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