic orbits have a strong influence on the waves,which was not expected,We say the wave is "scarred" by the periodic orbit.
Caustic I
Three-dimensional caustics formed on a flat sea bottom by light passing through two consecutive wavy surfaces.
Caustics are places where things accumulate; in this case light is accumulating,We often think of focal points as places where light gathers after passing through a lens,but more generally,for "random" lenses,there are many more interesting patterns to examine,Caustics are seen abundantly in the two-dimensional electron flow Transport series; there,we are looking at the flow as it moves along in two dimensions,Here,we are seeing the flow of light in three dimensions as it is interrupted by the sea bottom surface,This pattern is not possible for a true sea bottom,because the light has passed through seven consecutive surfaces,being refracted twice,(Caustic I uses two such wavy surfaces to refract the light)
Caustic II
Three-dimensional caustics formed on a flat sea bottom by light passing through seven consecutive wavy surfaces.
Caustics are places where things accumulate; in this case light is accumulating,We often think of focal points as places where light gathers after passing through a lens,but more generally,for "random" lenses,there are many more interesting patterns to examine,Caustics are seen abundantly in the two-dimensional electron flow Transport series; there,we are looking at the flow as it moves along in two dimensions,Here,we are seeing the flow of light in three dimensions as it is interrupted by the sea bottom surface,This pattern is not possible for a true sea bottom,because the light has passed through seven consecutive surfaces,being refracted twice,(Caustic I uses two such wavy surfaces to refract the light,Caustic II uses seven).
Caustic IV
Caustic IV displays folds,cusps,and swallowtails,which are typical caustic structures,here formed by looking through a ruled,transparent colored three-dimensional curved sheet.
Caustic IV displays folds,cusps,and swallowtails,which are typical caustic structures,here formed by looking through a ruled,transparent colored three-dimensional curved sheet,The sheet itself is smooth (but not flat); when we project it onto a plane (by looking through it from a certain angle) we see accumulation regions where material builds up along the line of sight,One of the most common caustics is called a cusp,Cusps result when a flat part of a sheet develops a fold somewhere along the sheet,At a definite point,we can see two new edges or caustics where before none existed,Several of these can be seen in Caustic IV,Caustic IV emphasizes the appearance of caustics in projections of higher dimensional objects onto lower dimension,a property also present in Torus III and Torus IV,A caustic is a region where the higher dimensional surface lies tangent to the projection,thus it’s shadow,piles up” along a caustic,Color hue and value in this image are determined in part by color subtraction of overlapping parts of the sheet.
Interpenetrating Surfaces
We peer through several interpenetrating approximately ellipically shaped surfaces.
Electrons in molecules normally move a lot faster than the nuclei,This leads to the concept of effective potential energy surfaces that govern the motion of nuclei,These surfaces interpenetrate,and intersect in various ways,Their intersection actually causes a breakdown in the assumption that the motion of the nuclei can take place on the surfaces in the first place,The breakdown leads to interesting experimental consequences,Here we peer through several interpenetrating approximately ellipically shaped surfaces,transparent enough to see through into the next surface,Caustics develop as the viewing direction goes parallel to the surfaces,The breakdown of the so-called "adiabatic" approximation is suggested by the broken nature and color chaos of the surfaces.
2DEG
Variation and elaboration of an illustration for an article on two dimensional electron gases in Physics Today.
This image began as an illustration for an article on two dimensional electron gases in Physics Today,Depicting various layers in the micron sized heterostructures,and especially the electron donating atoms together with the effective potential energy landscape which induce on the electrons living in the interfacial layer between Gallium Arsinide and Aluminum Gallium Arsinide semiconductor crystals,the original image invited variation and elaboration.
Analyzed Collision
Collisions between polyatomic and diatomic molecules,with acceleration vectors included.
Two different collisions are shown,one behind the other,Each collision is performed in time steps,and at each step the atoms making up the molecules are drawn,For example,the twisting red and green track on the lower left is a diatomic molecule,vibrating and rotating as it moved toward the edge of the image,Each of the two collisions actually took place in two dimensions,i.e,the plane of the image,Therefore,when the track of one atom is hiding another,it is because that atom appeared there after the other had passed by,The overall effect is three dimensional,but the third dimension is really time,not depth! Knowing this,it is possible to reconstruct much of the history of the collisions from the image,The acceleration arrows show how much and in what direction each atom was accelerating at each step,The collision in the foreground proceeded from top to bottom,the molecules entering the scene from the top and upper left,and after colliding in the middle,exit on the bottom and lower right.
Collision II
Stroboscopic record of a collision between four diatomic molecules,a tetra-atomic molecule,and an atom.
This is the stroboscopic record of a collision between four diatomic molecules,a tetra-atomic molecule,and an atom,The vibrations and rotations of the molecules before and after collision are seen in the peculiar intertwining paths executed by the atoms in the molecule,This is a classical picture of the collision showing chaos in the very complicated and sensitive dependence on exactly where the atoms are as they begin to collide,The collision proceeded from top to bottom,the molecules entering the scene from the top and upper left,and after colliding in the middle,exit on the bottom and lower right,although two diatomic molecules started within the scene.
Rotating Rotator I
Tracks of three different four segmented rotators are seen in the foreground,as they proceeded from lower part of the image towards the upper.
