Category Archives: Mathematical Art

Lifted Domain Coloring

This year, one of the pictures at the International Science & Engineering Visualization Challenge, caught my interest.

Poelke and Polthier’s Lifted Domain Coloring is a coloring scheme for visualizing properties of complex functions: it maps numbers in the complex plane stereographically to the Riemann sphere, and assigns a hue based on the inclination angle (though I’m not sure that much is gained by the stereographic projection, since the polar representation of the complex numbers seems to provide all the needed information). Saturation and Brightness are controlled by the modulus of the number: when the modulus goes towards infinity, the color turns white, and for numbers close to zero, the color turns black. The exact radial mapping used by the authors is not specified in the paper, but I think my implementation is quite close:

The visualization scheme makes it possible to visually identify different properties, such as zeroes and poles in complex functions.

One of the ways, I think such a visualization may be improved is by using a heightmap:

Here I’ve raised the poles and lowered the zeroes: first, I made the poles and zeroes appear symmetric, by transforming the modulus: r = abs(r + 1/r). Then I applied a sigmoid function to tame the infinities, and finally another sigmoid transformation was applied to change the sign of the zeroes. This technique will only work for somewhat well-behaved functions (meromorph functions – functions with a countable number of zeroes and poles).

Of course, I’ve also tested the Lifted Domain Coloring scheme on fractals.

Here is a Mandelbrot and Julia plot:

Usually Mandelbrot visualizations focus on coloring the outside of the set, but since the exterior of the Mandelbrot set has infinite modulus, only the interior (with its zeroes) are visualized here. The zeroes are visualized as peaks for better graphical clarity.

I also tested the coloring scheme on Samuel Monnier’s Ducks fractal:


Here, the coloring scheme does a decent job for low iteration counts, but for higher iterations the images become messy, so for pure aestethic purposes there are probably better coloring schemes around.

Distance Estimated Polychora

My last post mentioned some scripts that Knighty (a Fractal Forums member) made for distance estimated rendering of many types of polyhedra, including the Platonic solids. Shortly after, Knighty really raised the bar by finding a distance estimator for four dimensional polytopes. In this post, I’ll show some images of a subset of these, the convex regular polychora.

There are several ways to depict four dimensional structures. The 4D Quaternion Julia, one of the first distance estimated fractals, simply showed 3D slices of 4D space. Another way would be to project the shadow of a 4D object onto a 3D space. Ideally, a proper perspective projective would be preferable, but this seems to be complicatated with distance estimation techniques.

The technique Knighty used to create a 3D projection, was to place the polychoron boundary on a 3-sphere, and then stereographically project the 3-sphere surface onto a 3-dimensional space. For a very thorough and graphical introduction to stereographic projection of higher-dimensional polytopes, see this great movie: Dimensions. It should be noted that the Jenn3D program also uses this projection to depict a variety of polytopes (using polygonal rendering, not distance estimated ray marching).

Back to the structures:

The convex regular 4-polytopes are the four-dimensional analogs of the Platonic solids. They are bounded by three-dimensional cells, which are all Platonic solids of the same kind (similar to the way the Platonic solids are bounded by identical regular 2D polygons). The convex regular polytopes are consistently named by the number of identical cells (Platonic solids) that bounds them. In four dimensions, there are six of these, one more than the number of Platonic solids.

5-cell

The 5-cell (or 4-simplex, or hypertetrahedron) is the simplest of the convex regular polytopes. It is composed of 5 three-dimensional tetrahedrons, resulting in a total of 5 vertices and 10 edges. It is the 4D generalisation of a tetrahedron.

The curved lines are a consequence of the stereographic projection. In 4D the lines would be straight.

8-cell

The 8-cell (or hypercube, or Tesseract) is also simple. Composed of 8 3D-cubes, it has 16 vertices, and 32 edges. It is the 4D generalisation of a cube.

16-cell

Things start to get more complicated with the 16-cell (or hyperoctahedron). It is composed of 16 tetrahedrons, and has 8 vertices and 24 edges. It is the 4D generalisation of an octahedron.

