Eric J Heller Gallery

ART AND SCIENCE

I do both, I really can't help it. I feel deep connections, and sometimes can't tell whether I'm doing one or the other. The reason is my art all derives from my research. I "paint" with electron flow. Chaos in its dynamical forms (not pretty picture hunting in fractals) can become an artists medium. I feel an urge to convey to others, non-scientists and scientists alike, where I've been and how I feel about it, almost as if I were a landscape painter.

The viewing public has various reactions to my work, as with any artist. Mostly positive, but the negative ones strangely delight me the most. Like the art critic in DC who critiqued my work in a show at the National Academy of Science building, by stating "...Heller's rainbow-colored designs, based on computer algorithms, reek a little too much of a bongwater-soaked college dorm room". I love that one and quote it playfully when I give talks on my work. Another was from a reviewer of a National Science Foundation proposal of mine, for work in quantum mechanics. You are asked to do "public outreach" these days, which I do gladly through my art. One reviewer said "I have seen Heller's exhibits, and they are not art." He/she gave me a low rating on the rest of the proposal too (surprise), and it was not funded. With some people, science and art don't mix all that well.

I am proud to be an elected member of the National Academy of Sciences, the American Philosophical Society, and the American Academy of Arts and Sciences.

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1 Two different collisions are shown, one behind the other. Each collision is performed in 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 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.	2 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. Caustic IV emphasizes the appearance of caustics in projections of higher dimensional objects onto lower dimension, a property also present in Heller’s images, 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.	3 Banyan is a member of the Transport series, which render electron flow paths in a micron-sized "two dimensional electron gas" (2DEG) as they cross over the device. The electron experience small deflections from smooth bumps in their path, due to positive donor atoms near the electron gas. In Banyan, electrons are launched from the lower left. There is also an extra hill to climb in this case, due to a voltage applied from top to bottom, slowing the electrons down as the approach the top, 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.
4 Looking like some sort of musical notation, this is a sonogram of a Bach Chorale, played on a piano. Time rises vertically here, and low frequency is to the left. The decay of the piano notes, and their harmonic structure (together with the rigid musical structure of almost mathematical precision reflecting Bach's genius) are evident	5 Bessel 21 shows an addition of 21 plane waves (perfect sets of parallel wave-fronts) Imagine taking one such plane wave and pivoting it about a point, copying it at 21 equal angles. All these copies are then added together. The pivot point is the center of the "sun", and the mathematical result is an approximation to what is known as a Bessel function. But the approximation breaks down near the edge and everywhere outside the yellow sun region. Thus, the region of breakdown is most of the image. The slice shown in lighter colors is one repeating unit, which if copied 20 times would regenerate the whole image. This image is a quasicrystal too, meaning that at some great distance, outside the image (which continues forever in all directions) the "sun" and its surroundings would be approximately recapitulated. All additions of more than three such plane waves are quasicrystals, but the near-repetition distance moves out rapidly as the number of plane waves used increases. Bessel 21 is part of the traveling exhibit Approaching Chaos: Visions from the Quantum Frontier, funded by MIT and the National Science Foundation. It appeared in Harvard Magazine 1/01 and in Computer Graphics World, 10/01 and in the Italian magazine La Republica 10/01. Ballantine Books selected this image for the cover of a science fiction book, Up the Line, by Robert Silverberg, 2002.	6
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$Caustic I Caustics are places where things accumulate; in this case it is light which is accumulating. We often think of focal points as places where light gathers after passing through a lens, but more generally for “random” lenses there are many more interesting patterns to examine. In Caustic I, rays of light from a point source have passed through two imaginary, successive layers of water, each with a wavy surface, i.e, a random lens refracting the light. A “sea bottom” ultimately interrupts the refracted rays, which is the plane of the image. The bright patches, or “caustics”, are similar to those produced when sunlight shines on a swimming pool, giving the familiar ribbed pattern on the pool bottom made famous by the painter David Hockney. The same exact thing happens with sound in the atmosphere, refracted by thermal and velocity turbulence,which is why the sound from a jet a few thousand feet up can fluctuate so much from moment to moment on a day with a disturbed atmosphere. Listen for it next time.$ 10 Caustics are places where things accumulate; in this case it is light which 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. In Caustic I, rays of light from a point source have passed through two imaginary, successive layers of water, each with a wavy surface, i.e, a random lens refracting the light. A “sea bottom” ultimately interrupts the refracted rays, which is the plane of the image. The bright patches, or “caustics”, are similar to those produced when sunlight shines on a swimming pool, giving the familiar ribbed pattern on the pool bottom made famous by the painter David Hockney. The same exact thing happens with sound in the atmosphere, refracted by thermal and velocity turbulence,which is why the sound from a jet a few thousand feet up can fluctuate so much from moment to moment on a day with a disturbed atmosphere. Listen for it next time.	11 Caustics are places where things accumulate. In this case, it is light which is accumulating. We often think of focal points as points where light gathers after passing through a lens. But for "random" lenses there are much more interesting caustic patterns to examine. In a three-dimensional flow, as is shown here, we must intercept the flow with a plane surface (such as a projection screen) to see the caustics easily. Caustic II depicts three-dimensional caustic patterns formed on a flat surface after light passes through seven consecutive wavy layers.	12 We often think of focal points as places where light gathers after passing through a lens, but a true focus requires a perfect lens. More generally, for "random" lenses, like the curved and somewhat irregular surface of a swimming pool with people in it, there are many more interesting patterns to examine. Light from the sun passing through such a surface and interrupted by a plane some distance below gives rise to a characteristic caustic pattern which most people have noticed on the bottom of swimming pools. Caustics are places where things accumulate; in this case light is accumulating. In Caustic Sunset, the light intensity passing through two wavy layers of water and captured by a plane was used to "color" the sky; here the darker shades represent more light, with the red highlights added. The image is completed at the base with a version of the chaotic Rotating
