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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

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polyhedra
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
in two dimensions polygons
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
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
Some Dodecahedra

12 isosceles triangles

12 rhombi

Regular: 12 pentagons

“rhombic dodechedron”

12 isosceles triangles

12 kites

12 irregular pentagons

some non convex dodecahedra
Some Non-convex Dodecahedra

“small stellated dodecahedron”

(12 pentagrams)

A torus is not convex

concave dodecahedron

historical examples
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
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
Roman Dice

ivory

stone

roman dodecahedra
Roman Dodecahedra

Bronze, unknown function

paolo uccello 1397 1475
Paolo Uccello (1397-1475)

Small stellated dodecahedron mosaic

mazzocchio (donut hat)

piero della francesca 1410 1492
Piero della Francesca (1410? - 1492)

Truncated tetrahedron

Icosahedron in cube

leonardo da vinci 1452 1519
Leonardo da Vinci (1452-1519)

Illustrations for Luca Pacioli's 1509 book The Divine Proportion

leonardo da vinci
Leonardo da Vinci

Illustrations for Luca Pacioli's 1509 book The Divine Proportion

leonardo da vinci19
Leonardo da Vinci

“Elevated” Forms

leonardo
Leonardo

Cube structure

leonardo s ludo geometrico
Leonardo’s Ludo Geometrico

ludo geometrico = “geometry game”

= “make systematic modifications”

leonardo24
Leonardo

Torus variations

luca pacioli 1445 1514
Luca Pacioli (1445-1514)

Portrait of Pacioli, by Jacopo de Barbari, 1495

the divine proportion
The Divine Proportion

“Golden ratio”

pacioli leonardo
Pacioli + Leonardo

Printed as woodcuts in 1509

albrecht durer 1471 1528
Albrecht Durer (1471-1528)

Melancholia I, 1514

albrecht durer
Albrecht Durer

Painter’s Manual, 1525

Net of snub cube

albrecht durer34
Albrecht Durer

Find the error!

Painter’s Manual, 1525

daniele barbaro 1513 1570
Daniele Barbaro (1513-1570)

La Practica della Perspectiva, 1568

wentzel jamnitzer 1508 1585
Wentzel Jamnitzer (1508-1585)

Perspectiva Corporum Regularium, 1568

wentzel jamnitzer39
Wentzel Jamnitzer

(oldest chiral icosahedral image)

johannes kepler 1571 1630
Johannes Kepler (1571-1630)

(detail of inner planets)

johannes kepler
Johannes Kepler

Harmonice Mundi, 1619

kepler archimedean solids
Kepler: Archimedean Solids

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

johannes kepler43
Johannes Kepler

Regular Dodecahedron

Rhombic Dodecahedron

johannes kepler44
Johannes Kepler

Symbolism from Plato:

Octahedron = air

Tetrahedron = fire

Cube = earth

Icosahedron = water

Dodecahedron =

the universe

lorenz stoer
Lorenz Stoer

Geometria et Perspectiva, 1567

lorenz stoer47
Lorenz Stoer

Geometria et Perspectiva, 1567

jean cousin
Jean Cousin

Livre de Perspective, 1560

nicolas neufchatel
Nicolas Neufchatel

Portrait of Johann Neudorfer and his Son, 1561

hans lencker
Hans Lencker

Perspectiva, 1571

hans lencker51
Hans Lencker

Perspectiva, 1571

lorenzo sirigatti
Lorenzo Sirigatti

La pratica di prospettiva, 1596

paul pfinzing
Paul Pfinzing

Optica, 1616

jean francois niceron
Jean-Francois Niceron

Thaumaturgus Opticus, 1638

jean dubreuil
Jean Dubreuil

La Perspective Pratiq, 1642

jean dubreuil56
Jean Dubreuil

La Perspective Pratiq, 1642

tomb of sir thomas gorges
Tomb of Sir Thomas Gorges

Salisbury Cathedral, 1635

jacques ozanam
Jacques Ozanam

Geometrie pratique, 1684

alain manesson mallet
Alain Manesson Mallet

La Geometrie Pratique, 1702

abraham sharp 1651 1742
Abraham Sharp (1651-1742)

Geometry Improv'd, 1718

brook taylor
Brook Taylor

New Principles of Linear Perspective, 1719

brook taylor63
Brook Taylor

New Principles of Linear Perspective, 1719

paul heinecken
Paul Heinecken

Lucidum Prospectivae Speculum

1727

thomas malton
Thomas Malton

Compleat Treatise on Perspective, 1779

christoph nilson
Christoph Nilson

Anleitung zur Linearperspective,

c. 1800

m c escher
M.C. Escher

Double Planetoid, 1949

m c escher70
M.C. Escher

Waterfall, 1961

m c escher71
M.C. Escher

Reptiles, 1943

conclusions
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