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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

• 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

• 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

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

• 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?

Carved stone from circa 2000 B.C.E.

Hundreds known.

Most are cube-based.

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

ivory

stone

Bronze, unknown function

Small stellated dodecahedron mosaic

mazzocchio (donut hat)

Truncated tetrahedron

Icosahedron in cube

Illustrations for Luca Pacioli's 1509 book The Divine Proportion

Illustrations for Luca Pacioli's 1509 book The Divine Proportion

“Elevated” Forms

Cube structure

Leonardo’s Ludo Geometrico

ludo geometrico = “geometry game”

= “make systematic modifications”

Torus variations

Portrait of Pacioli, by Jacopo de Barbari, 1495

“Golden ratio”

Printed as woodcuts in 1509

Melancholia I, 1514

Painter’s Manual, 1525

Net of snub cube

Find the error!

Painter’s Manual, 1525

La Practica della Perspectiva, 1568

Perspectiva Corporum Regularium, 1568

(oldest chiral icosahedral image)

(detail of inner planets)

Harmonice Mundi, 1619

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

Regular Dodecahedron

Rhombic Dodecahedron

Symbolism from Plato:

Octahedron = air

Tetrahedron = fire

Cube = earth

Icosahedron = water

Dodecahedron =

the universe

Geometria et Perspectiva, 1567

Geometria et Perspectiva, 1567

Livre de Perspective, 1560

Portrait of Johann Neudorfer and his Son, 1561

Perspectiva, 1571

Perspectiva, 1571

La pratica di prospettiva, 1596

Optica, 1616

Thaumaturgus Opticus, 1638

La Perspective Pratiq, 1642

La Perspective Pratiq, 1642

Salisbury Cathedral, 1635

Lorenz Zick (1594-1666)Turned Ivory Spheres

Modern Asian example

Geometrie pratique, 1684

La Geometrie Pratique, 1702

Geometry Improv'd, 1718

New Principles of Linear Perspective, 1719

New Principles of Linear Perspective, 1719

Lucidum Prospectivae Speculum

1727

Compleat Treatise on Perspective, 1779

Anleitung zur Linearperspective,

c. 1800

Max BrücknerVielecke und Vielflache, 1900

Stars, 1948

Double Planetoid, 1949

Waterfall, 1961

Reptiles, 1943

• 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