13 We often think of focal points as places where light gathers after passing through a lens, but a true focus requires a perfect lens. More generally, for "random" lenses, like the curved and somewhat irregular surface of a swimming pool with people in it, there are many more interesting patterns to examine. Light from the sun passing through such a surface and interrupted by a plane some distance below gives rise to a characteristic caustic pattern which most people have noticed on the bottom of swimming pools. Caustics are places where things accumulate; in this case light is accumulating. In Caustic Sunset, the light intensity passing through two wavy layers of water and captured by a plane was used to "color" the sky; here the darker shades represent more light, with the red highlights added. The image is completed at the base with a version of the chaotic Rotating	14 We often think of focal points as places where light gathers after passing through a lens, but a true focus requires a perfect lens. More generally, for "random" lenses, like the curved and somewhat irregular surface of a swimming pool with people in it, there are many more interesting patterns to examine. Light from the sun passing through such a surface and interrupted by a plane some distance below gives rise to a characteristic caustic pattern which most people have noticed on the bottom of swimming pools. Caustics are places where things accumulate; in this case light is accumulating. In Caustic Sunset, the light intensity passing through two wavy layers of water and captured by a plane was used to "color" the sky; here the darker shades represent more light, with the red highlights added. The image is completed at the base with a version of the chaotic Rotating	15 manyConcentricCircles An homage to Sol Lewitt, an artist of immense imagination and importance
16 This is the result of a buggy sonogram of a constant complex tone and a rising compex tone. They sound together, revealing wonderful consonances (at the more or less clear vertial streaks) and irritaing dissonances. The streaks are an artifact of the algorithm, but turned out to reveal the consonances, where low order integer ratios prevail relating the partials. Pythagoras all over.	17 Crystal I is a perfectly regular array involving three “atoms” in three dimensions. Diophantine is a two dimensional array of only one “atom”. Along with regularity goes predictability: knowing where a few objects are allows us to predict with ease the positions of all the others. We can check, and they will be where they are supposed to be. Crystal I also shows that even regular patterns can appear complicated. If you look at some directions through the crystal you again see the orchard effect, and a rather confusing array of atoms in the line of sight. Actually this is a harbinger of quasicrystals (next). Note too the different crystal faces have different atoms populating them.	18
19	20 Everyone has looked down rows in an orchard or some regular array of repetitive 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, i.e. 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 Diophantine. Diophantine is a perfectly rectangular two dimensional array of only one “atom”. Imagine you have your camera set on a road that runs alongside 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. Perspective and camera parallax distort the image in specific ways. Now, even though the road and the first row are also a straight line, this 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 as seen from the vantage of the camera are rendered perfectly straight.	21 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 to from every point. 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 stars.
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31 Rendered in 3D, with lighting, the pattern of density of electron flow into a semiconductor with an effective low amplitude random potential looks a lot like a Western fluvial landscape.	32 At the top of the image, “the sky” is a random wave corresponding to the quantum manifestation of classical chaos, as in Linear Ramp. 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	33 One of the best ways of studying chaos is through maps. A map is a simply a rule for jumping from place to place within the image. Maps are used as surrogates for more complicated dynamical systems. Although they are simple in concept and execution, they are complex upon repeated iteration, showing chaos in all its forms. The “dynamics” of a map is just the iteration of the rule defining the map. Each point in the plane of the image has a single point that was visited in the step before, and a point to go to in the step following. There are infinitely many points with coordinates (x,y) on the image, and every point is associated with just two others (one you came from, one you go to). Starting at one point, you have a definite path you must follow. Each move is predetermined. It isn’t as boring as it sounds, because even with very simple prescriptions for the map one get fantastically intricate patterns which result after many steps. Once initiated, a classical trajectory also has no choice or uncertainty, and must follow certain laws that take it from one time and place to the next. Given the rules above, the question arises as to what pattern the point will make in the course of many moves. There are three possibilities. An initial point could return exactly to itself after some finite number of moves. This is very rare, and is called a periodic orbit. It is happening at the center of all the “islands” in Suris. Second, the point can wander around on a curved line or disjoint set of lines closed on themselves. This is happening, for example, in the “sun” region, and all the islands, which really consists of sets of concentric distorted circles, each a line closed on itself, one curved line for each point separately initiated in that region. The islands correspond to resonances, of the sort which cause musical instruments to sound and old cars to rattle. The colors of the resulting dots made by the successive moves are arbitrary in Suris, but are be chosen according to a different prescription in Chaos I. Finally, it is possible for a given point to cover a whole region, as in the blue “sky”, which resulted from 100,000 successive moves with a single blue point to start with. This region is chaotic. If a slightly different initial point is chosen, the location after a few moves is different, although the same blue region is covered after many moves. The colors in Chaos I are chosen according to the de Broglie relation, in much the same way as was described in connection with Rosetta Stone. Thus, the colors correspond to crests and troughs of waves in the quantum version of the map. Chaos II shows most clearly the successive iterations of a map. Starting with a region of points in the upper left, that region becomes the shape (and color) to the right, etc., and so on as we would read a page. The pattern that develops is called a homoclinic oscillation, and is characteristic of chaotic motion. Map I and Chaos I are similar to Chaos II, except that the region shown is zoomed in, showing only the region of the periodic orbit around which the homoclinic oscillations develop. These maps show structures ubiquitous in nature.