Notice, that if we rotate the 3-sphere, we can get interesting depictions, with edges getting infinitely long in 3-space:

24-cell

The 24-cell is exceptional, since it has no 3D analogue. It is built from 24 octahedrons, has 24 vertices and 96 edges.

120-cell

The 120-cell is a beast, built from 120 three-dimensional dodecahedra. With 600 vertices and 1200 edges, it is the most complex of the convex, regular polychora. It is the 4D generalisation of a dodecahedron.

Zooming in, it is easier to see the pentagons making up the dodecahedra.

600-cell

Finaly, the 600-cell is built from an even larger number of polyhedra: 600 three-dimensional tetrahedrons. However, the simpler polyhedra means there is only a total of 120 vertices and 720 edges. It is the 4D generalisation of an icosahedron.

Knighty’s great Fragmentarium scripts can be found in this thread at Fractal Forums.

Knighty’s script is not limited to regular convex polychora. Many types of polytopes can be made. Here are the parameters used for the images in this post:

polychora06.frag parameters:

Type,U,V,W,T
3,0,1,0,0 - 5-cell
4,0,1,0,0 - 8-cell
4,0,0,1,0 - 24-cell
4,0,0,0,1 - 16-cell
5,0,1,0,0 - 120-cell
5,0,0,0,1 - 600-cell

And for completeness, here are the parameters for the 3D polyhedron script:

Type,U,V,W
3,0,1,0 Tetrahedron
4,0,0,1 Cube
3,1,0,0 Octahedron
5,0,0,1 Dodecahedron
5,0,1,0 Icosahedron

A Few Links

…some old, some new.

The Demoscene

It was only a matter of time, before a Mandelbox would show up on the Demoscene:

Hochenergiephysik by Still is a 4K demo, featuring the Mandelbox. If 4KB sounds bloated, Still has also created a 1K demo: Futurism by Still.

And while we are at it, may I suggest these as well: The Cube by Farbrausch, Rove by Farbraush, and Agenda Circling Forth by Fairlight & Cncd.

New software

NodeBox 2.0 is out in a beta version. The big news is that it is now available for Windows. It also sports a graph-based GUI for patching nodes together.

Tacitus is a GUI for creating per-pixel GPU effects. In concept it is similar to Pixel Bender. It has a very nice look and feel, but a big short-coming is that it is not possible to directly edit the GPU scripts in the GUI – you have to compile your script to a plugin via an included compiler. Another feature I miss, is the ability to directly navigate the camera using the mouse on the viewport, instead of using sliders (something Pixel Bender also doesn’t support). But Tacitus is still in beta, and it will be interesting to see where it is going. It comes with a single plugin, a port of Subblue’s Mandelbulb Pixel Bender plugin. Tacitus is Windows only.

NeuroSystems Substance is an ‘Evolutionary and Organic Art Creator’. Some interesting concepts here, including a real-time global illumination raytracer (video here). Unfortunately, the raytracer is not part of the free viewer. Surprisingly, NeuroSystems impressive visualization technology seems to originate from SIMPLANT, a real-time 3D breast implant simulator. Substance is Windows only, and the full (non-free) versions should be released very soon.

Gifts for Geeks

A Calabi-Yau Manifold Crystal sculpture.



A Gömböc. “The ‘Gömböc’ is the first known homogenous object with one stable and one unstable equilibrium point, thus two equilibria altogether on a horizontal surface. It can be proven that no object with less than two equilibria exists.”

The Reality of Fractals

“… no one, not even Benoit Mandelbrot himself […] had any real preconception of the set’s extraordinary richness. The Mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. If it has meaning to assign an actual existence to the Mandelbrot set, then that existence is not within our mind, for no one can fully comprehend the set’s endless variety and unlimited complication.”

Roger Penrose (from The Road to Reality)

The recent proliferation of 3D fractals, in particular the Mandelbox and Mandelbulb, got me thinking about the reality of these systems. The million dollar question is whether we discover or construct these entities. Surely these systems give the impression of exploring uncharted territory, perhaps even looking into another world. But the same can be said for many traditional man-made works of art.

