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Computer Art Now

Exact Aesthetics Practically
by

Tomáš Staudek

on 7 August 2013

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Transcript of Computer Art Now

Gerda de Vries
Cyclic, 2005
Computer
Art
holistic thinking
feelings
intuition
right hemisphere
subjectivity
synthetic mind
analogies
science
objectivity
abstraction
logic
analytic mind
linear thinking
left brain hemisphere
Analog graphics
Solving equations — nonlinear, differential…

Scientific computations

Simulation and automatization
Krystollos, 1968
Graphical output :
plotter + oscilloscope
1st International CalComp
Awards, Los Angeles
Ben Laposky
Oscillon, 1960
High speed photographs of oscillograms
projected through color filters
Cybernetic Serendipity, London

“Acclaimed as the best math art of the time”
Ivan Moscovich
Harmonograms of Moscovich, 1968
Aesthetic functions
Jack Tait
Light Wave Invert, 2006
Light Sinewave 01, 2006
Sinewave machine drawings
Periodical aesthetics
Trigonometric functions
Epicycloids & hypocycloids
http://en.wikipedia.org/wiki/Trigonometric_functions
http://en.wikipedia.org/wiki/Cycloid
Mickey Shaw
Spirals in Chaos, 2009
Transformed spiral drawings on a moving drawing board
Parametrical shape
a, b – radius of the inner / outer circle

d – stylus distance from the center
a/b is integer : closed curve in a single iteration, a/b = number of cusps

a/b is rational = p/q : curve closed after q iterations with p cusps

a/b is irrational : infinite cusps
Lamé’s curve generalization
2/3 : asteroid

1 : rhomboid

2 : ellipse

> 2 : oval

5/2 : super-egg
Piet Hein
http://en.wikipedia.org/wiki/Superegg
Parametrized super-ellipse in polar coordinates
Superformula
Superformula
http://en.wikipedia.org/wiki/Superformula
Artistic rendering of f (x, y)
Sophisticated modelling of natural phenomena
Ornamental rhythm of complex & composite functions
Tomáš Mrkvička
2009
*
Stephen Luecking
Cornrow, 2008
“Images begin as super-ellipses constructed from
Bezier curves in which the weight and position of
control points are randomized.

The goal is to seek out visual tensions implicit in the
relationship between the wholeness of the circle
and the fragmentation in its interior.”
Victor Vasarely
Squares, 1973
Op-art
Koert Feenstra
Moire 6, 2009
Tomáš Mrkvička
2009
*
*
Petr Springl
2009
*
Jan Konvalina
2009
Fractals
Image quantization
2d b&w / color palette mapping
*
Ivo Serba
Goniometrics, 1993
= Algorithmic Art Course / Výtvarná informatika
Ivo Serba • Tomáš Staudek • Jiří Chmelík + cca 600 students
since 1992 – Faculty of Fine Arts, Brno University of Technology

since 1999 – Faculty of Informatics, Masaryk University

since 2006 – Faculty of Information Technology, Brno University of Technology
Ivo Serba
Hurricane Eye, 1992
*
Rhythm, 1994
Homage to Kandinsky, 1993
Tomáš Staudek
Talking Heads, 2007
*
Jan Novák
2002
*
Tomáš Slavíček
2003
*
Jiří Hýsek
2006
*
Kaz Maslanka
Dog Dream, 2006
“Dog dream is a colloquial Korean expression
for a goofy dream of little significance.
The Korean words say:
Dog dream = Irrationality / Importance”
Function systems
Iterated transformations
http://en.wikipedia.org/wiki/Iterated_function_system
Affine transformations:
scaling, rotation, reflection & shear followed by translation
—Wacław Sierpiński, 1916
Michael Barnsley, 1985
Katsushika Hokusai
Kanagawaoki namiura, 1831
IFS reconstruction
Promising technique for real-world
scene compression
…until JPEG conquered the scene
(1987–1992)
Philip van Loocke
Iterated Folding of Hexagonal Twist
2009
“The illuminator’s craft is replaced with the
craft of manipulating coordinates on the
complex number plane to create the image.”
*
Tomáš Mrkvička
2009
Kerry Mitchell
KM 1, 2004
Anita Chowdry
Fractal Shamsa, 2009
Mathematical chaos ≠ accidence
Chaos theory :
unpredictable output of models
with no random parameters
—Edward Lorenz, 1963
Michel Hénon, Otto Rössler, 1976
Chaotic attractors
Deterministic nonlinear dynamic systems
with unstable, nonperiodic behavior
http://en.wikipedia.org/wiki/Chaos_theory
Attractor = a phase diagram with implicit time
Frequent occurence of fractal dimension
Population growth, climate modelling,
hydrodynamics, cosmic bodies motion,
stock market fluctuation…
Tomáš Staudek
Orderisles, 2004
*
—Steven Whitney
Quadratic 3d map with 30 coefficients