The simplest rotator consists of two rigid bars pivoted together end to end,The bars are free to rotate around the pivot like the segments of an old-fashioned carpenter's ruler,only without the friction,If you throw such a rotator into the air,the segments will pivot around each other in interesting ways,while the object as a whole flies smoothly through the air,If there are three or more segments,the pivoting is chaotic,First one segment may spin wildly,then all three segments may rotate as a unit,then perhaps the two end segments spin in opposite directions,etc,No matter how many segments there are,the wild rotations of the individual segments,together with the overall rotation as a whole,proceed independently of the smooth motion of what is called the "center of mass" of the object,The reason for this is that the rotator is acting only on itself,The forces,which cause the segments to rotate at different rates,are exerted by the rotator and not by some outside agent,If there is no outside agent,the center of mass moves uniformly,according to Newton's laws,
In Rotating Rotator I,the tracks of three different four segmented rotators are seen in the foreground,as they proceeded from lower part of the image towards the upper,In the background,more rotator paths are shown,The difference is that in the background three sets of rotators actually collided near the middle of the picture,leading to changes in the pattern of rotation and changes in the directions of the center of mass of each of the rotators.
Rotating Rotator II
Three sets of four-segmented rotators were set spinning and traveling from left to right.
The simplest rotator consists of two rigid bars pivoted together end to end,The bars are free to rotate around the pivot like the segments of an old-fashioned carpenter's ruler,only without the friction,If you throw such a rotator into the air,the segments will pivot around each other in interesting ways,while the object as a whole flies smoothly through the air,If there are three or more segments,the pivoting is chaotic,First one segment may spin wildly,then all three segments may rotate as a unit,then perhaps the two end segments spin in opposite directions,etc,No matter how many segments there are,the wild rotations of the individual segments,together with the overall rotation as a whole,proceed independently of the smooth motion of what is called the "center of mass" of the object,The reason for this is that the rotator is acting only on itself,The forces that cause the segments to rotate at different rates are exerted by the rotator and not by some outside agent,If there is no outside agent,the center of mass moves uniformly,according to Newton's laws,
In Rotating Rotator II,three sets of four-segmented rotators were set spinning and traveling from left to right,The images are stroboscopic; meaning that after very short intervals a new picture of what the rotator is doing is taken and added to existing pictures,The rule is that the current image of the rotator overwrites and overlays all prior images,In this way,the history of the rotation and the progress of the rotator can be deduced from the image,
The tracks of three different four segmented rotators are seen in the foreground,as they proceeded from lower part of the image towards the upper,In the background,more rotator paths are shown,The difference is that in the background three sets of rotators actually collided near the middle of the picture,leading to changes in the pattern of rotation and changes in the directions of the center of mass of each of the rotators,
Banyan
Electrons are injected at the lower left and ride over a bumpy landscape,with the elevation slowly rising toward the top of the image.
Electrons are launched from the lower left over a bumpy landscape simulating donor atoms near an electron gas,There is also a potential gradient in this case,slowing the electrons down as the approach the top of the image,and turning them back so that they head toward the bottom again,As they slow down,the bumps become too high to surmount,and several hilltops are visible as excluded holes in the upper third of the picture
Dendrite
Electrons injected at the top of the image rain down in the branched flow pattern seen,as they ride over a bumpy landscape.
These images render electron flow paths in a "two dimensional electron gas",Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,The electron flow images are excellent examples of the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories looks like looking through translucent kelp! Or,like ridges on a mountain.
End of Coherence
Electron flow launched from the upper center in a weakly random potential,showing quantum phase as color,and the decay of the coherence of the phase to the lower right.
Electron flow launched from the upper left in a weakly random potential is colored according to quantum phase,Trajectories were written by color addition,so that color saturation represents coherence of the trajectories,they get farther from the source,they loose coherence with each other,The color banding diminishes,A data subtraction was done using saturation as the marker,leaving the least coherent region in the lower right transparent and showing the artificial color gradient beneath,
Exponential
Electrons launched from the upper right fan out and then form branch,as indirect effects of travelling over bumps.
These images render electron flow paths in a "two dimensional electron gas." Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,These two images are excellent examples of the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories is like looking through translucent kelp,or like ridges on a mountain.
Nanowire
Electron paths in a nanowire,including imperfections in the wire.
As components of electronic devices get ever smaller,wires connecting those components must also shrink in proportion,At the micron to nanometer scale,where devices are now being built,the wave nature of matter is becoming critical,This may be an advantage or it may be a problem,Making and understanding nanowires is certainly a challenge,Real nanowires have imperfections,The image Nanowire grew out of a study of electron flow in a wire riddled with random imperfections,It shows electrons injected at one point contact,the,sun,” flowing out from there to all regions of the wire,The disturbance of the electron tracks by the imperfections is shown in their somewhat unruly paths,The quantum aspect of the electrons is shown in color,we can follow the wave nature of the electrons by assigning yellow to the crest of the wave,blue to a trough,continuously around the color circle,The creative process leading to Naonwire is typical of my artwork,a synthesis of research and artistic creation,each one enhancing the other,Experiments conducted by M,Topinka,B,LeRoy and B,Westervelt measuring electron transport in semiconductor microstructures led to scientific illustrations of electrons riding over bumpy landscape potentials,Experimentation with various methods of recording individual electron tracks (overwrite,transparency,color combination) led to a variety of effects and expanded the horizon of the medium,The resulting Transport series is the first of large format high resolution electron flow images using branched flow physics,These images revealed the caustics formed when electrons flow from a particular point over a hilly landscape.
Transport II
Electrons launched from the center in all directions fan and then form branches,as indirect effects of travelling over bumps.
These images render electron flow paths in a "two dimensional electron gas",Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,These two images are excellent examples of the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories looks like looking through translucent kelp,or like ridges on a mountain.
Transport III
Electron flow with folds and caustics,resulting from injection at the lower right followed by branching due to the rough landscape over which the electrons are travelling.