34 One of the best ways of studying chaos is through maps. A map is a simply a rule for jumping from place to place within the image. Maps are used as surrogates for more complicated dynamical systems. Although they are simple in concept and execution, they are complex upon repeated iteration, showing chaos in all its forms. The “dynamics” of a map is just the iteration of the rule defining the map. Each point in the plane of the image has a single point that was visited in the step before, and a point to go to in the step following. There are infinitely many points with coordinates (x,y) on the image, and every point is associated with just two others (one you came from, one you go to). Starting at one point, you have a definite path you must follow. Each move is predetermined. It isn’t as boring as it sounds, because even with very simple prescriptions for the map one get fantastically intricate patterns which result after many steps. Once initiated, a classical trajectory also has no choice or uncertainty, and must follow certain laws that take it from one time and place to the next. Given the rules above, the question arises as to what pattern the point will make in the course of many moves. There are three possibilities. An initial point could return exactly to itself after some finite number of moves. This is very rare, and is called a periodic orbit. It is happening at the center of all the “islands” in Suris. Second, the point can wander around on a curved line or disjoint set of lines closed on themselves. This is happening, for example, in the “sun” region, and all the islands, which really consists of sets of concentric distorted circles, each a line closed on itself, one curved line for each point separately initiated in that region. The islands correspond to resonances, of the sort which cause musical instruments to sound and old cars to rattle. The colors of the resulting dots made by the successive moves are arbitrary in Suris, but are be chosen according to a different prescription in Chaos I. Finally, it is possible for a given point to cover a whole region, as in the blue “sky”, which resulted from 100,000 successive moves with a single blue point to start with. This region is chaotic. If a slightly different initial point is chosen, the location after a few moves is different, although the same blue region is covered after many moves. The colors in Chaos I are chosen according to the de Broglie relation, in much the same way as was described in connection with Rosetta Stone. Thus, the colors correspond to crests and troughs of waves in the quantum version of the map. Chaos II shows most clearly the successive iterations of a map. Starting with a region of points in the upper left, that region becomes the shape (and color) to the right, etc., and so on as we would read a page. The pattern that develops is called a homoclinic oscillation, and is characteristic of chaotic motion. Map I and Chaos I are similar to Chaos II, except that the region shown is zoomed in, showing only the region of the periodic orbit around which the homoclinic oscillations develop. These maps show structures ubiquitous in nature.	35 A coding mistake in making a mesh for another purpose led to thsi image, or something close to it. A little manipulation and MeshMan resulted.	36 A quantum eigenstate, i.e. a stable, unchanging pattern of a wave bouncing off the walls of the lemon shaped billiard. Even thoguh this is a stationary wave, classical point particles bounce and for certain starting positions and velocities, they take on the same pattern, without the nodes and wave behavior.
37 If we think of a set of parallel waves on the ocean for example, we notice that the energy of the wave, and the direction of travel, is perpendicular to the wave crests and troughs. We can think of this direction of travel as the “ray” or trajectory. The rule is that the wave tags along, with its crests and troughs perpendicular to the trajectory. Suppose the rays or trajectories are heading every which way. (This actually corresponds to classical chaos, with characteristic wild motion of ray trajectories). Then wave sets would be going everywhere. We have to add them up to see what overall wave pattern results. In fact, the sum of all the randomly directed waves is the pattern seen in Random Sphere II, and Monolith. The waves lie on the surface of a sphere in Random Sphere II, just as the oceans waves do on earth. In Monolith, the waves are living on a flat surface, and are plotted as a landscape in perspective. Random waves are random sums of many plane waves, first studied by SIr MIchael Berry, and are the quantum manifestation of classical chaos.	38 Nodal 8 is one of a series of random quantum waves trapped in a parabolic bowl. All the waves have exactly the same energy. Quantum tunneling allows some of the wave to rise up the sides of the bowl, to energies much higher than the waves actually possess. The bottom of the bowl is at the upper left, and the jumble of waves seen there is inside the "allowed" region where no tunneling is occuring. The jumble is a random wave in the sense of Sir Michael Berry; a random jumble of waves going in all directions, witout bias.	39 The sum of many randomly directed waves on the surface of a sphere in seen. Color boundaries reveal nodal lines, where the wave is zero. It changes sign on either side of the boundaries. The jumble is a random wave in the sense of Sir Michael Berry; a random jumble of waves going in all directions, witout bias.
