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The History of Polyhedra, 4th day

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Daniel Kage

on 5 February 2015

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Transcript of The History of Polyhedra, 4th day

The History of Polyhedra
Neolithic Carved Stone Polyhedra
Petrosphere (from Greek πέτρα (petra), "stone", and σφαῖρα (sphaira), "ball")
Dated to around 2000 BC, Scotland.
Wales, Hungary, Italy, German and France.
Origin of the Carved Stones Polyhedra, Scotland
Paolo Uccello (1397-1475)
Pythagoras 570 – c. 495 BC
(Phytagorean School, Harmonics of the Quadrivium, Pythagorean theorem)

Plato 350 BC
(Platonic Academy, Natural philosophy, The 5 Platonic Solids)

Archimedes of Syracuse 287 BC – c. 212 BC
(The 13 Archimedean Solids)

Polyhedra in Ancient Greece
Plato 360 B.C.
The 5 Platonic Solids & the 5 elements
Archimedes of Syracuse
287 BC – c. 212 BC
13 Archimedean Solids
Semi-regular convex polyhedron
Pythagoras of Samos 570 – c. 495 BC
The Phytagorean Theorem
Harmonics of the Quadrivium

- Mathematics, Geometry, Music and Astronomy.
Tetrahedron (The shape of the global cosmic world)

- Harmony is based upon ratio (Golden Proportion)
Truncated Tetrahedron
8 faces: 4 Triangles,
4 Hexagons
Edges: 18
Vertices: 12
14 faces: 8 triangles,
6 Squares
Edges: 24
Vertices: 12
Truncated Cuboctahedron
26 Faces: 12 Squares,
8 Hexagons, 6 Octagons
Edges: 72
Vertices: 48
Snub Cube
38 Faces: 32 Triangles,
6 Squares
Edges: 60
Vertices: 24
62 Faces: 20 Triangles,
30 Squares, 12 Pentagons
Edges: 120
Vertices: 60
Truncated Icosidodecahedron
62 Faces: 30 squares,
20 hexagons, 12 decagons
Edges: 180
Vertices: 120

Snub Dodecahedron
92 Faces: 80 triangles,
12 pentagons
Edges: 150
Vertices: 60
Truncated Cube
14 faces: 8 Triangles,
6 Octagons
Edges: 36
Vertices: 24
32 Faces: 20 Triangles,
12 Pentagons
Edges: 60
Vertices: 30
Truncated Dodecahedron
32 Faces: 20 Triangles,
12 Decagons
Edges: 90 Edges
Vertices: 60 Vertices
Truncated Icosahedron
32 Faces: 12 Pentagons,
20 Hexagons
Edges: 90
Vertices: 60
Technical terms in Polyhedra
Snub Polyhedron: A polyhedron with extra triangular faces

Truncated Polyhedron: A truncation "cut" of one Polyhedron

Rhombic Polyhedron: A polyhedron with extra square faces
School of Athenas. Raphael, year 1500
Roman Dodecahedra 2nd or 3rd Century B.C.
Johannes Kepler. Year 1570, Germany
The Model of Solar System. Year 1596
Kepler-Pointsot Polyhedra:
Regular Star Polyhedra
Mysterium Cosmographicum
Crystallography: "On the Six-Cornered Snowflake"
Paulo Uccello's Polyhedra. Italy, Year 1400
Leonardo da Vinci's Polyhedra. Italy, Year 1500
Luca Pacioli's Polyhedra. Italy, Year 1450
The Divine Proportion book, Year 1509
Fra Giovanni's Intarsia. Italy, 15th-16th Century
German Renaissance
Snub Cube
Truncation of aTruncated Cube
Albrecht Dürer (Year 1525, book Unterweysung der Messung )
Wentzel Jamnitzer's Polyhedra
1568 book Perspectiva Corporum Regularium
1619 book, Harmonice Mundi
Salisbury Cathedral, England. Year 1635
Jean-Francois Niceron. Book Thaumaturgus Opticus (1638)
Twentieth Century
Richard Buckminster Fuller July 1895 — July 1983
Geodesic Domes
Maurits Cornelis Escher June 1898 — March 1972)
Metamorphose drawings
Polyhedra art
The Vector Equilibrium
The Cuboctahedron (V.E.) & the 64 Tetrahedra
Polyhedra in the Modern World
3D printed Sculptures

Max Bruckner 1906 polyhedra & icosahedron models
Über die gleichecking-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder.
Truncated Octahedron
14 Faces: 6 Squares,
8 Hexagons
Edges: 36
Vertices: 24
26 Faces: 8 Triangles,
18 Squares
Edges: 48
Vertices: 24
Basilica of St. Mark, Venice- Italy.
Small Stellated Dodecahedron
Tomb of Sir Thomas Gorges and his wife Helena
Waterfall (1961)
Order and Chaos (1950)
Compound of three Cubes
The first stellation of Rhombic Dodecahedron
Small Stellated Dodecahedron
Stars (1948)
Study for stars (1948)
Charles O. Perry
(1973). Hyatt Regency Hotel, USA.
John Robinson (2003). Institute of Astronomy, England
Keneth Snelson, USA
Stu's Atom 2002
Superstar (1960)
My art
Lamp shade
Deltoidal Hexecontahedron (Catalan Solid)
Eternamente (2011).
Cesar Franzoi
height: 1,70 m
Vida (2014)
Peter Gric
Psychanodia Ochema by Laurence Caruana
Space-Warp Machine II
Geron's Pentagonal Simplex
Deltoidal Hexacontahedron
The Father of Polyhedra nets
Dürer's Polyhedra Paintings
Johannes Kepler
the arrangement of atoms in the crystalline solids
Book: De nive sexangula
12 regular pentagrams 12 regular pentagons 12 regular pentagrams 20 equilateral triangles
Saturn (Outside sphere)
Mars (inscribed in the Tetrahedron)
Mercury (inscribed in the Octahedron)
Jupiter (Inscribed in the Cube))
Venus (inscribed on the Icosahedron)
Earth (inscribed in a Dodecahedron)
Nassim Haramein
Euclid or Archimedes
Polyhedra in Renaissance
The first illustrations of polyhedra ever in the form of "solid edges"
Intarsia are mosaics made of pieces of inlaid wood
Tesselation's Diagram
Truncated Icosahedron
- 4 to 11 cm
- Function not determined.
faces: 12
edges: 30
vertices: 20
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