Testing chaos by co-occurence of fractal dimension and positive Lyapunov exponent
— Julien C. Sprott, 1994
Jiří Hutárek
2009
*
*
Jakub Obr
2009
*
David Blaheta
2009
Jack Tait
TT LT Mono Grey Inversion 2007
Jack Tait
TT Col Ch 08, 2007
“Turntable machine with digital camera recording light pen with separate
neutral density / color R, G, B filters moving indpendently to the rotating
light slit image.”
Jan Pinter
2009
*
Peter Jansen
3b, 2007
Sortie de Secours, 2007
*
Libor Ryšavý
2009
Tomáš Maršák
2009
*
http://en.wikipedia.org/wiki/Attractor
Tessellation
Regular isomorph periodical tiling
vertices of same type
tiles of same type
repetition in n dimensions
3 regular polygons, 17 symmetry groups
—György Pólya, 1924
Regular polymorph tiling
8 more configurations
46 dichromatic groups, 6 trichromatic…

230 monochromatic spatial symmetry groups,
1191 dichromatic spatial groups…
Gerda de Vries
Cyclic, 2005
*
Martin Ptašek
2009
Margaret Kepner
17 Book, 2007
“The 17 Book is a visual exploration of the 2d symmetry groups— the so-called ‘wallpaper groups’. These 17 groups have interesting mathematical properties, and the associated patterns are widely used in the decorative arts.

A symmetry pattern can be transformed by (one or more) of the motions of translation, reflection, or glide reflection, while still preserving the overall pattern.”
Neda Yavari Rad
Mosaic, 2008
Alhambra decorations
Granada, Spain
13th–15th century
One of the oldest means of systematic embellishment,
from cca 20 000 BCE
Periodic tiling
Ornamental design
Interlocking pieces with rich segmentation between vertices
Escher’s tiling
Regular division of the plane
Wallpaper groups
http://en.wikipedia.org/wiki/Wallpaper_group
Maurits C. Escher
p6 symmetry
http://en.wikipedia.org/wiki/Regular_Division_of_the_Plane
Sky and Water II, 1938
Robert Fathauer
Marathon, 2004
Robert Fathauer
Drawing on Glide Reflection Symmetry, 2005
Behrooz Zabihian
Fish in an Islamic Mausoleum, 2006
“The background is an original tiling from
Hazrat-e-Mausome, an Islamic mausoleum
in Qom, Iran. I found a fish shape inside it.”
*
Jan Bílek
2009
*
Jan Kopidol
2009
Islamic ornaments
Sheikh Lotf Allah Mosque,
Isfahan, Iran, early 17th century
Alhambra
Granada, Spain
1. Constructing a rosette
from a regular polygon

2. Inscribing the shape into
polymorph tiling

3. Removing tile edges and
extrapolating shape lines

4. Artistic rendering
Nathan Edward Voirol
Sa'odat—Happiness, 2007
John Rigby
Interlacing Pattern with Birds, 2004
Václav Bubník
2009
*
*
Tomáš Ptáček
2009
Hyperbolic geometry
Non-Euclidean tiling
“Parallel” lines can intersect

