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Algorithmic Mathematical Art

Xah Lee, 2004-01. Last revision: 2006-08.

Goto=20 Page: 1,=20 2= , 3=

With the inception of computers, there began a movement of = mathematical=20 computer art. In the beginning, they were merely visualization aids in = the study=20 of mathematics. Gradually, the complexity and artistry of the images = become a=20 end itself.

In this exposition, i try to examine the various methods of = algorithmic=20 mathematical art, and indicate the various states of the art and = possibilities.=20 In the end, i give a definition of Algorithmic Mathematical Art.


Geometric Surfaces

3D"borg=20

Above: =93Borg cube=94, a plot of the equation:=20 Sin[x*y]+Sin[y*z]+Sin[z*x]=3D=3D0.

3D""=20=20

above: a section of Riemann's Surface. A minimal surface from Richard = Palais's 3D-Xpl= orMath↗=20 software.

Plots of arbitrary 3D surfaces for visual arts purposes has not been = explored=20 much. The exception is visualization of some special mathematical = surfaces,=20 especially in differential geometry. (for example: Klein bottle, minimum=20 surfaces.)

As of today, it is scarcely known how surfaces look like of arbitrary = equations of 3 variables. Equations with more than 3 variables can be = projected=20 or sliced to 3 dimensional space, and this is completely unexplored.


Regular Solids of 3 or more dimensions

3D"linked=20

above: A dodecahedron formed by pentagonal rings, by Michael Trott. = (http://www.graphica.com/gallery= /).

In 2 dimensions we have regular polygons, such as square and pentagon = and=20 hexagon. The 3-dimensional version is called regular polyhedron. For = example:=20 cube, octahedron, icosahedron. The general name for such objects in any=20 dimension is called polytope.

Polyhedrons↗ = as=20 artistic themes have been explored greatly, way back in the Greek times = and in=20 different cultures, often exhibited as toys, sculptures, paper models = (origami),=20 or hanging decoration. In modern times, polyhedrons have been exhibited = as=20 architecture element as well. (Examples: the Disney's geodesic = dome, Eden Project = hemi-spheres,=20 Atomium, Biosphere2↗ ) =

Above left: paper models by Robert Webb↗. = Above right: A=20 Islamic lamp in the shape of a stellated polyhedron. (at the entrance of = Aladdin=20 hotel resort in Las Vegas).

Computer models as visual arts have been very popular. However, most = of them=20 are direct rendering of the mathematical object with almost no artistic = value.=20 Commonly seen are polyhedrons rendered with latest ray-tracing computer = graphics=20 advances, with some stellation or truncation. Highly artist algorithmic=20 exploration of polytopes is almost non-existent. The image by Michael = Trott=20 above, of a dodecahedron rendered as linked pentagonal rings, is a basic = artistic exploration.

For some examples of artistic algorithmic exploration, one can = systematically=20 examine ways to render a space by linked rings. For the centrally = symmetric=20 dodacahedron, for example, by some cleverness in linking, the links can = extend=20 to infinity as to fill the whole space with these links, and a cut-away = view can=20 be made to result in a artistic image. Or, polygonal tubes can be made = to have=20 larger radius, so that each polygonal =93side=94 of the tube is a large = flat area=20 that a Maze↗ can = be incribed=20 upon. The number of sides of the ring can increase or decrease according = to=20 certain parameters, such as symmetry location or distance from origin. = Or, one=20 can use spheres instead of rings. The coloring of balls around the = dodecahedron=20 can be exploited systematically, so to result in different symmetry = groups of=20 different groups of balls. Imagine a 3-D space with a cluster of = differently=20 colored balls that exhibits subtly different symmetry groups as in an = elaborate=20 intricate tiling work. Or, imagine 3-D space tiled uniformly by regular=20 polyhedrons, with some walls open, as to form a 3-dimensional maze. If = well=20 done, a cut-away view or translucent view of this would be fantastic. A=20 walk-thru as implemented in video games also has great potential. Solid = stacks=20 of cubes is called Soma-Cubes (a 3-dimensional version of domino or = polymino).=20 Solid block sculptures with other regular polyhedron are basically = unexplored.=20 Block stacks of tetrahedron are particularly interesting.

Below are some examples of polytopes:

  • Sandor Kabai has created many fantastic algorithmic mathematical = art, many=20 are themed around regular polyhedrons. His gallery is at: http://www.kabai.hu/=20
  • Robert Webb's polyhedron gallery: http://www.software3d.co= m/MyModels.html=20
  • Beautiful intricate physical models of polyhedrons by Ulrich = Mikloweit. http://www.polyed= ergarten.de/e_polyseite10.htm=20 (it would be great if these can be generated on a computer = algorithmically.)=20
  • Russell Towle's study of polytopes: http= ://home.inreach.com/rtowle/Mathematica/Mathematica.html;=20 http://h= ome.inreach.com/rtowle/Zonohedra/zonohedra.html=20
  • HyperSpace Polytope Slicer Java applet by Mark Newbold, 2000. http://dogfeathers.c= om/java/hyperslice.html;=20 and Hyperspace Star Polytope Slicer: http://dogfeathers.co= m/java/hyperstar.html=20

Plane Curves

3D"trignometry=20

Above: =93fiery=94: plots of the equation = Sin[x*Sin[y]]-Cos[y*Cos[x]]=3D=3D0.

=20

Above: A density plot version.

Two-dimensional plots for visual art purposes have not been explored = much.=20 Algebraic curves of more than 4 degrees are almost unexplored. One could = create=20 a program that systematically generates and plots all possible equations = by=20 degree or type, including non-algebraic ones.