Loosely speaking,caustics are edges,lines along which one object or space ends and another begins,But edges are usually much more,In a drawing,caustics determine where a line should fall,and where it should begin and end,If the object being rendered is a smooth,three dimensional form,light will usually collect or diminish rapidly at an edge,and detail will accumulate there,This is because the caustic of a curved surface is where we look tangent to (i.e,along) the surface,If we imagine the surface as a thin shell of smoky plastic in front of a uniform gray sky,then the edges (caustics) will be very dark,because there light must pass through much more material to get through than at a typical place,This points to the more general concept of caustics,places where accumulation occurs,
Whether by training or by instinct,we associate a line in a simple drawing with a caustic in the real world,But caustics are not always found at the obvious places,For example,look at the "proper" way to draw a torus (donut) from a certain perspective,This simple drawing is much more convincing than the way that might seem more obvious,The reason is that the top edge persists up to a certain point and then just vanishes; where it vanishes we stop the line,There is no reason for the top edge to stop where the bottom one meets it,as in the incorrect drawing,Caustics are found everywhere,and nowhere are they as beautiful as when looking through a thin folded translucent sheet,such as translucent kelp,One of the caustics we are bound to see is called a cusp It happens when a flat part of the kelp develops a fold as we follow along a blade,At a definite point,we start to see two new edges or caustics arise where before there were none,Light coming through folded kelp is an example of projection,in this case from three dimensions into two (think of taking a picture of the kelp,it is three dimensional,but its image will have to live on the two dimensional surface of the film).
Transport IV
Electrons launched from the bottom fan out and then form branch,as indirect effects of travelling over bumps.
These images render electron flow paths in a "two dimensional electron gas",Inspired by the experiments of Mark Topinka,Brian Leroy,and Prof,Robert Westervelt at Harvard,Theory performed by Scot Shaw of my group,and me,
These two images are based on the actual electron flow patterns for electrons riding over bumpy landscape,The electrons have more than enough energy to ride over any bump in the landscape,and the concentrations of electron flow (white/pink in Dendrite,and darkest streaks in Transport III) are newly discovered indirect effects of that bumpy ride,The channeling or branching was unexpected and has serious implications for small electronic devices of the future,These two images are excellent examples of 1) the degree to which I manipulate the raw data to get an image,and 2) more important still,the wonderful way nature emulates herself in different contexts,Thus,the folding of the electron trajectories looks like looking through translucent kelp! Or,like ridges on a mountain.
Transport IX
Paths of electrons followed for a short time,representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live,
In Transport IX we see the paths of electrons followed for a short time,representing the effect of starting the electrons in a narrow beam at two different places on the random potential landscape on which they live,The distinct and overlapping patterns resulted from the particular hills and valleys encountered from a new location,It is seen that the branches emanating from different initial launch points cross and seem independent,confirming that they are not due to any fixe features in the potential landscape but rather are due to the history of encounters with hills and valleys "upstream".
Transport VI
Electrons launched from the upper left fan out and then form branches,as indirect effects of travelling over bumps.
Transport VII
Electron flow over a two dimensional hilly terrain.
Electron flow over a two dimensional bumpy surface; the electrons were injected from the top in a uniform sheet,all initially heading straight down,The focussing and defocussing effects of the hilly terrain are clearly seen.
Transport XI
Electron tracks in nano device launched at various places,The bunching or branching of electron tracks depends on where the electrons are launched.
Electron flow started at several points on a single random potential surface,some moved and copied Small-scale electronic devices,the size of a bacterium or even a hundred times smaller,inevitably have minute imperfections in them which cause electrons to scatter and spread out as they progress through the device,We recently discovered that the electrons tend to bunch up and form branches,as is seen in many of the Transport images,This image shows that the branching is not a matter only of the landscape over which electrons are traveling,but also depends on where the electrons begin their journey,In this image,the electrons are launched at different places,over a very small range of initial angles,represented by the narrow "Stems",Smaller initial differences in angle grow quickly,as evidenced by the fanning out of electron paths,This is the beginning of the eventual chaotic motion of these electrons,However,the branching is also evident,Note that the branches cross,This means that the branches are not following specific valleys in the landscape,but are rather indirect effects caused by focusing as electrons travel over bumps and hill
Transport XII
Electrons launched from a particular point ride over a bumpy landscape,forming cusps and branches as they move away from launch point.
Electrons launched from a particular point ride over a bumpy landscape,forming cusps and branches as they move away from launch point,The color of the electron tracks in this image is keyed to the initial angle of launch of the electrons,This was done to see whether the branches which form are all coming from a small range of angles,or are collected from several different angles,Orange and predominately yellow branches show that these branches are coming from a very small range of initial angles of launch,Shortly after the electrons are launched,bombs and dips in the landscaped plaza to focus into V shaped regions called cusps,The edges of these regions often become branches,as is clearly seen in this image.
Transport XIII
This image shows two kinds of chaos,a random quantum wave on the surface of a sphere,and chaotic electron paths launched over a range of angles from a particular point,
This image shows two kinds of chaos,a random quantum wave on the surface of a sphere,and chaotic electron paths launched over a range of angles from a particular point,Transport 13 juxtaposes two kinds of chaos,One is quantum mechanical,and one is classical,The quantum mechanical chaos is represented by the random wave confined to the surface of a sphere,This random wave corresponds to a classical particle moving wildly on the surface of the sphere,The foreground we see the tracks electrons launched from a particular point with a range of angles,The different angles were assigned colors,so that one could see more readily the mechanisms by which the electrons' spread out,The increasingly random paths of the electrons as they travel farther from the launching point are the classical analog of the chaos seen on the sphere.