40 The sum of many randomly directed waves on the surface of a sphere in seen. Color boundaries reveal nodal lines, where the wave is zero. It changes sign on either side of the boundaries. The jumble is a random wave in the sense of Sir Michael Berry; a random jumble of waves going in all directions, witout bias.	41 The sum of many randomly directed waves on the surface of a sphere in seen. Color boundaries reveal nodal lines, where the wave is zero. It changes sign on either side of the boundaries.	42 A quantum electron wave that was once very tidy and collected at the top of the frame was dropped onto the rough surface in a gravitational field. Starting compactly in the sky, it accelerated toward the mountain, wavelengths growing shorter as it sped toward the mountain below. It then bounced elastically off the mountain, and reached its former height (on average) at the moment this image was recorded,, i.e. after one bounce. The wave now looks disheveled. This work is part of an ongoing study of scattering of quantum waves from rough surfaces
43	44	$Quantum Resonators (triptych) A quantum wave begins as a disturbance in two resonators present in this image. Quantum waves bounded by walls reflect and diffract, and build up resonances, in this sequency of three snapshots of the quantum time evolution.$ 45 A quantum wave begins as a disturbance in two resonators present in this image. Quantum waves bounded by walls reflect and diffract, and build up resonances, in this sequency of three snapshots of the quantum time evolution.
46 Quasicrystals lie between order and chaos, and are regular sums of a few plane waves (e.g. 5 or 9). The waves propagate at angles uniformly spaced around the compass.	47 Quasicrystals lie between order and chaos, and are regular sums of a few plane waves (e.g. 5 or 9). The waves propagate at angles uniformly spaced around the compass. Bessel 21 is a quasicrystal which is an addition of 21 plane waves “pivoted” about the center of the “sun”. Not only is this a quasicrystal, but it is also an approximation to a mathematical shape called a Bessel function. The approximation breaks down outside a radius inside the sun region. If we had used more than 21 waves, the approximation would still break down, but it would break down farther away from the center. As it is, the region of breakdown is most of the image. The slice shown in lighter colors is one repeating unit, which if copied 20 times around a circle would regenerate the whole image. This quasicrystal has approximate repetitions of the sun region which are very far away, outside the image.	48 The sum of many randomly directed waves is the pattern seen in Random Sphere and Random Synthesis, and Monolith. The waves lie on the surface of a sphere in Random Sphere, just as the oceans waves do on earth. In Monolith, the waves are living on a flat surface, and are plotted as a landscape in perspective.
49 The sum of many randomly directed waves is the pattern seen in Random Sphere and Random Synthesis, and Monolith. The waves lie on the surface of a sphere in Random Sphere, just as the oceans waves do on earth. In Monolith, the waves are living on a flat surface, and are plotted as a landscape in perspective.	50 manyConcentricCircles An homage to Sol Lewitt, an artist of immense imagination and importance	$ReflectionDiffraction1$ 51
$ReflectionDiffraction2$ 52	$ReflectionRefractionDiffraction This image says it all: reflection, refraction, and diffraction of burst of plane wavefrom the upper left impinging on the rectangle of medium with a different refractive index. The impedance mismatch causes reflections; some wave amplitude penetrantes and refracts, and some "misses" near the edge but diffracts abount the corners.$ 53 This image says it all: reflection, refraction, and diffraction of burst of plane wavefrom the upper left impinging on the rectangle of medium with a different refractive index. The impedance mismatch causes reflections; some wave amplitude penetrantes and refracts, and some "misses" near the edge but diffracts abount the corners.	$ReflectionRefractionDiffraction2 This image has it all: reflection, internal reflection, refraction, interference, and diffraction from a plane wave pulse (still seen more or less intact at the bottom right), having arrived some time ago at the block of material with higher refractive index than the surrounding medium. The pulse arrives from the upper left. The impedance mismatch causes reflections; some wave amplitude penetrates and refracts, and some "misses" near the edge but diffracts abount the corners. The circular backscattered wave in the upper zone is diffraction off the corner of the block. Internt reflection is commencing inside the block.$ 54 This image has it all: reflection, internal reflection, refraction, interference, and diffraction from a plane wave pulse (still seen more or less intact at the bottom right), having arrived some time ago at the block of material with higher refractive index than the surrounding medium. The pulse arrives from the upper left. The impedance mismatch causes reflections; some wave amplitude penetrates and refracts, and some "misses" near the edge but diffracts abount the corners. The circular backscattered wave in the upper zone is diffraction off the corner of the block. Internt reflection is commencing inside the block.
