**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

Cuboctahedron

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

Rhombicosidodecahedron

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

Icosidodecahedron

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

Architecture

3D printed Sculptures

Max Bruckner 1906 polyhedra & icosahedron models

Book:

Über die gleichecking-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder.

Truncated Octahedron

14 Faces: 6 Squares,

8 Hexagons

Edges: 36

Vertices: 24

Rhombicuboctahedron

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)

ECLIPSE

Charles O. Perry

(1973). Hyatt Regency Hotel, USA.

Pulse

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)

Paintings

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)

Pos-Renaissence

Nassim Haramein

Softwares

Phytagoras

Plato

Aristotle

Euclid or Archimedes

Polyhedra in Renaissance

Rombhicuboctahedron

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.

Dodecahedron

faces: 12

edges: 30

vertices: 20