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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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Cse325 computer science and sculpture l.jpg

CSE325 Computer Science and Sculpture

Prof. George Hart



Polyhedra l.jpg
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 l.jpg

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 l.jpg
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 l.jpg
Some Dodecahedra

12 isosceles triangles

12 rhombi

Regular: 12 pentagons

“rhombic dodechedron”

12 isosceles triangles

12 kites

12 irregular pentagons


Some non convex dodecahedra l.jpg
Some Non-convex Dodecahedra

“small stellated dodecahedron”

(12 pentagrams)

A torus is not convex

concave dodecahedron


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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?


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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.


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Roman Dice

ivory

stone


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Roman Dodecahedra

Bronze, unknown function



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Paolo Uccello (1397-1475)

Small stellated dodecahedron mosaic

mazzocchio (donut hat)


Piero della francesca 1410 1492 l.jpg
Piero della Francesca (1410? - 1492)

Truncated tetrahedron

Icosahedron in cube


Leonardo da vinci 1452 1519 l.jpg
Leonardo da Vinci (1452-1519)

Illustrations for Luca Pacioli's 1509 book The Divine Proportion


Leonardo da vinci l.jpg
Leonardo da Vinci

Illustrations for Luca Pacioli's 1509 book The Divine Proportion



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Leonardo da Vinci

“Elevated” Forms




Leonardo l.jpg
Leonardo

Cube structure


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Leonardo’s Ludo Geometrico

ludo geometrico = “geometry game”

= “make systematic modifications”


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Leonardo

Torus variations


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Luca Pacioli (1445-1514)

Portrait of Pacioli, by Jacopo de Barbari, 1495



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The Divine Proportion

“Golden ratio”



Pacioli leonardo l.jpg
Pacioli + Leonardo

Printed as woodcuts in 1509




Albrecht durer 1471 1528 l.jpg
Albrecht Durer (1471-1528)

Melancholia I, 1514


Albrecht durer l.jpg
Albrecht Durer

Painter’s Manual, 1525

Net of snub cube


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Albrecht Durer

Find the error!

Painter’s Manual, 1525


Daniele barbaro 1513 1570 l.jpg
Daniele Barbaro (1513-1570)

La Practica della Perspectiva, 1568


Wentzel jamnitzer 1508 1585 l.jpg
Wentzel Jamnitzer (1508-1585)

Perspectiva Corporum Regularium, 1568




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Wentzel Jamnitzer

(oldest chiral icosahedral image)


Johannes kepler 1571 1630 l.jpg
Johannes Kepler (1571-1630)

(detail of inner planets)


Johannes kepler l.jpg
Johannes Kepler

Harmonice Mundi, 1619


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Kepler: Archimedean Solids

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


Johannes kepler43 l.jpg
Johannes Kepler

Regular Dodecahedron

Rhombic Dodecahedron


Johannes kepler44 l.jpg
Johannes Kepler

Symbolism from Plato:

Octahedron = air

Tetrahedron = fire

Cube = earth

Icosahedron = water

Dodecahedron =

the universe



Lorenz stoer l.jpg
Lorenz Stoer

Geometria et Perspectiva, 1567


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Lorenz Stoer

Geometria et Perspectiva, 1567


Jean cousin l.jpg
Jean Cousin

Livre de Perspective, 1560


Nicolas neufchatel l.jpg
Nicolas Neufchatel

Portrait of Johann Neudorfer and his Son, 1561


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Hans Lencker

Perspectiva, 1571


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Hans Lencker

Perspectiva, 1571


Lorenzo sirigatti l.jpg
Lorenzo Sirigatti

La pratica di prospettiva, 1596


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Paul Pfinzing

Optica, 1616


Jean francois niceron l.jpg
Jean-Francois Niceron

Thaumaturgus Opticus, 1638


Jean dubreuil l.jpg
Jean Dubreuil

La Perspective Pratiq, 1642


Jean dubreuil56 l.jpg
Jean Dubreuil

La Perspective Pratiq, 1642


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Tomb of Sir Thomas Gorges

Salisbury Cathedral, 1635


Lorenz zick 1594 1666 turned ivory spheres l.jpg
Lorenz Zick (1594-1666)Turned Ivory Spheres

Modern Asian example


Jacques ozanam l.jpg
Jacques Ozanam

Geometrie pratique, 1684


Alain manesson mallet l.jpg
Alain Manesson Mallet

La Geometrie Pratique, 1702


Abraham sharp 1651 1742 l.jpg
Abraham Sharp (1651-1742)

Geometry Improv'd, 1718


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Brook Taylor

New Principles of Linear Perspective, 1719


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Brook Taylor

New Principles of Linear Perspective, 1719


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Paul Heinecken

Lucidum Prospectivae Speculum

1727


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Thomas Malton

Compleat Treatise on Perspective, 1779


Christoph nilson l.jpg
Christoph Nilson

Anleitung zur Linearperspective,

c. 1800


Max br ckner vielecke und vielflache 1900 l.jpg
Max BrücknerVielecke und Vielflache, 1900



M c escher l.jpg
M.C. Escher

Double Planetoid, 1949


M c escher70 l.jpg
M.C. Escher

Waterfall, 1961


M c escher71 l.jpg
M.C. Escher

Reptiles, 1943


Conclusions l.jpg
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


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