Any line has at least two distinct “parallels”

The angles of a triangle add to less than straight angle
http://en.wikipedia.org/wiki/Hyperbolic_geometry
http://euler.slu.edu/escher
http://www.peda.com/tess/contest.html
Mehrdad Garousi
The Carpet, 2007
Irene Rousseau
Hyperbolic Diminution — Blue, 2003
Recursive geometry
a.k.a. “Droste effect”
http://en.wikipedia.org/wiki/Droste_effect
Maurits S. Escher
Print Gallery, 1956
http://www.josleys.com/article_show.php?id=82
Michael LaPalme
John Sommers
References
Books
The MIT Press, 2006
ISBN 0-262-06250-X
Paul A. Fishwick : Aesthetic Computing
Kostas Terzidis : Algorithmic Architecture
Elsevier, 2006
ISBN : 0-7506-6725-7
Edward Rodriguez : Computer Graphic Artist
Global Media, 2007
ISBN 8189940422
Intellect Books, 2007
ISBN 978-1-84150-168-0
Anna Bentekowska, ed. : Futures Past
(Thirty Years of Arts Computing)
Springer-Verlag, 2005
ISBN 3-540-21368-6
Michele Emmer: Mathematics and Culture II
The MIT Press, 2005
ISBN 0-262-05076-5
Michele Emmer: The Visual Mind II
Internet
http://dataisnature.com
DATAISNATURE
Subblue
http://www.subblue.com
Cyberxaos
http://cyberxaos.blogspot.com
http://lapin-bleu.net/riviera
Riviera Blog
http://studioseen.blogspot.com
Studio Seen
http://svvvvn.blogspot.com
SVVN
Leonardo Solaas @ Delicious
Tomáš Staudek @ Delicious
Bridges
http://www.delicious.com/lsolaas
http://www.delicious.com/tom.staudek
http://bridgesmathart.org
Winquant
Try it yourself…
GraphEQ
http://www.peda.com/grafeq
http://dl.dropbox.com/u/2526200/mathart_winquant.zip
Art
Computer
Computer
New media art

Computational art

Algorithmic art

Generative art

Mathematical art

Computer aided art
http://www.rchoetzlein.com
Rama Hoetzlein
http://deitchman.com/mcneillslides/units.php?unit=%2020th%20Century%20Art%20%281945-pres%29
Art
fine arts
mathematics & informatics
here we are
a e s t h e t i c s
Exact Aesthetics Practically
Tomáš Staudek
Computer Art Now
computer graphics
design
cybernetics
psychology & communication
http://en.wikipedia.org/wiki/Computer_art
Jackson Pollock
№No. 5, 1948
Action painting
Bézier curves
A cubic curve driven by four control points

Semi–random curve settings :

control point position

brush style & color
Tomáš Daněk
2009
*
Tomáš Pafčo
2009
*
*
Jozef Mlich
2009
PseudoPollock
http://dl.dropbox.com/u/2526200/mathart_pollock.zip
Try it yourself…
“I began by converting a drawing of a two-component
link into a symmetric collection of points. By treating
the points as the cities of a Traveling Salesman Problem
and adding constraints that forced the tour to be
symmetric, I constructed a simple-closed curve that
divides the plane into two symmetric pieces.”
Jan Dvořák
2009
*
Jan Horák
2009
*
Francesco de Comite
Doyle Spiral + Circle Inversion, 2008
Spatial curves
Edmund Harriss
3d Cog Spirographs, 2010
Jan Kratochvíla
2009
*
Robert Bosch
Embrace, 2009
Polynomiography
Solving algebraic 2d / complex equations
Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
—Fundamental Theorem of Algebra
http://en.wikipedia.org/wiki/Polynomial
Bahman Kalantari
Circus, 2000
Bahman Kalantari
Mona Lisa 01, 2001
Geometrical Approach
“Every line is a circle, there are no endpoints, no lines may cross,
each circle’s diameter is associated with a particular color, and
the diameters relate to each other proportionally.”
Aurora
Quantum Froth, 2009
Giac / Xcas
GNUplot
Graphing Calculator
http://www.gnuplot.info
http://calculator.runiter.com/graphing-calculator
http://www-fourier.ujf-grenoble.fr/~parisse/giac.html
Try it yourself…
Jindřich Morávek
2003
*
Javier Barallo
Cthulhu Mythos, 2009
Variation of the Mandelbrot set formula raised to the 8th degree (z⁸+ c) instead of being quadratic
Chaoscope
Try it yourself…
GNU XaoS
Ultrafractal
http://www.ultrafractal.com
Iterated functions with unusual behavior
http://en.wikipedia.org/wiki/Complex_numbers
Sequence of iterations = orbit

