PLATONIC SOLIDS: Plato, 387 B.C. wrote of the five “elements” in his work “the Timaeus”. The cube was the shape of the element making up “earth”; tetrahedron “fire; octahedron “air”; icosahedron “water”; and the dodecahedron was his model for the whole universe. In 300 B.C. Euclid proved in his manuscript the cosmic solids were as Plato deducted and named them the “Platonic Solids” after Plato. Mathematical law requires the faces must be “regular” polygons and exactly congruent to each other. This characteristic gives the Platonic Solids cosmic uniqueness.

Tetrahedron
(tet-tra-he-dron)
4 equilateral triangles

Hexahedron
(hex-a-he-dron)
“cube” 6 squares

Dodecahedron
(do-dec-a-he-dron)
12 pentagons

Icosahedron
(i-caw-sa-he-dron)
20 equilateral triangles

Octahedron
(oct-a-he-dron)
8 equilateral triangles


ARCHIMEDEAN SOLIDS: About 250 B.C. Archimedes of Syracuse, who calculated the first accurate estimation of pi, proclaimed there are exactly 13 semi-regular convex prototypes having a variety of “regular” polygons that had congruent vertices in that each vertex had the same arrangement of faces. All the vertices would land on the same spheroid, as would the vertices of the Platonic Solids. Pappus of Alexandria first wrote about Archimede’s finding in the 4th century A.D. in Book V.

Cuboctahedron Truncated Cube
(cube-oct-a-he-dron) (trun-cat-ed cube)
8 triangles, 6 squares 8 triangles, 6 octagons

Icosidodecahedron Truncated Dodecahedron
(i-caw-si-do-dec-a-he-dron) (trun-cat-ed do-dec-a-he-dron)
20 triangles, 12 pentagons 20 triangles, 12 decagons

Rhombicuboctahedron Truncated Icosahedron
(rom-bi-cube-oct-a-he-dron) (trun-cat-ed i-co-sa-he-dron)
8 triangles, 18 squares 12 pentagons, 20 hexagons

Rhombicosidodecahedron Truncated Octahedron
(rom-bi-ico-si-do-dec-a-he-dron) (trun-cat-ed oct-a-he-dron)
20 triangles, 30 squares, 12 pentagons 6 squares, 8 hexagons

Snub Cube Truncated Tetrahedron
(snub cube) (trun-cat-ed tet-tra-he-dron)
32 triangles, 6 squares 4 triangles, 4 hexagons

Great Rhombicuboctahedron Snub Dodecahedron
(great-rom-bi-cube-oct-a-hed-ron) (snub do-dec-a-he-dron)
30 squares, 20 hexagons, 12 decagons 80 triangles, 12 pentagons

Great Rhombicosidodecahedron
(great-rom-bi- i-caw-si-do-dec-a-he-dron)
12 squares, 8 hexagons, 6 octagons
 

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