I started out by citing Roger Penrose. He is a mathematical Platonist, and believes that both the fractals worlds (such as the Mandelbrot set) and the mathematical truths (such as Fermat’s last theorem) are discovered. In his view, the mathematical truths have an eternal, unchanging, objective existence in some kind of Platonic ideal world, independent of human observers.

In Penrose’s model, there are three distinct worlds: the physical world, the mental world (our perception of the physical world), and the cryptic Platonic world. Even Penrose himself admits that the connections and interactions between these worlds are mysterious. And personally I cannot see any kind of evidence pointing in favor of this third, metaphysical world.


Designer World by David Makin

Roger Penrose is a highly renowned mathematician and physicist, and I value his opinions and works highly. In fact, it was one of his earlier books, The Emperors New Mind, that in part motivated me to become a physicist myself. But even though he is probably one of the most talented mathematicians living today, I am not convinced by his Platonist belief.

Personally, I subscribe to the less exotic formalist view: that the mathematical truths are the theorems we can derive by applying a set of deduction rules to a set of mathematical axioms. The axioms are not completely arbitrary, though. For instance, a classic mathematical discipline, such as Euclidean geometry, was clearly motivated by empirical observations of the physical world. The same does not necessarily apply to modern mathematical areas. For instance, Lobachevsky’s non-Euclidean geometry, was conceived by exploring the consequences of modifying one of Euclid’s fundamental postulates (interestingly non-Euclidean geometry later turned out to be useful in describing the physical world through Einstein’s general theory of relativity).

But if modern mathematics has become detached from its empirical roots, what governs the evolution of modern mathematics? Are all formal systems thus equally interesting to study? My guess is that most mathematicians gain some kind of intuition about what directions to pursue, based on a mixture of trends, historical research, and feedback from applied mathematics.


Mandelballs by Krzysztof Marczak [Mandelbox / Juliabulb mix]

Does my formalist position mean that I consider the Mandelbrot set to be a man-made creation, in the same category as a Picasso painting or a Bach concert? Not exactly. Because I do believe in a physical realism (in the sense that I believe in a objective, physical world independent of human existence), and since I do believe some parts of mathematics is inspired by this physical world and tries to model it, I believe some parts of mathematics can be attributed an objective status as well. But it is a weaker kind of objective existence: the mathematical models and structures used to describe reality are not persistent and ever-lasting, instead they may be refined and altered, as we progressively create models with greater predictive power. And I think this is the reason fractals often resemble natural structures and phenomena: because the mathematics used to produce the fractals was inspired by nature in the first place. Let me give another example:


Teeth by Jesse

Would a distant alien civilization come up with the same Mandelbrot images as we see? I think it is very likely. Any advanced civilization studying nature, would most likely have created models such as the natural numbers, the real numbers, and eventually the complex numbers. The complex numbers are extremely useful when modeling many physical phenomena, such as waves or electrodynamics, and complex numbers are essential in the description of quantum mechanics. And if this hypothetical civilization had computational power available, eventually someone would investigate the convergence of a simple, iterated system like z = z2 + c. So there would probably be a lot of overlapping mathematical structures. But there would also be differences: for instance the construction of the slightly more complex Mandelbox set contains several human-made design decisions, making it less likely to be invented by our distant civilization.

I think there is a connection to other areas of generative art as well. In the opening quote Penrose claims that no-one could have any real preconception of the Mandelbrot sets extraordinary richness. And the same applies to many generative systems: they are impossible to predict and often surprisingly complex and detailed. But this does not imply that they have a meta-physical Platonic origin. Many biological and physical systems share the same properties. And many of the most interesting generative systems are inspired by these physical or biological systems (for instance using models such as genetic algorithms, flocking behavior, cellular automata, reaction-diffusion systems, and L-systems).

Another point to consider is, that creating beautiful and interesting fractal images as the ones above, requires much more than a simple formula. It requires aesthetic intuition and skills to choose a proper palette, find an interesting camera location, and it takes many hours of formula parameter tweaking. I know this from my experiments with 3D fractals – I’m very rarely satisfied with my own results.