Predictable orbit for c = 0 :
Complex parabola
| z₀ | < 1 : singularity in complex zero

| z₀ | > 1 : spiral divergence to infinity

| z₀ | = 1 : circular border between the finite and the infinite
More interesting orbits for c ≠ 0 :

c ≠ 0 const. < 2, z₀ var. (Julia)

c ≠ 0 var., z₀ = 0 (Mandelbrot, using a computer)
The area of ‘strange values’ is relatively small



The map of nondivergent points is self-similar,
creating a scale-independent shape
The Mandelbrot set
http://en.wikipedia.org/wiki/Mandelbrot_set
Complex fractals
http://en.wikipedia.org/wiki/Iterated_function
Continuous in every point, but no derivations

Infinite length of contour, finite area / volume

Discrepancies in topological dimension
Investigation of quadratic functions in complex numbers
—Gaston Julia, Pierre Fatou, Benoît Mandelbrot, Adrien Douady
1917–1980
Ivan Nejezchleb
2009
Quaternion fractals
Expansion of complex numbers into 3d is not possible,
the nearest field is four-dimensional
http://en.wikipedia.org/wiki/Quaternion
Ivo Serba
Wood Worm Quat, 1998
*
Peter Jansen
Julia 6002, 2006
Ľubomír Hurtečák
2009
*
Peter Lukáč
2003
*
Michal Minárik
2009
*
Tomáš Staudek
Quatermorphosis, 2007
*
—Alan Norton, 1982
*
Ivan Nejezchleb
2009
Quat
Try it yourself…
http://www.physcip.uni-stuttgart.de/phy11733/quat_e.html
Mystica
http://www.dawntec.com/mystica
http://xaos.sourceforge.net
http://www.chaoscope.org
Fractal flames
Scott Draves, 1992
nonlinear transformations

elaborate symmetries

color as a separate dimension

computation histograms

layers — filtration and translucency

slow exposition

fractal morphing
http://en.wikipedia.org/wiki/Fractal_flame
Jan Horáček
2009
*
*
Tomáš Mrkvička
2009
Adam Vlček
2009
*
*
Radek Černobíla
2009
Electric sheep
Distributed computing of nonlinear IFS fractal animation
http://electricsheep.org
Iterating (non-) linear 3d transformations
Transformations in space
Tom Beddard
Subblue, 2009
Mark J. Stock
Sunset on Squares, 2004
“Seven-level refinement of Vicsek-like fractal
contains 105 million unique cubes and reveals
surprising symmetries.

An accurate light interreflection simulation
(particle-based radiosity) during the rendering
elevates the geometry above the virtual.”
http://www.subblue.com
Ferhan Kiziltepe
Colli-Sculpture 09, 2009
“The motifs are stylisations of major themes of the
16th–17th century Ottoman tiles.