Transport XVI
Electron flow in a two-dimensional electron gas
Electrons flow in a sheet from a line on the top of the image; quantum phase is shown as color,The electrons are experiencing a random potential that defects them and causes the "curtains" of caustics to form.
Transport XVII
Electron flow launched from the upper center in a weakly random potential.
Electron flow launched from the upper center in a weakly random potential,Trajectories launched evenly over 180 degrees,using strict overwrite of trajectories,This gives a hidden surface effect to the caustics (cusps) which has much the same topology as a true erosion landscape,Color assgned from direction of trajectory.
Transport XVIII
Electron flow launched from the upper center in a weakly random potential.
Electron flow launched from the upper center in a weakly random potential,Trajectories launched evenly over 180 degrees,using strict over write of trajectories,This gives a hidden surface effect to the caustics (cusps) that has much the same topology as a true erosion landscape,Color assigned from direction of trajectory.
Dissippation
Classical dissipoative map with whorls.
Dissipation is an image of a mathematical map,The term "map" is used differently here than the usual meaning of the word,Rather than showing a representation of geographic features,a mathematical map assigns a new point to go on a region to from every point in the region shown on the map,Starting in one place,or point,the map specifies the next point to go to,But that point has another that it goes to,and so on,This is repeated many times,yielding figures of fantastic complexity,
We can imagine mapping every point inside some region,say a circle,which will result in a new region with a new distorted shape and more than likely centered somewhere else,If the area of the mapped circle remains the same,and just the shape changes,we have an area preserving map,If the area contracts,it might after many steps settle on a region of lower dimensionality than the one it started in; this is the case for Dissipation,
In Dissipation,the mapped region is the surface of a sphere,several circles of different colors on a black field were mapped 16 times,There are attracting vortices (whirlpools),where area is destroyed,Some places become so stretched and thinned out that they are transparent,revealing the surface of a "sun" below,The net effect is not unlike the convective cells seen on the surface of star
Homoclinic
Early evolution of a homoclinic tangle ina chaotic map.
Map I
Homoclinic oscillations from iteration of a chaotic map,with color chosen to represent quantum phase.
This image results from a chaotic "map",We start with the yellow region at the upper left,which maps onto the yellow region to its immediate right,and so on as we would read a book,This map has been pixelized so that if any point in a pixel is accessed the whole pixel lights up,The colors are given by the deBroglie wavelength,using a so-called semiclassical approximation to what the quantum map would be.
Map II
An iterated classical map with color chosen by quantum considerations.
An iterated classical map with color chosen by quantum considerations,In the top half,an entire region has been mapped and we are looking at one iteration,in the lower half,a single point has been mapped many times and we have recorded everywhere it went.
Nodal V
Rapid variation of a quantum wave function (middle) gives way to slow variation in the "forbidden" regime.
Tunneling is one of the starkest manifestations of the strange quantum realm,Particles can get through energy barriers that cannot penetrate classically,It is theoretically possible for a baseball to pass through a window or a wall without breaking anything,although the chances of this are so astronomically tiny that it may as well be impossible,However,smaller,lighter particles,such as carbon 60,have been seen to tunnel through barriers,
Nodal V is inspired by this tunneling phenomenon,It is an abstracted version of a quantum wave function confined to an infinitely deep pit,The bottom of the pit is the cent of the images,and the energy of the particle is such that classically it could not progress beyond the radius of the red flame region,The wave is sometimes positive,sometimes negative,and when it changes sign from plus to minus there is always an abrupt color shift,The lines along which the sign change takes place are called nodal lines,Beyond the red zone,the particle is in a "classically forbidden" realm,and the nodal lines change qualitatively,This images shows that some nodal lines persist to infinity,and it indicates haw many such lines do so,The wave function seen in the center is an example of a random quantum wave,and now we know how this randomness is manifested in the tunneling region.
Suris
A point-to-point map of the plane onto itself iterated many times with 500 initial point of various colors initiated at many locations.
This image results from a mathematical "map",which is like each house in a development having a unique "sister" house,There is a funny address book,which just says for each house what its sister house is that it must send letters to,Each house receives letters from just one other house and sends them to just one house,Either house can be itself,which is very rare,or the sending and receiving houses could be the same house,which is also rare,Typically the letter goes on a long journey,which either closes on itself in a simple way or fills a whole region of houses,Suppose a letter is sent from a house to its sister house,The sister sends it on to its sister,etc,All sister houses are somewhere in the same development,Now,it is not obvious (and not true in general) that the letter will reach all the houses in the development,Eventually however it must come back to the house where it originated,even if it has to go to every other house first,Even the same adress book,or map,can differ wildly as to how many houses are reached from a given house,If there are 1000 houses in the development,some houses would reach only a few others before the letter came back,and others could reach 100 or 500,What you see here is the result of sending about 500 "letters" in a development of about 5 million houses,The letter were colored differently to see where they went,There aren't that many strongly distinguishable colors,so sometimes its hard to tell,For example,in the "sun" region,the letters all went around in little ovals,but all were colored only slightly differently in that region,A blue "letter" went to about a million houses,as you can see,These maps show structures which are ubiquitous in nature,The islands are resonances,and the filled in regions (e.g,"sky") are regions of chaos where the letter covers every house in certain areas,but doesn't visit other areas.
Crystal I
Looking corner-on at a small cubic sample of a perfect crystal consisting of a periodic array of three different atoms.