But to sum it all up: Even though fractals (and generative systems) may posses endless variety and unlimited complication, there is no need to call upon metaphysical worlds in order to explain them.

Mandelbulb

A lot of sites have reported that a new, interesting 3D version of the Mandelbrot set has been discovered. The Mandelbulb has aesthetic qualities similar to Quaternion-Julia sets, but seems more diverse and suited for exploration.


“Cave of Lost Secrets” from Skytopia.

Skytopia has a great overview complete with many stunning images.

A good way to view the basic structure is this 56 Megapixel render from Skytopia (using the Seadragon viewer – requires Silverlight):


As of now, I do not know of any released software capable of generating Mandelbulbs, but it probably won’t be long:

Recent posts by Iñigo Quílez (who produced the Kindernoiser Quaternion-Julia set GPU renderer) indicate that he is very to close to completing a fast GPU implementation. These posts also include the basic source-code, which I believe should make it possible to port to other targets, for instance Pixel Bender. Apparently Quílez has cooked up a distance estimator, and a fake ambient occlusion scheme (based on orbit traps) for these Mandelbulbs, which sounds very promising.

Quaternion Julia sets and GPU computation.

Subblue has released another impressive Pixel Bender plugin, this time a Quaternion Julia set renderer.

The plugin can be downloaded here.

Quaternions are extensions of the complex numbers with four independent components. Quaternion Julia sets still explore the convergence of the system z ← z2 + c, but this time z and c are allowed to be quaternion-valued numbers. Since quaternions are essentially four-dimensional objects, only a slice (the intersection of the set with a plane) of the quaternion Julia sets is shown.

Quaternion Julia sets would be very time consuming to render if it wasn’t for a very elegant (and surprising) formula, the distance estimator, which for any given point gives you the distance to the closest point on the Julia Set. The distance estimator method was first described in: Ray tracing deterministic 3-D fractals (1989).

My first encounter with Quaternion Julia sets was Inigo Quilez’ amazing Kindernoiser demo which packed a complete renderer with ambient occlusion into a 4K executable. It also used the distance estimator method and GPU based acceleration. If you haven’t visited Quilez’ site be sure to do so. It is filled with impressive demos, and well-written tech articles.

Transfigurations (another Quaternion Julia set demo) from Inigo Quilez on Vimeo.

In the 1989 Quaternion Julia set paper, the authors produced their images on an AT&T Pixel Machine, with 64 CPU’s each running at 10 megaFLOPS. I suspect that this was an insanely expensive machine at the time. For comparison, the relatively modest NVIDIA GeForce 8400M GS in my laptop has a theoretical maximum processing rate of 38 gigaFLOPS, or approximately 60 times that of the Pixel Machine. A one megapixel image took the authors of the 1989 paper 1 hour to generate, whereas Subblues GPU implementation uses ca. 1 second on my laptop (making it much more efficient than what would have been expected from the FLOPS ratio).

GPU Acceleration and the future.

These days there is a lot of talk about using GPUs for general purpose programming. The first attempts to use GPUs to speed up general calculations relied on tricks such as using pixel shaders to perform calculations on data stored in texture memory, but since then several API’s have been introduced to make it easier to program the GPUs.

NVIDIAs CUDA is currently by far the most popular and documented API, but it is for NVIDIA only. Their gallery of applications demonstrates the diversity of how GPU calculations can be used. AMD/ATIs has their competing Stream API (formerly called Close To Metal) but don’t bet on this one – I’m pretty sure it is almost abandoned already. Update: as pointed out in the comments, the new ATI Stream 2.0 SDK will include ATIs OpenCL implemention, which for all I can tell is here to stay. What I meant to say was, that I don’t think ATIs earlier attempts at creating a GPU programming interface (including the Brook+ language) are likely to catch on.