The motifs are subjected to elementary isometric
transformations (translation, rotation, reflection and
glide reflection) to create 3d steel sculptures.”
Cayetano Ramirez
Fractal 3D2, 2007
Stan Goldade
The Sierpinski Tetrahedron, 1995
Try it yourself…
Apophysis
http://www.apophysis.org
Try it yourself…
Attract
Chaoscope
http://www.chaoscope.org
http://dl.dropbox.com/u/2526200/mathart_attract.zip
Recursive rewriting systems
for embranchment modelling
Algoritmization of Carl von Linné´s taxonomy (18th century)
L–systems
Realistic plant growth simulation
http://en.wikipedia.org/wiki/L-system
—Aristid Lindenmayer, 1968
Przemysław Prusinkiewicz, 1978
Robert Fathauer
Fractal Trees, 2007
“A digital artwork constructed by iterating an arrangement of a photograph of a tree.
The original photograph was digitally altered to allow smooth joining of the smaller copies.”
Parametric L–systems
shape parameters, randomness, season terms, age of growth, interaction with environment
*
Jan Halamíček
2009
Robert Fathauer
Tree of Knowledge, 2004
L-Studio
Fractal Life Engine
Try it yourself…
Contextfree
http://flea.sourceforge.net
http://algorithmicbotany.org/lstudio
http://www.contextfreeart.org
Try it yourself…
IFS Tools
IFS Construction Kit
http://ifs-tools.sourceforge.net/
http://ecademy.agnesscott.edu/~lriddle/ifskit
Tomáš Staudek
Homage to Drella, 2002
*
http://dl.dropbox.com/u/2526200/mathart_ornament.zip
http://www.peda.com/tess
Ornament
Tess
Try it yourself…
Platonic solids
Spatial tessellation
http://en.wikipedia.org/wiki/Platonic_solid
Sections of regular convex polyhedra can tessellate 3d space
Bob Rollings
Fun with Polyhedra, 2010
Ulrich Mikloweit
Snub Dodecadodecahedron, 2008
Piotr Pawlikowski
90 Squares and 40 Triangles, 2004
Anna Virágvölgyi
Rubik’s New Clothes, 2009
Felicity Wood
Cubes Wrapped on the Skew, 2010
“Example of extending 4×4×4 pattern over the surface
of Rubik’s cube. Each square of the set appears twice on
the 96 tiles of the cube. The are various symmetries on the
sides of the cube and between the sides also. So there is
more than one coherent and continuous arrangement.”
Briony Thomas
Reidun #1, 2010
“Rhombic tricontahedron with faces
tessellated with kites, darts and rhombs. ”
“The three intersecting planes are
Golden rectangles. Their intersection
creates 20 equilateral triangles –
an icosahedron.”
Jeff Chyatte
Elements, 2009
“The sculpture is inspired by 3d origami
construction. 30 identical ribbons bent
around the surface of a cylinder are joined
together to form the shape with the
rotational symmetry of an icosahedron.”
Vladimir Bulatov
Origami I, 2008
Briony Thomas
Contercharge Icosahedron #2, 2009
Rinus Roelofs
26 Tetrahedra, 2005
Dániel Erdely
Dodeca Spidroball Lampshader, 2006
Ergun Akleman
Twirling Sculptures, 2006
Jiří Chmelík
2009
*
“One part of the sculpture is a Moebius band with three
one-half left-hand twists. The other part is a helical ribbon
of six right-hand turns wrapped into a toroidal shape..”
Tom Longtin
Moebius–Helix, 2001
Xavier de Clipperleir
Transforming Cube, 2007
“Each edge of the cube is an elliptic cylinder with two
circular sections with rotation axes. This allows the cube
to rotate ina solid with 24 faces — icositetrahedron.”
Edmund Harris
Scuplture System 5, 2010
Fractal landscape
Parametrized mid-point displacement
Segment breaking algorithm, diamond–square algorithm, plasma fractals
—Gavin Miller, 1986
http://en.wikipedia.org/wiki/Diamond-square_algorithm
Ivo Serba
Vysoká, 1991
*
*
Jan Zelený
2009
Tomáš Kuřina
2009
*
Anne Burns
Fractal Scene, 2006
Mingjang Chen
Chaotic Landscape Painting, 2010
Saurav Subedi
2008
Try it yourself…
Landscape Studio
Terragen
http://landscapestudio.omgames.co.uk
http://www.planetside.co.uk
StructureSynth
Try it yourself…
http://structuresynth.sourceforge.net
Fract-o-rama
Gnofract 4d
http://gnofract4d.sourceforge.net
http://fractorama.com
http://en.wikipedia.org/wiki/Tessellation
staudek@gmail.com
http://dl.dropbox.com/u/2526200/mathart_kaleidomania.zip
Kaleidomania
GraphEq
http://www.peda.com/grafeq
http://en.wikipedia.org/wiki/Bezier_curve
http://en.wikipedia.org/wiki/Action_painting
http://en.wikipedia.org/wiki/Color_quantization
http://en.wikipedia.org/wiki/Op_art
http://en.wikipedia.org/wiki/Islamic_interlace_patterns
Arabeske
Taprats
Try it yourself…
http://www.cgl.uwaterloo.ca/~csk/washington/taprats/
http://www.wozzeck.net/arabeske
*
IFS Lab
http://www.nzeldes.com/Fractals/Fractals_core.htm
http://prezi.com/py9fkx0tzitd
Allows both periodic and nonperiodic layouts
Semi-periodic tiling
Polyominal tiling
domino triomino tetromino
Pentominal tiles cover rectangles 3 × 20 (2 solutions),
4 × 15 (368 solutions), 5 × 12 (1 010 solutions)
and 6 × 10 (2 339 solutions)