$Resonator I* Resonator 1, 2 and 3 show a Westervelt resonator (a form of Helmholtz resonator). Resonance is one of the key concepts in wave theory and in physics. Resonance can be achieved by sympathetically driving or pumping a system at one of its natural frequencies. Here, a quantum wave builds up in a resonant cavity between the straight and curved walls, when waves are arriving from below. There is a horizontal reflecting wall with a small hole. Most of the wave energy is reflected back, however 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 these images 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 semicircular mirror, and so forth. Since not many electrons can get through the hole, you would think not many would be found in the “cavity” between the wall and the mirror. But electrons are really waves, not particles. As the speed of the electrons is increased, their wavelength gets shorter, according to the de Broglie 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, i.e., 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 occurs at only occasional frequencies (or velocities in the case of an electron) and usually it is pretty quiet in the cavity. Even though classical electrons would be trapped in the cavity (except possibly for going back through the hole), the quantum waves manage to leak out the sides. It is easy to see the waves spilling out the sides of the cavity, and crawling along the back wall, only to escape finally and head toward the top. This is diffraction; waves do this, but particles cannot.$ 55 Resonator I* Resonator 1, 2 and 3 show a Westervelt resonator (a form of Helmholtz resonator). Resonance is one of the key concepts in wave theory and in physics. Resonance can be achieved by sympathetically driving or pumping a system at one of its natural frequencies. Here, a quantum wave builds up in a resonant cavity between the straight and curved walls, when waves are arriving from below. There is a horizontal reflecting wall with a small hole. Most of the wave energy is reflected back, however 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 these images 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 semicircular mirror, and so forth. Since not many electrons can get through the hole, you would think not many would be found in the “cavity” between the wall and the mirror. But electrons are really waves, not particles. As the speed of the electrons is increased, their wavelength gets shorter, according to the de Broglie 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, i.e., 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 occurs at only occasional frequencies (or velocities in the case of an electron) and usually it is pretty quiet in the cavity. Even though classical electrons would be trapped in the cavity (except possibly for going back through the hole), the quantum waves manage to leak out the sides. It is easy to see the waves spilling out the sides of the cavity, and crawling along the back wall, only to escape finally and head toward the top. This is diffraction; waves do this, but particles cannot.	$Resonator2 Resonator 1, 2 and 3 show a Westervelt resonator (a form of Helmholtz resonator). Resonance is one of the key concepts in wave theory and in physics. Resonance can be achieved by sympathetically driving or pumping a system at one of its natural frequencies. Here, a quantum wave builds up in a resonant cavity between the straight and curved walls, when waves are arriving from below. There is a horizontal reflecting wall with a small hole. Most of the wave energy is reflected back, however 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 these images 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 semicircular mirror, and so forth. Since not many electrons can get through the hole, you would think not many would be found in the “cavity” between the wall and the mirror. But electrons are really waves, not particles. As the speed of the electrons is increased, their wavelength gets shorter, according to the de Broglie 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, i.e., 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 occurs at only occasional frequencies (or velocities in the case of an electron) and usually it is pretty quiet in the cavity. Even though classical electrons would be trapped in the cavity (except possibly for going back through the hole), the quantum waves manage to leak out the sides. It is easy to see the waves spilling out the sides of the cavity, and crawling along the back wall, only to escape finally and head toward the top. This is diffraction; waves do this, but particles cannot.$ 56 Resonator2 Resonator 1, 2 and 3 show a Westervelt resonator (a form of Helmholtz resonator). Resonance is one of the key concepts in wave theory and in physics. Resonance can be achieved by sympathetically driving or pumping a system at one of its natural frequencies. Here, a quantum wave builds up in a resonant cavity between the straight and curved walls, when waves are arriving from below. There is a horizontal reflecting wall with a small hole. Most of the wave energy is reflected back, however 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 these images 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 semicircular mirror, and so forth. Since not many electrons can get through the hole, you would think not many would be found in the “cavity” between the wall and the mirror. But electrons are really waves, not particles. As the speed of the electrons is increased, their wavelength gets shorter, according to the de Broglie 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, i.e., 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 occurs at only occasional frequencies (or velocities in the case of an electron) and usually it is pretty quiet in the cavity. Even though classical electrons would be trapped in the cavity (except possibly for going back through the hole), the quantum waves manage to leak out the sides. It is easy to see the waves spilling out the sides of the cavity, and crawling along the back wall, only to escape finally and head toward the top. This is diffraction; waves do this, but particles cannot.	$Resonator3 Resonator 1, 2 and 3 show a Westervelt resonator (a form of Helmholtz resonator). Resonance is one of the key concepts in wave theory and in physics. Resonance can be achieved by sympathetically driving or pumping a system at one of its natural frequencies. Here, a quantum wave builds up in a resonant cavity between the straight and curved walls, when waves are arriving from below. There is a horizontal reflecting wall with a small hole. Most of the wave energy is reflected back, however 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 these images 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 semicircular mirror, and so forth. Since not many electrons can get through the hole, you would think not many would be found in the “cavity” between the wall and the mirror. But electrons are really waves, not particles. As the speed of the electrons is increased, their wavelength gets shorter, according to the de Broglie 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, i.e., 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 occurs at only occasional frequencies (or velocities in the case of an electron) and usually it is pretty quiet in the cavity. Even though classical electrons would be trapped in the cavity (except possibly for going back through the hole), the quantum waves manage to leak out the sides. It is easy to see the waves spilling out the sides of the cavity, and crawling along the back wall, only to escape finally and head toward the top. This is diffraction; waves do this, but particles cannot.$ 57 Resonator 1, 2 and 3 show a Westervelt resonator (a form of Helmholtz resonator). Resonance is one of the key concepts in wave theory and in physics. Resonance can be achieved by sympathetically driving or pumping a system at one of its natural frequencies. Here, a quantum wave builds up in a resonant cavity between the straight and curved walls, when waves are arriving from below. There is a horizontal reflecting wall with a small hole. Most of the wave energy is reflected back, however 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 these images 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 semicircular mirror, and so forth. Since not many electrons can get through the hole, you would think not many would be found in the “cavity” between the wall and the mirror. But electrons are really waves, not particles. As the speed of the electrons is increased, their wavelength gets shorter, according to the de Broglie 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, i.e., 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 occurs at only occasional frequencies (or velocities in the case of an electron) and usually it is pretty quiet in the cavity. Even though classical electrons would be trapped in the cavity (except possibly for going back through the hole), the quantum waves manage to leak out the sides. It is easy to see the waves spilling out the sides of the cavity, and crawling along the back wall, only to escape finally and head toward the top. This is diffraction; waves do this, but particles cannot.