Almost all ornamental elements in architecture or interior = decorations are=20 based in geometry, and most of them based on curves. Examples of curve = based=20 traditional art include: dome, arch, = vault↗ (based on arcs of circle), volutes↗ (as = scrolls on columns=20 or violins) (based on = various=20 spirals), curlicue=20 (e.g. iron wrought filigree found on gates) (based on Cornu = spiral↗, = lituus),=20 toy Spirograph↗ = (based on=20 rose= s,=20 cycloidal=20 curves).

Methods of artwork based on curves can include: tangents, envelope,=20 caustics=20 (String art); osculating circles and inversion,=20 evolute/involute,=20 pe= dal=20 curves, parallel=20 curves, pursuit=20 curves.

  • Dens= ity=20 Plots of Trig expressions

Plane Geometry and Processes

=20

Above: Nested inversion of circles. Nested circles can result in many = esthetic images.

=20

Above: Stereographic projection of a circles on a sphere to the plane = (produced by software KriviznaPlus↗ = by Viktor=20 Massalogin).

3D"nested =20

Above left: Nested inversion of circles. (by William=20 Gilbert↗). Above right: Circles generated by Mobius transform = by Ed Pegg Jr=20 in his M= ath Games=20 column↗.

Traditional artworks based on geometric process include that of = circles of=20 tangency ( Appollonius=20 circles↗ ), various plane transformations applied to a grid or = regular=20 tilings (linear, affine, projective, geometric inversion, mobi=FCs = transformation,=20 stereographic projection...). A particular example in recreation is=20 anamorphosis, which is a drawing that is not recognizable until its = reflection=20 is viewed on a shiny cylinder or metal ball. (See artwork = =93Trash=20 Reflection=94)

3D""=20=20

Above: A wallpaper pattern with star motif, cut into a diamond = outline, then=20 a =93fish-eye=94 transformation is applied.

For some images of traditional geometric processes, see:

  • Inversion=20 of circles gallery=20
  • Transformation=20 of the Plane; =95 Transformation=20 of the Plane II=20
  • Stereographic=20 Projected Beauties=20
  • Anamorphosis explained by Paul Bourke: http://astronomy.swin.edu.au/~pbourke/projection/anamorphism/inde= x.html.=20

L-Systems, Turtle Graphics

3D"mushroom=20

Above: =93mushroom triangle=94, generated by recursive = line-replacement

=20

Above: A snowflake generated by recursive line-replacement. (from An = eye for=20 Fractals, Michael McGuire. 1991. p.16.)

3D"pinwheel=20

Above: a =93Pinwheel tiling=94: recursive dissection of a right = triangle with=20 sides 1,2 Sqrt[5] (~2.236). (Two photos of the Federation = Square↗ ,=20 Melbourne, Australia, where the entire building's flat surfaces are = rendered as=20 pinwheel tiling: ph= oto=20 1;ph= oto=20 2)

L-System↗= ; is=20 a recursive string replacement system originally devised to model plant = growth.=20 It is often used to generate self-similar images by interpreting the = string as=20 drawing commands or geometric objects. Turtle Graphics is from the = programing=20 language LogoU= 99;, which=20 algorithmically controls the movement of a pen, by specifying directions = and pen=20 down or up.

L-System and Turtle Graphics have been somewhat popular among = recreational=20 programers. However, there have not been serious studies of visual art=20 possibilities with these methods. Commonly found are illustration of = famous plane-filling=20 curves↗ in mathematics, plant-growth modeling, or simplistic = symmetric=20 drawings for children as a demo of Logo. Few of them are ingenious.

  • William McWorter's L-System tutorial. 1997: http://span= ky.triumf.ca/www/fractint/lsys/tutor.html=20
  • Pinwheel Aperiodic Tiling, Attributed to Charles Radin, by Paul = Bourke,=20 June 2002. Includes photos of the Federation Square buildings in = Melbourne,=20 Australia, where the entire building's flat surfaces are rendered as = pinwheel=20 tiling. http://a= stronomy.swin.edu.au/~pbourke/texture/pinwheel/=20
  • Non Periodic Tiling of the Plane. by Paul Bourke. October 1995. http:= //astronomy.swin.edu.au/~pbourke/texture/nonperiodic/=20
  • Robert M Dickau has a gallery of well-known plane-filling curves. = http://mathfor= um.org/advanced/robertd/lsys2d.html=20

Plotting of Functions and Processes

Many functions in mathematics can be visualized as a plot. For curves = and=20 surfaces, the plotting scheme is simple. Often just on a 2D grid or 3D = grid,=20 with marked axes as coordinate. For other functions, such as vector = valued=20 functions, complex valued functions, mathematicians have developed other = schemes=20 to visualize these functions. Here are some exapmles.

3D""=20=20

above: =93The real part of the discriminant as a function of the nome = q on the=20 unit disk=94, by Linas Vepstas, 2005. (gall= ery↗)=20

3D""=20=20

A plot of the complex-valued function (z-2)^2(z+1-2*i)(z+2+2*i)/z^3. = By Hans=20 Lundmark, 2004. (gallery↗)=20

Goto=20 Page: 1,=20 2= , 3=


Page created: 2004-01
=A9 copyright 2004 by Xah Lee. (mailto:xah%40xahlee.org) (excluding =
mirrored pages or images.)
3D"Xah
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------=_NextPart_000_0029_01C73FFA.11A294F0--