In Crystal I we are looking corner-on at a small cubic sample of a perfect crystal consisting of a periodic array of three different atoms,The top layer (red) and the other two surfaces visible of Crystal I shows the orchard effect,with a much more complicated pattern looking through the bulk of the crystal,which is a three dimensional orchard,In spite of the complexity seen peering through the crystal,Crystal I is a perfectly regular array of three different atoms,This reminds us that even perfectly ordered systems may be quite complex,Note the different atoms that appear on different faces,only red atoms on the top,red and green on the right face,and all three on the left face.
Crystal III
Looking top down into a three-dimensional crystal,quasi-crystalline elements are seen in projection.
Looking top down into a three-dimensional perfect crystal,both crystalline and quasi-crystalline elements are seen,For,if we look through a perfect 3D crystal at irrational angles,we see a quasi-crystalline (non-repeating but non-random space filling pattern) structure,The various "diophantine" solutions of number theory are seen as clear paths through the crystal.
Diophantine
Rectangular ordered array rendered with panoramic "camera".
This image records several interesting phenomena,The crystalline images (Orchard,Crystal I) are the antithesis of chaos,although both images illustrate how perfectly "simple" periodic arrays can have their own intricacies and complexities,Orchard is truly simple,a perfectly rectangular array of dots,This array was "photographed" at close range by a panoramic camera,giving the apparent distortion and distinctive shape,Everyone has had the pleasure of looking down rows in an orchard or field with a repetitive array of objects,The more perfect the array and the thinner the objects,the more we can see through the orchard at various angles,Seeing through the orchard is a matter of looking in directions where a straight line never touches one of the objects in the orchard,Imagine standing near the orchard,and taking a wide angle photograph of it,You (and the camera) see straight paths,lines leading from your eye or the lens through the orchard,The camera records such paths also as straight lines,as you can see in Orchard,Imagine you have your camera set on a road which runs along the orchard,On one side of the road is the first row of trees,and the road stretches to your right and to your left,Now,even though the road and the first row are also a straight line,it is not a line that leads from the camera lens in one direction,and the camera is under no obligation to record it as straight! Thus,the distortions caused by the panoramic camera are very specific,and in spite of the serious camera distortion the paths through the orchard are rendered perfectly straight,
The possible paths through the orchard without hitting a tree are limited by the size of the trees,All of the directions possible through the orchard are given by two integers,n and m,which specify the number of rows n "over" and the number of rows m "up" that we travel as we proceed along the path,So,every direction is specified by a rational fraction n/m,because any common factor of m and n doesn't matter,(For example,going over 1 and up 2 is the same as going over 3 and up 6),For a given size tree,there are a finite number of paths,because if n or m gets too large you hit a tree.
Pyramid
A perfectly regular and predictable crystalline array displays surprising complexity when viewed at a typical angle
Quantize I
Color Quantization
If we can directly construct the wave behavior from the rays,then we understand the relation of classical chaos and wave,or quantum chaos,An example of such a construction in the case of non-chaotic ("regular") motion is given in Quantize I and Rosetta Stone,The separate dots represent the location of a classical particle at equally spaced times,The particle is oscillating in a somewhat asymmetrical bowl,The colors are taken to correspond to the phase of the wave,Phase corresponds to whether you are at a crest or a trough or somewhere in between,A simple sine wave for example oscillates from crest to trough and back to crest again,We represent the phases as colors,so that red corresponds to a crest,and cyan to a trough,When the trajectory (the track the particle or marble takes) overlaps itself,we do not just overwrite what was there before,but rather we add to it,so that we get the effect of interference of waves,In Rosetta Stone we see the wave pattern (see "Chladni") which is constructed out of the trajectory,right before our eyes.
Torus I
Track left by resonant energy transfer between two different types of motion
This image is a three dimensional image (plotted in two dimension) of a four dimensional object,When classical motion of particles is not chaotic,we say it is integrable,it can be confined to the surface of donut-shaped objects or "tori" which live in four or more dimensions,We cannot accurately represent such objects on the two-dimensional surface of an image,But we can try,The twisting surface seen here follows a torus wrapped around a simpler,larger torus,This corresponds to a classical resonance,in which one kind of motion efficiently exchanges energy back and forth with another.
Torus II
Track left by resonant energy transfer between two different types of motion
This image is a three dimensional image (plotted in two dimension) of a four dimensional object,When classical motion of particles is not chaotic,we say it is integrable; it can be confined to the surface of donut-shaped objects or "tori" which live in four or more dimensions,We cannot accurately represent such objects on the two dimensional surface of an image,But we can try,The torus appears to intersect itself,but this is because we are pretending it exists in three dimensions,In the four dimension space,it does not intersect,The surface of the torus was made partial transparent to reveal the structure within.
Torus III
Apparently self-intersecting tori (Torus III and IV).
The dynamics of a system of two degrees of freedom involves two positions and their rate of change,i.e,two velocities; altogether four coordinates,Thus a system of two degrees of freedom,lives” in four mathematical dimensions,If it is non-chaotic,it lives on the surface of two-dimensional torus,embedded in the four dimensional space,In projecting this motion into the plane of the image we have only a,shadow” of the full four dimensional space onto two chosen dimensions,The shadow appears to self-intersect (Torus III) or to turn itself inside out (Torus IV),but in the full four-dimensional space it does no such thing,
TorusIV
A four-dimensional tranparent torus is projected onto two dimensions,showing characteristic caustic structures,caused by viewing tangent to surfaces at some place
Bessel 21
A superposition of 21 plane waves,travelling in 21 evenly space directions around the compass.