Far more important is the emerging OpenCL standard (which is being promoted in Apples Snow Leopard, and is likely to become a de facto standard). Just as OpenGL, it is managed by the Khronos group. OpenCL was originally developed by Apple, and they still own the trademark, which is probably why Microsoft has chosen to promote their own API, DirectCompute. My guess is that CUDA and Brook+ will slowly fade away, as both OpenCL and DirectCompute will come to co-exist just the same way as OpenGL and Direct3D do.

For cross-platform development OpenCL is therefore the most interesting choice, and I’m hoping to see NVIDIA and AMD/ATI release public drivers for Windows as soon as possible (as of now they are in closed beta versions).

GPU acceleration could be very interesting from a generative art perspective, since it suddenly becomes possible to perform advanced visualization, such as ray-tracing, in real-time.

A final comment: a few days ago I found this quaternion Julia set GPU implementation for the iPhone 3GS using OpenGL ES 2.0 programmable shaders. I think this demonstrates the sophistication of the iPhone hardware and software platform – both that a hand-held device even has a programmable GPU, but also that the SDK is flexible enough to make it possible to access it.

Generative Bots

GroBoto is a commercial 3D modeling tool built around the concept of bots. Bots are small iterated systems, with a few selected variables that can be customized. Bots are selected from a list of presets – more than 100 are available. Some of the Bots are very similar to what can be accomplished in Structure Synth.

GroBoto is a very polished product. The GUI is slick, and there are loads of advanced visualization customizations: textures, lightning and animation. When moving and rotating objects an OpenGL view is used, but the scene is always automatically rendered using an internal raytracer, which is really amazingly fast (typically less than a second).

My only complaint is that you are somewhat limited by the presets offered by GroBoto. It would be amazing to be able to completely script the objects. Yet again, that would make GroBoto a tough competitor to Structure Synth :-)

GroBoto is available for $59 (using the coupon offer) for Windows and Mac OS X.

Be sure to a look at their gallery for more images or try the demo.

Communications of the ACM

I did the cover for the April issue of Communications of the ACM and a few illustrations inside as well.

CACM Readers: for more Structure Synth pictures see either my personal Flickr account or the public Structure Synth pool. If you want to try out the program for youself, it is freely available from SourceForge.

The structures were created in Structure Synth, and raytraced in SunFlow in high resolution (the largest picture was 6000×6000 pixels).

I was about to leave for Japan, when I was asked to make the cover image, so I had a very tight deadline. Despite this, the actual work process went fine, thanks to clear artistic guidance from the graphical editor (Alicia Kubista from Andrij Borys Associates).

By the way, even though the cover notes state that I’m a computer scientist, and even though I’ve worked professionally with software development for the last eight years or so, I am a physicist. Really.

Escher’s Droste Effect

On Subblue’s blog I stumbled upon his Adobe Pixel Blender filter that recreates Escher’s Droste effect.

First a few words about Adobe Pixel Blender which was new to me. It is a toolkit that allows you to write filters in a C-like language, that are compiled and executed on either the GPU or CPU. They can be used in Photoshop, After Effects and even Flash 10 – making it possible to create very powerful Flash content. It is interesting to see that general-purpose computing on graphics processing units now are becoming mainstream, and no longer just an academic exercise.

Prentententoonstelling by M.C. Eschers (1956)

Prentententoonstelling by M.C. Escher (1956) together with its grid transformation.

The story behind Escher’s Droste effect is very interesting. It refers to the particular transformation Escher used in his ‘Prentententoonstelling’ lithograph (pictured above). The mathematics behind the picture was unraveled by a team of Dutch mathematicians in 2003 (it is ‘…drawn on a certain elliptic curve over the field of complex numbers…’) and is excellently described on their website. Several other software implementations exists, including one for GIMP and a plugin for Paint.net.


Double Droste Clock by fpsurgeon

Above is a modern example of using this effect, found on the Escher’s Droste Print Gallery Flickr group. By coincidence, the structure of this particular picture reminds me of the very first structure synth image I created.

To be honest, I never liked Escher’s original painting much. I’ve always found it to be too mathematical and fabricated and not very interesting. But the math behind it is interesting and the Flickr group really contain many great pictures.

Be sure to check out Subblue’s many excellent examples of his filter (note the video at the bottom!).