PLATONIC SOLIDS

Tetrahedron Cube Octahedron Dodecahedron Icosahedron
Four faces Six faces Eight faces Twelve faces Twenty faces
Tetrahedron.svg
(Animation)
Hexahedron.svg
(Animation)
Octahedron.svg
(Animation)
Dodecahedron.svg
(Animation)
Icosahedron.svg
(Animation)
Polyhedron Vertices Edges Faces Schläfli symbol Vertex configuration
tetrahedron Tetrahedron 4 6 4 {3, 3} 3.3.3
cube Hexahedron (cube) 8 12 6 {4, 3} 4.4.4
octahedron Octahedron 6 12 8 {3, 4} 3.3.3.3
dodecahedron Dodecahedron 20 30 12 {5, 3} 5.5.5
icosahedron Icosahedron 12 30 20 {3, 5} 3.3.3.3.3

Cartesian coordinates[edit]

For Platonic solids centered at the origin, simple Cartesian coordinates are given below. The Greek letter φ is used to represent the golden ratio 1 + √5/2.

Cartesian coordinates
Figure Tetrahedron Octahedron Cube Icosahedron Dodecahedron
Faces 4 8 6 20 12
Vertices 4 6 (2 × 3) 8 12 (4 × 3) 20 (8 + 4 × 3)
Orientation
set
1 2 1 2 1 2
Coordinates (1, 1, 1)
(1, −1, −1)
(−1, 1, −1)
(−1, −1, 1)
(−1, −1, −1)
(−1, 1, 1)
(1, −1, 1)
(1, 1, −1)
(±1, 0, 0)
(0, ±1, 0)
(0, 0, ±1)
(±1, ±1, ±1) (0, ±1, ±φ)
(±1, ±φ, 0)
φ, 0, ±1)
(0, ±φ, ±1)
φ, ±1, 0)
(±1, 0, ±φ)
(±1, ±1, ±1)
(0, ±1/φ, ±φ)
1/φ, ±φ, 0)
φ, 0, ±1/φ)
(±1, ±1, ±1)
(0, ±φ, ±1/φ)
φ, ±1/φ, 0)
1/φ, 0, ±φ)
Image CubeAndStel.svg Dual Cube-Octahedron.svg Icosahedron-golden-rectangles.svg Cube in dodecahedron.png

 

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REGULAR TESSELLATIONS

The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the sphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. There are three possibilities:

The three regular tilings
 *442

Uniform tiling 44-t0.svg

(3)p6m

Uniform tiling 63-t2.png

*632p4m

Uniform tiling 63-t0.png

{4, 4}

Vertex type 4-4-4-4.svg

44
(t=1, e=1)1-uniform n5.svg

{3, 6}

Vertex type 3-3-3-3-3-3.svg

36
(t=1, e=1)

1-uniform n11.svg

{6, 3)

Vertex type 6-6-6.svg

63
(t=1, e=1)

1-uniform n1.svg

 

 

 

Polygon nets around a vertex
Polyiamond-3-1.svg
{3,3}
Defect 180°
Polyiamond-4-1.svg
{3,4}
Defect 120°
Polyiamond-5-4.svg
{3,5}
Defect 60°
Polyiamond-6-11.svg
{3,6}
Defect 0°
TrominoV.jpg
{4,3}
Defect 90°
Square tiling vertfig.png
{4,4}
Defect 0°
Pentagon net.png
{5,3}
Defect 36°
Hexagonal tiling vertfig.png
{6,3}
Defect 0°
A vertex needs at least 3 faces, and an angle defect.
A 0° angle defect will fill the Euclidean plane with a regular tiling.
By Descartes’ theorem, the number of vertices is 720°/defect.

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The 19th century gum bichromate process in 21st century concept and techniques « Gum Bichromates « Formulas And How-To « AlternativePhotography.com

Christina Z. Anderson gives us the “why and how” of the gum process, including making negatives.Share this:Click to email this to a friend (Opens in new window)Click to print (Opens in n…

Source: The 19th century gum bichromate process in 21st century concept and techniques « Gum Bichromates « Formulas And How-To « AlternativePhotography.com