Transcript Polyhedra in Art - Colorado College

The Other Polyhedra
Steven Janke
Colorado College
Five Regular Polyhedra
Dodecahedron
Tetrahedron
Icosahedron
Octahedron
Cube
Prehistoric Scotland
Carved stones from about 2000 B.C.E.
Roman Dice
ivory
stone
Roman Polyhedra
Bronze, unknown function
Radiolaria drawn by Ernst Haeckel (1904)
Theorem: Let P be a convex polyhedron whose faces are
congruent regular polygons. Then the following are equivalent:
1.
2.
3.
4.
5.
The vertices of P all lie on a sphere.
All the dihedral angles of P are equal.
All the vertex figures are regular polygons.
All the solid angles are congruent.
All the vertices are surrounded by the same number of faces.
Plato’s Symbolism
(Kepler’s sketches)
Octahedron = Air
Tetrahedron = Fire
Cube = Earth
Icosahedron = Water
Dodecahedron = Universe
Theorem: There are only five convex regular polyhedra.
(Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron)
Proof:
In a regular polygon of p sides, the angles are (1-2/p)π.
With q faces at each vertex, the total of these angles must
Be less than 2π:
q(1-2/p)π < 2π
1/p + 1/q > 1/2
Only solutions are: (3,3) (3,4) (4,3) (3,5) (5,3)
Johannes Kepler (1571-1630)
(detail of inner planets)
Golden Ratio in a Pentagon
Three golden rectangles inscribed in an icosahedron
Euler’s Formula: V + F = E + 2
Vertices
Faces
Edges
Tetrahedron
4
4
6
Cube
8
6
12
Octahedron
6
8
12
Dodecahedron
20
12
30
Icosahedron
12
20
30
Duality: Vertices
Faces
Regular Polyhedra Coordinates:
Cube:
(±1, ±1 , ±1)
Tetrahedron:
(1, 1, 1) (1, -1, -1) (-1, 1, -1) (-1, -1, 1)
Octahedron:
(±1, 0, 0) (0, ±1, 0) (0, 0, ±1)
Iscosahedron: (0, ±φ, ±1) (±1, 0, ±φ) (±φ, ±1, 0)
Dodecahedron: (0, ±φ-1, ±φ) (±φ, 0, ±φ-1) (±φ-1, ±φ, 0) (±1, ±1, ±1)
Where φ2 - φ - 1 = 0 giving φ = 1.618 … (Golden Ratio)
Portrait of Luca Pacioli (1445-1514)
(by Jacopo de Barbari (?) 1495)
Basilica of San Marco (Venice)
(Floor Pattern in Marble)
Possibly designed by Paolo Uccello in 1430
Albrecht Durer
(1471-1528)
Melancholia I, 1514
Church of Santa Maria in Organo, Verona
(Fra Giovanni da Verona 1520’s)
Leonardo da Vinci (1452-1519)
Illustrations for Luca Pacioli's 1509 book The Divine Proportion
Leonardo da Vinci
“Elevated” Forms
Albrecht Durer
Painter’s Manual, 1525
Net of snub cube
Wentzel Jamnitzer (1508-1585)
Perspectiva Corporum Regularium, 1568
Wentzel Jamnitzer
Theorem: The only finite rotation groups are:
Cyclic
Dihedral
Tetrahedral (alternating group of degree 4)
Octahedral
Icosahedral (alternating group of degree 5)
Lorenz Stoer
Geometria et Perspectiva, 1567
Lorenz Stoer
Geometria et Perspectiva, 1567
Jean Cousin
Livre de Perspective, 1560
Jean-Francois Niceron
Thaumaturgus Opticus, 1638
Tomb of Sir Thomas Gorges
Salisbury
Cathedral,
1635
M.C. Escher (1898-1972)
Stars, 1948
M.C. Escher
Waterfall, 1961
M.C. Escher
Reptiles, 1943
Order and Chaos
M.C. Escher
Regular Polygon with 5 sides
Johannes Kepler
Harmonice
Mundi, 1619
Theorem: There are only four regular star polyhedra.
Small Stellated Dodecahedron (5/2, 5)
Great Dodecahedron
(5, 5/2)
Great Stellated Dodecahedron (5/2, 3)
Great Icosahedron
(3, 5/2)
Kepler: Archimedean Solids
Faces regular, vertices identical, but faces need not be identical
Lemma: Only three different kinds of faces can occur at each vertex
of a convex polyhedra with regular faces.
Theorem: The set of convex polyhedra with regular faces and
congruent vertices contains only the 13 Archimedean polyhedra
plus two infinite families: the prisms and antiprisms.
Max Brückner
Vielecke und Vielflache, 1900
Historical Milestones
1. Theatetus (415 – 369 B.C.): Octahedron and Icosahedron.
2. Plato (427 – 347 B.C.): Timaeus dialog (five regular polyhedra).
3. Euclid (323-285 B.C.): Constructs five regular polyhedra in Book XIII.
4. Archimedes (287-212 B.C.): Lost treatise on 13 semi-regular solids.
5. Kepler (1571- 1630): Proves only 13 Archimedean solids.
6. Euler (1707-1783): V+F=E+2
7. Poinsot (1777-1859): Four regular star polyhedra. Cauchy proved.
8. Coxeter (1907 – 2003): Regular Polytopes.
9. Johnson, Grunbaum, Zalgaller (1969): Prove 92 polyhedra with regular faces.
10. Skilling (1975): Proves there are 75 uniform polyhedra.
Retrosnub Ditrigonal Icosidodecahedron
(a.k.a. Yog Sothoth)
(Vertices: 60; Edges:180; Faces: 100 triangles + 12 pentagrams
References:
Coxeter, H.S.M. – Regular Polytopes 1963
Cromwell, Peter – Polyhedra 1997
Senechal, Marjorie, et. al. – Shaping Space 1988
Wenninger, Magnus – Polyhedron Models 1971
Cundy,H. and Rollett, A. – Mathematical Models 1961
Polyhedra inscribed in other Polyhedra