A superposition of 21 plane waves,traveling in 21 evenly space directions around the compass,results in and approximation to a Bessel function near the center of the "comet",and increasingly disordered structure away from there,The resulting image,if displayed over a much wider area,would reveal its quasi-crystalline structure,The approximate Bessel function breaks down outside a radius inside the sun region,The region of breakdown is most of the image,The slice shown in lighter colors is one repeating unit as if copied 20 times would regenerate the whole image,This image is a quasi-crystal as well,except that the approximate repetitions of the sun region are very far away,outside the image,All additions of more than three plane waves are quasi-crystals,but the near-repetition distance moves out rapidly as the number of plane waves used increases.
Quasicrystal II
A quasi-crystal generated as an addition of 5 plane waves equally spaced in angle.
This image is a quasi-crystal,showing some aspects of crystalline order,but missing crucial long range order we expect of a crystal,That is,with an ordinary crystal we can always move around by multiples of the repeating distances ("lattice constants") and come to an repeated atom or structure; not so in this quasi-crystal,It looks like a crystal at first glance,but then stubbornly refuses to yield to our human tendency to search for a pattern,There is no repeating pattern,not ever!
It is easy to "tile" a floor with simple triangles,squares,or hexagons (three,four or six sided objects) with no gaps,but it is a maddening fact that you can't tile a floor with pentagons without gaps,Adding two sets of plane waves at right angles gives a square array,just like a square tiling of a floor,Similarly,three wave sets at 120 degrees from each other gives a perfect equilateral triangular,crystalline array,So the idea occurs naturally to "force" a pentagonal tiling with FIVE wavesets equally spaced in angle,that is,at 72 degrees from one another,What we get is not a regular tiling at all,but a quasi-crystal,It is frustrating to look too long at this image! Inspection shows that again and again similar structures appear,some with five-fold symmetry,but no two are exactly alike,and the similar structures do not occur on a regular lattice of positions,The penta-quasicrystal is the beginning of an infinite variety of quasi-crystals,since no regular n-sided tile with n greater than 6 can tile the plane without gaps,As we make n bigger,taking more and more wave sets equally spaced in angle from each other,the 2n-sided "flowers" which are formed make a quasicrystalline array,The flowers grow much farther apart as n increases so that we see but one good flower in Comet and Quasicrystal III.
Quasicrystal III
In addition of 9 plane waves equally spaced in angle "pivoted" about the center of the "sun".
An addition of 9 plane waves equally spaced in angle,The plane waves were "pivoted" about the center of the "sun",giving an approximation to a Bessel function centered there,which breaks down outside a radius which is already inside the sun region,The region of breakdown is most of the image,The slice shown in lighter colors is one repeating unit which if copied 8 times would regenerate the whole image,This image is a quasicrystal as well,except that the approximate repetitions of the sun region are very farther away from each other than in Quasicrystal II,which has 5 waves added together,but not so far away as in Comet,which has 21 waves,All additions of more than three plane waves are quasicrystals,but the near-repetition distance moves out rapidly as the number of plane waves used increases,What you call a "near-repetition" is somewhat arbitrary,The more exact one makes the repetition requirement,the further away the repetitions become,There are no really good repetitions in the image,although there are many poor ones.
Chladni
Hand colored version of one of Chladni's original drawing of nodal lines.
The diagrams of Ernst Chladni (1756-1827) are the scientific,artistic,and even the sociological birthplace of the modern field of wave physics and quantum chaos,Educated in Law at the University of Leipzig,and an amateur musician,Chladni soon followed his love of science and wrote one of the first treatises on acoustics,"Discovery of the Theory of Pitch",Chladni had an inspired idea,to make waves in a solid material visible,This he did by getting metal plates to vibrate,stroking them with a violin bow,Sand or a similar substance spread on the surface of the plate naturally settles to the places where the metal vibrates the least,making such places visible,These places are the so-called nodes,which are wavy lines on the surface,The plates vibrate at pure,audible pitches,and each pitch has a unique nodal pattern,Chladni took the trouble to carefully diagram the patterns,which helped to popularize his work,Then he hit the lecture circuit,fascinating audiences in Europe with live demonstrations,This culminated with a command performance for Napoleon,who was so impressed that he offered a prize to anyone who could explain the patterns,More than that,according to Chladni himself,Napoleon remarked that irregularly shaped plate would be much harder to understand! While this was surely also known to Chladni,it is remarkable that Napoleon had this insight,Chladni received a sum of 6000 francs from Napoleon,who also offered 3000 francs to anyone who could explain the patterns,The mathematician Sophie Germain took he prize in 1816,although her solutions were not completed until the work of Kirchoff thirty years later,Even so,the patterns for irregular shapes remained (and to some extent remains) unexplained,Government funding of waves research goes back a long way! (Chladni was also the first to maintain that meteorites were extraterrestrial; before that,the popular theory was that they were of volcanic origin.) One of his diagrams is the basis for image,which is a playfully colored version of Chaldni's original line drawing,Chladni's original work on waves confined to a region was followed by equally remarkable progress a few years later.
Mixed Chamber
Standing wave in a lemon shaped cavity
Imagine splitting a circle in two equal halves,If we move the halves apart and connect the gaps with straight lines,we have a "stadium" shape,which is perhaps the most famous shape in all modern billiard history,The stadium is also called the Bunimovich stadium,after its inventor,who showed that all of its trajectories are chaotic,except a subset of 0% of them which are periodic,If instead of moving the half circles apart we push them together,the enclosed shape is a "lemon",seen here,The lemon billiard is a "typical" dynamical system,which combines with ordered and chaotic behavior,depending on how the system is set into motion,As seen in Correspondence,there are four classes of trajectories,three of which are ordered and one chaotic,(This changes if we make fatter or thinner lemon shapes by moving the circle halves in or out),This particular standing wave corresponds to the most complicated of the three ordered trajectories,billiard wave functions.
