CSE 325 Computer Science and Sculpture Prof George

CSE 325 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: 3 D object bounded by flat surfaces Many types: – Platonic solids – Archimedean solids – Convex / concave • Long history of use in 3 D 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…

Examples: Regular 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 concave dodecahedron “small stellated dodecahedron” (12 pentagrams) A torus is not convex

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

Roman Icosahedron

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

Compare “Solid Edges” to Lines

Leonardo da Vinci “Elevated” Forms

Leonardo Doodles

Leonardo Doodles

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

Luca Pacioli

The Divine Proportion “Golden ratio”

Luca Pacioli

Pacioli + Leonardo Printed as woodcuts in 1509

Fra Giovanni da Verona, 1520’s

Intarsia by Giovanni da Verona

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

Wentzel Jamnitzer

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

Augustin Hirschvogel (1503 -1553)

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

Lorenz Zick (1594 -1666) Turned Ivory Spheres Modern Asian example

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

Max Brückner Vielecke und Vielflache, 1900

M. C. Escher (1898 -1972) Stars, 1948

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