CSE325 Computer Science and Sculpture

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# CSE325 Computer Science and Sculpture - PowerPoint PPT Presentation

CSE325 Computer Science and Sculpture Prof. George Hart 2. Polyhedra in Art + Sculpture A Historical View Polyhedra From Greek: poly =many + hedra =seats Singular: Polyhedron Def: 3D object bounded by flat surfaces Many types: Platonic solids Archimedean solids Convex / concave

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### CSE325 Computer Science and Sculpture

Prof. George Hart

### 2. Polyhedra in Art + Sculpture

A Historical View

Polyhedra
• From Greek: poly=many + hedra=seats
• Singular: Polyhedron
• Def: 3D object bounded by flat surfaces
• Many types:
• Platonic solids
• Archimedean solids
• Convex / concave
• Long history of use in 3D design
Greek gon = “knee”

Regular polygon:

equal lengths

equal angles

Allow “stars”

Terminology:

corner = vertex

plural: vertices

Number prefixes:

3) tri-

4) tetra-

5) penta-

6) hexa-

7) hepta-

8) octa-

9) ennea-

10) deca-

In Two Dimensions: Polygons
Five “Regular” Polyhedra
• Every face identical
• Every face regular
• Every vertex identical
• Only 5 are possible
• Euclid gives proof
• “Platonic Solids”
• Plato described them
• (known earlier)

Dodecahedron=12 sides

Icosahedron=20

tetrahedron octahedron cube

Some Dodecahedra

12 isosceles triangles

12 rhombi

Regular: 12 pentagons

“rhombic dodechedron”

12 isosceles triangles

12 kites

12 irregular pentagons

Some Non-convex Dodecahedra

“small stellated dodecahedron”

(12 pentagrams)

A torus is not convex

concave dodecahedron

Historical Examples
• Stone, ivory, wood carving
• Bronze casting
• Drawing, woodcut, engraving, etc
• Painting
• Stone or wood tiling (mosaics = “intarsia”)
• Wood, glass, or metal assembly

Guess: How old is the oldest existing dodecahedron?

Prehistoric Scotland

Carved stone from circa 2000 B.C.E.

Hundreds known.

Most are cube-based.

I don’t know of any icosahedron-based examples.

Roman Dice

ivory

stone

Roman Dodecahedra

Bronze, unknown function

Paolo Uccello (1397-1475)

Small stellated dodecahedron mosaic

mazzocchio (donut hat)

Piero della Francesca (1410? - 1492)

Truncated tetrahedron

Icosahedron in cube

Leonardo da Vinci (1452-1519)

Illustrations for Luca Pacioli's 1509 book The Divine Proportion

Leonardo da Vinci

Illustrations for Luca Pacioli's 1509 book The Divine Proportion

Leonardo da Vinci

“Elevated” Forms

Leonardo

Cube structure

Leonardo’s Ludo Geometrico

ludo geometrico = “geometry game”

= “make systematic modifications”

Leonardo

Torus variations

Luca Pacioli (1445-1514)

Portrait of Pacioli, by Jacopo de Barbari, 1495

The Divine Proportion

“Golden ratio”

Pacioli + Leonardo

Printed as woodcuts in 1509

Albrecht Durer (1471-1528)

Melancholia I, 1514

Albrecht Durer

Painter’s Manual, 1525

Net of snub cube

Albrecht Durer

Find the error!

Painter’s Manual, 1525

Daniele Barbaro (1513-1570)

La Practica della Perspectiva, 1568

Wentzel Jamnitzer (1508-1585)

Perspectiva Corporum Regularium, 1568

Wentzel Jamnitzer

(oldest chiral icosahedral image)

Johannes Kepler (1571-1630)

(detail of inner planets)

Johannes Kepler

Harmonice Mundi, 1619

Kepler: Archimedean Solids

Faces regular, vertices identical, but faces need not be identical

Johannes Kepler

Regular Dodecahedron

Rhombic Dodecahedron

Johannes Kepler

Symbolism from Plato:

Octahedron = air

Tetrahedron = fire

Cube = earth

Icosahedron = water

Dodecahedron =

the universe

Lorenz Stoer

Geometria et Perspectiva, 1567

Lorenz Stoer

Geometria et Perspectiva, 1567

Jean Cousin

Livre de Perspective, 1560

Nicolas Neufchatel

Portrait of Johann Neudorfer and his Son, 1561

Hans Lencker

Perspectiva, 1571

Hans Lencker

Perspectiva, 1571

Lorenzo Sirigatti

La pratica di prospettiva, 1596

Paul Pfinzing

Optica, 1616

Jean-Francois Niceron

Thaumaturgus Opticus, 1638

Jean Dubreuil

La Perspective Pratiq, 1642

Jean Dubreuil

La Perspective Pratiq, 1642

Tomb of Sir Thomas Gorges

Salisbury Cathedral, 1635

Jacques Ozanam

Geometrie pratique, 1684

Alain Manesson Mallet

La Geometrie Pratique, 1702

Abraham Sharp (1651-1742)

Geometry Improv'd, 1718

Brook Taylor

New Principles of Linear Perspective, 1719

Brook Taylor

New Principles of Linear Perspective, 1719

Paul Heinecken

Lucidum Prospectivae Speculum

1727

Thomas Malton

Compleat Treatise on Perspective, 1779

Christoph Nilson

Anleitung zur Linearperspective,

c. 1800

M.C. Escher

Double Planetoid, 1949

M.C. Escher

Waterfall, 1961

M.C. Escher

Reptiles, 1943

Conclusions
• Polyhedra, especially the five Platonic solids, have been an element of Western art for centuries.
• Beauty of symmetry
• Challenging models to show mastery of perspective
• Symbolic meaning assigned by Plato
• Mathematical foundation for artistry
• Good starting point for computer constructions