Each pentominal tile can be assembled of 9 different tiles
Günter Albrecht Bühler
—Hans Voderberg
modifications of tiles from isosceles triangles
Spiral tiling
Aperiodic tiling
No transitional symmetry in any configuration of tiles
—Andrew Glassner
inflation of basic tile shape with its copies
Hierarchical tiling
Ivo Serba
2006
*
*
Ivo Serba
2006
H-Tiles
Try it yourself…
http://dl.dropbox.com/u/2526200/mathart_htiles.zip
Subject of interest for crystalographers and mathematicians :
Aperiodic tessellation cannot exist
—Hao Vang, 1961

104 tiles form aperiodic mosaics
—Robert Berger, 1964
94 tiles form aperiodic mosaics
—Donald Knuth, 1966

6 tiles form aperiodic tiling
—Robert Amman, Raphael Robinson, 1971
6 tile formations with inflation rules, 1974
—Roger Penrose, 1974
Only two tiles can form aperiodic mosaics
Darts and kites

—Roger Penrose, 1988
Thick and thin rhombie
Darts & kites are convertible
to thick & thin rhombie,
and vice versa
Penrose
Try it yourself…
Geometric sculpture
Thank you !
Computer Art Lecturer
Brno University of Technology
staudek@gmail.com
Tomáš Staudek
“Borromean rings are three rings
(zero knots) joined in such a way that
each ring goes through both the others
and if one is broken the piece falls apart.”
Louise Mabbs
Borromean Rings, 2005
“Two 4×4 endless knots (also konwn as
‘knot of life’ or ‘Shiri Vasta’, one of eigth
precious symbols of Tibetan Buddhism,
are weaved together with one 3×3 knot.”
Jeff Chyatte
Tribute to Escher’s “Three Worlds”, 2003
Jiří Chmelík
2008
*
Tomáš Pafčo
2008
*
Marián Krivda
2008
*
Pavel Bierza
2008
*
“A bronze rod ¼" in diameter was bent into a figure eight knot,
and welded into a continuous loop. This is the most complex
knot after the trefoil knot, and its complement has a hyperbolic
structure.”
Alex J. Feingold
Figure 8 Knot Rod, 2007
“This work consists of two separated corps of red and golden knots
The golden knots display a pentagonal pattern on the front vertex
of icosahedron. Groups of dual red knots are used to tie all the
pentagons, placed at basic icosahedron’s vertices.”
Merhard Garousi
Knots and Knots, 2008
Jacqui Carey
Memory Basket, 2003
“An example of ‘open hexagonal plaiting’.
Basketmasters in many parts of the world
employ this weave — notably Africa and
the Far East.”
Richard Ahrens
Bacteriophage, 2004
Decorative knots
Knot theory
often instigates 3d geometric pieces
http://en.wikipedia.org/wiki/Knot_theory
“The knot is built up from five
interlinked repetitions of a smaller
knot, which itself is an extension
of the ubiquitios King Solomon’s knot.”
Deborah Robinson
Celtic Interlacing 2, 2006
“A nonoriantable minimal surface spanning
(like a soap film) a certain knotted boundary
curve. The surface has a four-fold and two-fold
rotational symmetries, but no mirrors.”
John M. Sullivan
Minimal Flower 4, 2008
Carlo H. Sequin
Knot 5.2, 2009
“The traditional ball in cage theme
is combined with a classical motif
from renaissance art and inspiration
from Japanese netsuke carvings.
The cage is a rhombic dodecahedral
edge frame.”
Bjarne Jespersen
Memento Mori, 1981
“Tetrakis hexahedron (TH) is dual to
truncated dodecahedron. It is polyhedron
with 24 identical triangular faces with total
symmetry of cube. The sculpture is one of