58 This image related to the mechanism for generation of freak waves in the ocean. We have published this work in the Journal of Geophysical Research-Oceans. The freak waves are not shown-instead, current eddies which cause random concentrations of wave energy are shown as color patches; ray paths were launched from the top of the image, representing waves impinging on the eddies from above. There is a 6 degree spread of the wave directions; in this rendition, the redder and brighter parts of the images correspond to concentrations of wave energy where rogue or freak waves are much more likely. As the wave energy progresses toward the bottom of the image, it becomes more fragmented, eventually varying on a much finer scale than the eddies that produced the concentrations of wave energy. The horizontal black streaks are "phase space" images of the ray paths, taken at various slices which can be seen as brigher horizontal lines. The vertical direction in the black streakes is the direction of the energy sideways (right or left) at that point. The "runners" or streeaks which are heading off at high angle at the bottom of the image can be seen as little balck blotches above and below the main streaks in that region.	59	60 Rogue is based on a calculation of the energy lumps in the ocean created by storm wave passing through ocean current eddies. The lumps are not the shape of the sea surface but rather the average wave energy over large regions. It might be 20-40 km from the center of one lump to the center of the nearest one.
61 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.	62 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.	63 The Rosetta stone was found in Egypt in 1799. It recorded the same passage in Greek, hieroglyphics, and ancient Egyptian. In much the same way the image Rosetta Stone translates between classical mechanics and quantum mechanics. The classical trajectory being traced out by the colored path is that of a marble rolling back and forth in a distorted bowl. A direct link or translation between classical mechanics (the trajectory weaving its path) and quantum mechanics is provided by keeping track of the corresponding wave properties, using de Broglie’s relation (velocity) X (wavelength) = constant. Define red as the crest of the wave, and green as the trough. The crests and troughs accumulate rapidly as the trajectory moves. As the trajectory overlaps itself the colors are added so as to correspond to addition of waves. For example, if a crest is added to a trough, the two cancel each other. After more time has elapsed (reading the images as in a book), the trajectory has overlapped itself many times and a new pattern emerges. This pattern consists of two complimentary colors, which arise naturally via the addition mentioned above. The pattern is a patchwork, which accurately reflects the true quantum mechanical wave (not shown). Thus, a trajectory has been used to construct a quantum wave-a true translation between classical and quantum mechanics. This sort of idea goes all the way back to Bohr and his quantization of the hydrogen atom through classical mechanics.
64 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.	65 In the middle distance is a stroboscopic accumulation of the motion of a chaotic rotator, seen in Rotating Rotator II.	66 This images is the basis of the cover of my book, "Why You Hear What You Hear", an work on acoustics and psychoacoustics, published by Princeton University Press. It shows, in sequecce clockwise around the clock, the evolution of a "bang" near the apex of a wedge with hard walls. The evolution of the wave is shown in 12 equally spacd time steps, with the marvelous pattern of reflection off the walls. The sould pressure pattern was rotated in each case so as not to overlap the previous frame.
67 This image related to the mechanism for generation of freak waves in the ocean. We have published this work in the Journal of Geophysical Research-Oceans. The image is not a literal rendition of a freak wave, but rather shows the mixing occuring in what is called phase space - the simultaneous redition of the position and the momentum of the wave. This does lead to freak waves.	68	69 Blue Peter
70	71	72
73 The dynamics of a system of two degrees of freedom involves two positions and the rate of change of these positions, i.e. two velocities. Altogether, this makes four coordinates. Thus, a system of two degrees of freedom “lives” in four mathematical dimensions. If the motion of the system is not chaotic, it actually lives only on the surface of two-dimensional torus, which has the same topology, or "shape", as a donut. The torus is, however, embedded in all four dimensions (a real donut has its two dimensional surface embedded in three dimensions). We can imagine a shadow of a clear, plastic donut falling on a piece of paper; this would be a two dimensional projection of a two-dimensional object embedded in three dimensions.	74 The dynamics of a system of two degrees of freedom involves two positions and the rate of change of these positions, i.e. two velocities. Altogether, this makes four coordinates. Thus, a system of two degrees of freedom “lives” in four mathematical dimensions. If the motion of the system is not chaotic, it actually lives only on the surface of two-dimensional torus, which has the same topology, or "shape", as a donut. The torus is, however, embedded in all four dimensions (a real donut has its two dimensional surface embedded in three dimensions). We can imagine a shadow of a clear, plastic donut falling on a piece of paper; this would be a two dimensional projection of a two-dimensional object embedded in three dimensions.	75 The dynamics of a system of two degrees of freedom involves two positions and the rate of change of these positions, i.e. two velocities. Altogether, this makes four coordinates. Thus, a system of two degrees of freedom “lives” in four mathematical dimensions. If the motion of the system is not chaotic, it actually lives only on the surface of two-dimensional torus, which has the same topology, or "shape", as a donut. The torus is, however, embedded in all four dimensions (a real donut has its two dimensional surface embedded in three dimensions). We can imagine a shadow of a clear, plastic donut falling on a piece of paper; this would be a two dimensional projection of a two-dimensional object embedded in three dimensions. Similarly, Torus III and Torus IV are projections onto two dimensions, from the four dimensional space in which the two dimensional torus is embedded. 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 not.