Modes I
A series of standing waves in a lemon shaped cavity.
This image shows a series of standing waves in a lemon shaped cavity,Each standing wave is a different resonant frequency of the chamber,The modes closely follow the patterns of classical ray paths bouncing within the chamber,
Linear Ramp
Random wave on a linear ramp potential in two dimensions
Chaos and unpredictability are also universal,Put waves and chaos together and you get random waves,There is something immediately familiar about them; perhaps we know intuitively that this is the stuff of the universe,
In the atomic domain we must grapple with the fact that particles often act like waves instead of little points of mass,Imagine waves coming at you in a long "set",The energy of the wave and the waves themselves are traveling toward you,but the crests of the waves stretch out in long lines to your right and left,A particle traveling somewhere actually corresponds to the energy in the wave,and in some sense the particle is really a wave,not a particle at all,At the particle bounces around,its associated wave goes with it,and we are led to consider superpositions of many wave "sets" or as we say,wave trains,The rule is the wave crests and troughs are perpendicular to the path of the particle,and the faster the particle goes,the shorter the distance from crest to crest,When we add waves together,well,we just add them add each point,so that when two crests are in the same place we get an even higher crest,but if a crest meets a trough of equal and opposite magnitude,they cancel out and we are on a nodal point,where the wave is neither above or below the mean height,All this "waveology" applies to sound waves,water waves,and almost as simply to light waves,But it absolutely mysterious for "matter waves",or the waves belonging to particles,The mystery has to do with the fact that we have both wavelike and particle like properties in all matter,and we still don't know how to resolve this,because those properties are such opposites,
Linear Ramp shows a random quantum wave,made of waves heading in all directions as described above,The analogous particles find themselves on a ramp,and like marbles on a sheet of plywood they reach a point (actually a line) where they can go no further,Along this line the wavelength is the longest,where the wavelengths are the shortest,In effect,the waves are lapping up against a shoreline,The shoreline is the pink band at the top of the blue area.
Monolith
Approaching chaos,perspective of random wave in two dimensions.
Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy that corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986.
Random Sphere I
Random and repeated superposition of waves on the surface of a sphere,simulating extreme quantum chaos
Sphere and Random Sphere show random super positions of waves on the surface of a sphere,Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy which corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986.
Random Sphere II
Random superposition of waves on the surface of a sphere,simulating extreme quantum chaos
Blue moon and Random Sphere show random super positions of waves on the surface of a sphere,Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy that corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986.
Random Sphere II
Random superposition of waves on the surface of a sphere,simulating extreme quantum chaos
Random Sphere I and Random Sphere II show random superpositions of waves on the surface of a sphere,Random waves are the paradigm for quantum chaos,This is as close as quantum mechanics can come to chaos,Classical chaos amounts to trajectories heading in every possible direction randomly; corresponding to this is the random addition of wave sets traveling in all directions,The particle executing the trajectory has a definite energy which corresponds to the waves all having the same wavelength,The result is the type of wave shown here,with characteristics not discovered until about 1986,
There is an eerie familiarity with the strange and yet strangely familiar structures you see in random waves,Knowing that chaos is the norm in nature,especially at the level of molecular motion,and knowing that everything is really quantum and that random waves are the quantum manifestation of chaos,it is perhaps not too melodramatic to say that we are made of the stuff you see in Random Wave II.
Random Synthesis
Simulation of quantum chaos,Blurred and colored sets of parallel lines ruled on a plane,overlapping in the central region.
By blurring and coloring sets of parallel lines ruled on a plane,we simulate the addition of plane waves,Here the wave sets converged in the center of the "flower",each wave having a limited width (the rule lines were each only about 1/4 the length of the image; but each set of ruled lines extends across the whole "flower",The sets all overlapped in the flower region,and were randomly chosen in direction and offset ("phase"),By blurring and coloring the lines we see an excellent random wave produced in the central region,Addition of such sets of lines was once a favorite motif of Sol Le Witt,Sometimes he used parallel lines as seen here,sometimes concentric circles,It turns out not to matter that is used as far as the qualitative result goes,This is in agreement with the mathematical fact that random addition of plane waves sets gives the exact same result as random addition of Bessel functions,using different places in the plane of the image (randomly) to locate the centers of the Bessel functions.
Trichaotic
Trichaotic is a composite three manifestations of chaos; two classical,one quantum.
Trichaotic is a composite three manifestations of chaos; two classical,one quantum,At the top of the image,the sky,is a random wave,corresponding to the quantum manifestation of classical chaos,In the middle distance is a stroboscopic accumulation of the motion of a chaotic rotator,as in Rotating Rotator I,In the foreground the motion of electrons in a nanowire is shown,The wire has some roughness in its shape (attributable to the method by which the wire is produced) which causes the trajectories to behave randomly over time.
Resonator I
A quantum wave builds up in a resonant cavity between the straight and curved walls.