TH stellations. The symmetry of the sculpture
is rotational symmetry of cube.”
Vladimir Bulatov
Stellation of tetrakis hexahedron, 2003
“A compound of four trefoil knots oriented
as the faces of a tetrahedron. Its symmetry
is that of the tetrahedron, but without
mirror reflections.”
Bjarne Jespersen
Great Tetraknot, 2002
“These streptohedrons only require basic
geometry to help construct regular
polygons, stars or other shapes which have
rotational symmetry. I use those shapes as
a base for solids of revolution which can be
split, twisted and rejoined.”
David Springett
Ziggurat, 2005
“The object is placed in the center of the
picture, and the surrounding three are
reflected images in mirrors. The object is
also an imaginary cube composed of the
16 representatives of the minimal convex
imaginary cube classes, but with a
different layout of the 16 components.”
Hideki Tsuiki
Imaginary Cube Sculpture, 2010
“A compound of five hamiltonian circuits
on a rhombicosidodecahedron.”
Francesco de Comite
Hamiltonian Circuits, 2009
“The base of this sculpture is polyhedron
with 12 rhombic faces with cubical
symmetry. Each of 12 faces was
transformed into curved shape with 4
twisted arms, which connects to other
shapes at vertices of valence 3 and 4. The
boundary of the resulting body forms
quite a complex knot.”
Vladimir Bulatov
Rhombic Dodecahedron I, 2008
“This sculpture was carved from a
circular piece of limestone. The form is
based on the shape of the soap film
minimal surface on a configuration of
a wire trefoil knot. There is a nice
interaction of the form and space with
light and shadow..”
Nat Friedman
Trefoil Knot Minimal Surface, 2006
Benjamin Storch
CMoebius, 2006
“I am not of course, suggesting that we discard our other modes of artistic expression

and replace them with computer graphics. Far from it. I am only proposing that we add

this new form to those we already have — much as we added photography a century and

a half ago. We need all the help we can get in understanding the world — both seen and

unseen — and every ounce of beauty we can find or help create. And since that’s so, why

shouldn’t the computer, that ubiquitous medium of information and communication,

also be called into service as an instrument of art?”
Theodore Wolff
“Digital is not here to put an end to anything. Rather it is here to expand all things,

to combine and to make more things attainable. For the artist, it is the edgiest work

of all; the biggest, most exciting challenge in a long history of the synthesis between

technology and hand and mind and heart.”
John D. Jarvis
“I write computer algorithms, i.e. rules that calculate and then generate a work

that could not be realized in any other way. It is not necessarily the system or

the logic I want to present in my work, but the visual invention that results from it.

My artistic goal is reached when a finished work can visually dissociate itself from

its logical content and convincingly stand as an independent abstract entity.”
Manfred Mohr
http://en.wikipedia.org/wiki/Aperiodic_tiling
http://dl.dropbox.com/u/2526200/mathart_penrose.zip
or establish a skeleton for twirling sculptures
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