76 The dynamics of a system of two degrees of freedom involves two positions and the rate of change of these positions, i.e. two velocities. Altogether, this makes four coordinates. Thus, a system of two degrees of freedom “lives” in four mathematical dimensions. If the motion of the system is not chaotic, it actually lives only on the surface of two-dimensional torus, which has the same topology, or "shape", as a donut. The torus is, however, embedded in all four dimensions (a real donut has its two dimensional surface embedded in three dimensions). We can imagine a shadow of a clear, plastic donut falling on a piece of paper; this would be a two dimensional projection of a two-dimensional object embedded in three dimensions. Similarly, Torus III and Torus IV are projections onto two dimensions, from the four dimensional space in which the two dimensional torus is embedded. 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 not.	77 The dynamics of a system of two degrees of freedom involves two positions and the rate of change of these positions, i.e. two velocities. Altogether, this makes four coordinates. Thus, a system of two degrees of freedom “lives” in four mathematical dimensions. If the motion of the system is not chaotic, it actually lives only on the surface of two-dimensional torus, which has the same topology, or "shape", as a donut. The torus is, however, embedded in all four dimensions (a real donut has its two dimensional surface embedded in three dimensions). We can imagine a shadow of a clear, plastic donut falling on a piece of paper; this would be a two dimensional projection of a two-dimensional object embedded in three dimensions. Similarly, Torus III and Torus IV are projections onto two dimensions, from the four dimensional space in which the two dimensional torus is embedded. 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 not.	78 Transport II renders electron flow paths in a "two-dimensional electron gas" (2DEG). The scale of the image is about the size of a bacterium, i.e. a few microns. The research leading to the Transport series was inspired by the experiments of Mark Topinka, Brian Leroy, and Prof. Robert Westervelt at Harvard, which actually measured the paths taken by the electrons. The theory was performed with the help of Scot Shaw, a member of my research group at the time. Transport II is based on flow patterns for electrons riding over bumpy landscape, which is what electrons experience in the two-dimensional electron gas (2DEG) that they dwell in. A 2DEG is a sea of electrons confined to a sheet, i.e. two-dimensions. The bumps they encounter are due to charged atoms lying above the sheet. The electrons have more than enough energy to ride over any bump, and the concentrations of electron flow into the branches seen here are recently discovered indirect effects of that bumpy ride. The channeling or branching was unexpected and has implications for small electronic devices of the future. This image arises from a numerical simulation which closely approximated what is seen experimentally, using extremely sensitive probes which can sample thousands of data points inside a space as small as a typical bacterium. About 100,000 individual electron tracks are shown here. Each track added grayscale density to nearby pixels as it passed by, so the dark areas depict where many electrons went, one at a time. All electrons were launched at the center and were sent in all directions equally. The existence of dark branches rather far from the launch point is surprising, as no valleys or other simple features of the landscape guide the branches. The electron tracks in Transport II are an excellent example of the wonderful way nature emulates herself in different contexts, as the branching pattern seen here is reminiscent of familiar natural forms.
79 Transport III, another image in the electron flow series, emphasizes the phenomenon of "caustics", or lines of accumulation where we look edge on. 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 caustics will be very dark, because there light must pass through much more material to get through than at a typical place. Whether by training or by instinct, we associate a line in a simple drawing with a caustic in the real world. Even the cave painters 50,000 years ago knew these tricks and rendered some images of animals with subtle use of line to represent caustics. But caustics are not always found at the obvious places. Caustics are found whenever there is "projection" to lower dimension. When we see something which is really three-dimensional, we automatically are projecting it onto the plane of our retina, using only two dimensions. Nowhere are caustics 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 up along a blade. At a definite point, we start to see two new edges or caustics arise where before there were none. Once again, nature has mimicked herself and given us the appearance of an underwater scene even though the medium is the flow of electrons on the micron (one millionth of a meter) scale. The fish was added to emphasize the aquatic allusions. This image is part of the museum exhibit “Approaching Chaos: Visions from the Quantum Frontier” that will travel the United States February 2001 through December 2003. It was selected at SIGGRAPH 2001 for the SIGGRAPH World Tour Exhibition, November 2001-November 2003. It has been featured in several publications including: Harvard Magazine (Jan/Feb 2001); Computer Graphics World (Oct 2001); Ny Teknik, (Oct 25, 2001); and IEEE Computer Graphics and Applications January/February 2002, in which articles describing Eric Heller’s work appears.	80 This image renders electron flow paths in a two dimensional electron gas. It is 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 or hill. In fact they deflection is so weak at each hill that several are encountered before much noticeable change of direction occurs. The hills and dales themselves are not seen here. The concentrations of electron flow which result from this kind of travel are newly discovered indirect effects of the deflections due to encounters with many hills and dales. The channeling or branching was unexpected and has implications for small electronic devices of the future. The Transport series was inspired by the electron flow experiments of Mark Topinka, Brian Leroy, and Professor Robert Westervelt at Harvard. Scot Shaw of my group and I did the theoretical work. 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 fixed features in the potential landscape but rather are due to the history of encounters with hills and valleys "upstream".	81 Transport VI is perhaps the most abstract of the Transport series. The image is based on the actual electron flow patterns for electrons riding over bumpy landscape in a "two dimensional electron gas". The electrons have more than enough energy to ride over any bump in the landscape. The concentrations of electron flow 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. In my work as a theoretical physicist, I have found that the attempt to produce art has led to new scientific discoveries and a deeper understanding of quantum mechanics. There is a connection, a feedback from the science to the art and back again. In Transport VI, I recorded a grayscale image made by accumulating tracks of individual electrons, and then color mapped, sharpened and gradient filled the image to evoke something beyond the data itself. I want the scene being rendered to evoke emotion and familiarity. In this image I see vaguely familiar things, such as pine trees with snow; a red sunset behind. The viewer can project these tangible objects back onto the science behind the image and thereby gain a sense of the power and mystery in the world of quantum mechanics and the domain of the atom and the electron. The transport series was 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, Eric Heller. Transport VI is in the MIT sponsored traveling exhibit Approaching Chaos (2001-2003) It was exhibited at SIGGRAPH (August, 2001) and selected for the SIGGRAPH World Tour (2002-2003). Transport VI appeared in Focus (Italy) October 2001 and IEEE Computer Graphics and Applications January/February 2002. A word about the process: Each image is an original created by sending a digital file to a LightJet imager. The LightJet writes to 50" wide photographic paper. The images are then developed through the normal photographic process. The resulting prints have a 60-year archival life under normal lighting conditions. AutumnColor, in Worcester, Massachusetts handles print management.