A quantum wave builds up in a resonant cavity between the straight and curved walls,when waves are arriving from below,Most of the wave energy is reflected back,but a surprisingly large fraction of it gets through the tiny hole if the wavelength is just right to make the cavity resonant,Westervelt Resonator-In this image,a quantum wave builds up in a resonant cavity between the straight and curved walls,when waves are arriving from below,Most of the wave energy is reflected back,but a surprisingly large fraction of it gets through the tiny hole if the wavelength is just right to make the cavity resonant,Prof,Robert Westervelt and his research group invented the "Westervelt resonator" around 1995 at Harvard University,for the purpose of investigating electron waves,In this picture you see various aspects of waves all acting together,reflection,diffraction,and resonance,The whole device is just a few microns across,or smaller than a bacterium,
The idea is to place a semicircular reflecting mirror facing a wall punctured by a little hole,Electrons are aimed at the wall,and most hit it and bounce back,But there are exceptions to this,Some electrons make it through the little hole,where they emerge and then get reflected back to the wall by the mirror,Most of these reflect back to the mirror,and so forth,Since not many electrons can get through the hole,you would think it would be dark (or quiet) in the "cavity" between the wall and the mirror (think of light,or sound,which are also waves),But as the speed of the electrons is increased,their wavelength gets shorter,according to the deBroglie formula l = h/(m v),where m is the mass,v is the velocity,and l is the wavelength,Some speeds are just right to fit a round number of wavelengths in the cavity,and a big buildup of waves takes place in there,the "resonance",If this were sound coming through a small hole in a wall,it would still be very loud in "room" at that one wavelength,or frequency of sound,The resonance is rare and usually it is pretty quiet in the cavity,Finally,even though the electrons would be trapped in the cavity (except for going back through the hole),they manage to leak out the sides,This is diffraction; waves do this,but particles cannot.
Resonance II
Westervelt Resonator
In this image,a quantum wave builds up in a resonant cavity between the straight and curved walls,when waves are arriving from below,Most of the wave energy is reflected back,but a surprisingly large fraction of it gets through the tiny hole if the wavelength is just right to make the cavity resonant,Prof,Robert Westervelt and his research group invented the "Westervelt resonator" around 1995 at Harvard University,for the purpose of investigating electron waves,In this picture you see various aspects of waves all acting together,reflection,diffraction,and resonance,The whole device is just a few microns across,or smaller than a bacterium,
The idea is to place a semicircular reflecting mirror facing a wall punctured by a little hole,Electrons are aimed at the wall,and most hit it and bounce back,But there are exceptions to this,Some electrons just miss the wall and wind up bending their paths AFTER they pass the wall,to fill in the shadow behind the wall,There would be a hard shadow behind the wall if we send particles at it from the same direction,Some electrons make it through the little hole,where they emerge and then get reflected back to the wall by the mirror,Most of these reflect back to the mirror,and so forth,Since not many electrons can get through the hole,you would think it would be dark (or quiet) in the "cavity" between the wall and the mirror (think of light,or sound,which are also waves),But as the speed of the electrons is increased,their wavelength gets shorter,according to the deBroglie formula l = h/(m v),where m is the mass,v is the velocity,and l is the wavelength,Some speeds are just right to fit a round number of wavelengths in the cavity,and a big buildup of waves takes place in there,the "resonance",If this were sound coming through a small hole in a wall,it would still be very loud in "room" at that one wavelength,or frequency of sound,The resonance is rare and usually it is pretty quiet in the cavity,Finally,even though the electrons would be trapped in the cavity (except for going back through the hole),they manage to leak out the sides.
Correspondence
A second "Rosetta Stone",showing the close correspondence between the quantum waves and classical motion
This is another "Rosetta Stone",showing the close correspondence between the standing waves in black (a standing wave keeps the same shape at time evolves) and classical trajectories seen in red,There are four types of classical motion for this lemon-shaped billiard,including a chaotic motion seen in the lower right,The motion depends only on the initial "launch" the trajectory is given,Other trajectories can differ in appearance somewhat but they always fall into one of the four classes,As seen in black,the standing waves (quantum wave functions; or alternately the sound level in a lemon shaped room at certain "resonant" frequencies) also fall into these four classes,Here,no attempt has been made to color the trajectories to reproduce the wave structure of the standing waves,
Double Diamond
Scarred quantum wave function in a stadium shaped enclosure.
Tradition had it years ago that the standing waves in chaotic cavities would be just like random waves,i.e,made of of wave sets added randomly from all directions,The reason is that the analogous classical trajectories bounce around wildly inside the stadium and suggest we add corresponding wave sets in all directions,The wave trapped in this cavity,shown here in perspective with the wave intensity as small hills ina very jumbled landscape,does share many features with random waves,However,there is an important difference,the concentration of strongest intensity lies along classical periodic orbits,here given by an orbit that makes a double diamond shape,The simple periodic orbits have a strong influence on the waves,which was not expected,We say the wave is "scarred" by the periodic orbit.
Rosetta Stone
A direct link or translation between classical mechanics (the trajectory weaving its path,like a marble rolling in a bowl) and quantum mechanics (the patchwork quilt effect seen at the lower right,exhibiting wavelike behavior)
The Rosetta stone was found in Egypt in 1799,and gave the same text in Greek,hieroglyphics,and ancient Egyptian,In much the same way,this image translates between classical mechanics of the 19th century and quantum mechanics of the 20th century,The classical mechanics (the trajectory weaving its path,like a marble rolling in a bowl) and quantum mechanics (the patchwork quilt effect seen at the lower right,exhibiting wavelike behavior) are connected by deBroglie's relation p l = h,where l is the wavelength of the particle and p is its momentum,h is called Planck's constant.
Scar
Scarred quantum wave function in a stadium shaped enclosure.
Tradition had it years ago that the standing waves in chaotic cavities would be just like random waves,i.e,made of of wave sets added randomly from all directions,The reason is that the analogous classical trajectories bounce around wildly inside the stadium and suggest we add corresponding wave sets in all directions,The wave trapped in this cavity does share many features with random waves,but there is an important difference,the concentration of strong waves along classical periodic orbits,here given by an orbit that makes an "X" in the center and bounces horizontally at the top and bottom of the stadium,The simple periodic orbits have a strong influence on the waves,which was not expected,We say the wave is "scarred" by the periodic orbit.