82 Transport VII shows the flow pattern for electrons riding over a two dimensional bumpy surface where the electrons were injected from the top in a uniform sheet, all initially heading straight down. The bumps are caused by the irregular arrangement of nearby atoms, some of which donated the electrons, and are thus positively charged. The electrons have more than enough energy to ride over the highest bumps in the landscape. The concentrations of electron flow into branches are newly discovered indirect effects of that bumpy ride. The branching seen here was not anticipated; it was thought that the flow would be more evenly spread out some distance from the center. This has significant implications for small electronic devices of the future. This image comes from a numerical simulation which closely approximated what is seen experimentally, using extremely sensitive probes which can sample thousands of data points inside a space as small as a typical bacterium. The whole picture occupies a hundredth of the width of a human hair.	83 Small-scale electronic devices, the size of a bacterium or even a hundred times smaller, inevitably have minute imperfections, 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. In this image, the electrons are launched over a very small range of initial angles, represented by the narrow "stems". Small initial differences in angle grow quickly, as evidenced by the fanning out and branching of electron paths. This is the beginning of the eventual chaotic motion of these electrons. Note that some branches cross. This implies that the branches are not following specific valleys in the landscape, but are subject to indirect effects caused by focusing as electrons travel over bumps and hills. In Transport XI individual electron paths can be seen as bright lines, but if many electrons go over the same region the coloring scheme makes the image darker.	84
85 Transport XIII 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. In 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 chaos depicted by the random paths of the electrons as they travel farther from the launching point is the classical analog of the chaos seen on the sphere. The Transport series was inspired by the electron flow experiments of Mark Topinka, Brian Leroy, and Professor Robert Westervelt at Harvard. Scot Shaw of my group and I did the theoretical work. There is a connection, a feedback from the science to the art and back again. In me, this has happened many times and has led to new scientific discoveries through the attempt to produce art. In the viewer and also in me, I strive for a feedback of a different kind, namely, I want the scene being rendered to evoke emotion and familiarity; this the viewer can project back onto the science behind the image to sense the power and mystery in the world of quantum mechanics and the microscopic chaos which is just under the surface.	86 This image renders electron flow paths in a two dimensional electron gas. It is 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 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 Transport series was inspired by the electron flow experiments of Mark Topinka, Brian Leroy, and Professor Robert Westervelt at Harvard. Scot Shaw of my group and I did the theoretical work. T	87 This image renders electron flow paths in a two dimensional electron gas. It is 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 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 Transport series was inspired by the electron flow experiments of Mark Topinka, Brian Leroy, and Professor Robert Westervelt at Harvard. Scot Shaw of my group and I did the theoretical work.
88 This image renders electron flow paths in a two dimensional electron gas. It is 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 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 Transport series was inspired by the electron flow experiments of Mark Topinka, Brian Leroy, and Professor Robert Westervelt at Harvard. Scot Shaw of my group and I did the theoretical work. T	89	90
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94	95	96 Trichaotic is a composite of three manifestations of chaos: two classical, one quantum.
97 This image related to the mechanism for generation of freak waves in the ocean. We have published this work in the Journal of Geophysical Research-Oceans. The image is not a literal rendition of a freak wave, but rather shows the mixing occuring in what is called phase space - the simultaneous redition of the position and the momentum of the wave. This does lead to freak waves.	98	99
100 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. He was educated in Law at the University of Leipzig, and an amateur musician. Chladni wrote one of the first treatises on acoustics, “Discovery of the Theory of Pitch”. Chladni invented the method of displaying the vibrations of objects like metal plates, by laying fine sand on them and then stroking them with a violin bow, which gets the plate vibrating at one of its resonant frequencies. As it vibrates, the movement pushes the sand around. But there are places, or lines, along which the plate is not moving at all. These are called nodes, and on either side of them the metal is moving up and down. The sand collects on these quiet spots, making them visible. At the same time, the plate rings with a loud, clear pure tone. Chladni’s “multimedia” lectures became a very popular. Napoleon, who was a strong supporter of science and scientific research, asked Chladni to give a private demonstration. He was very impressed, and offered Chladni a great deal of support for his research. He also offered a sizable prize to anyone who could explain the mathematical basis for the patterns of vibration. This was finally done some years later by the mathematician Sophie Germain. The image is taken directly from a drawing Chladni himself made, showing the various patterns he was able to get by bowing the plate at different resonant frequencies. The black lines inside each square show the location of the quiet regions. Each new frequency or tone has its own pattern. I found it irresitible to color the patterns, which I study in my own work.