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Identities of Platonic Solids

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Platonic Solids Relationship Table[1]

Table 1. Platonic Solids Relationship Table
Polyhedron Faces Vertices Edges Formulas
L=edge length
a=surface area
v=solid volume
Radii
cr=Circumradius
mr=Midradius
ir=Inradius
Unit Vertices
X, Y, Z
Φ=1.61803…
φ=0.61803…
Tetrahedron 4
3 sides
4 6

L=2√2
a=√3L2
a=8√3
v=(√2)/12L3
v=8/3

cr=
mr=1
ir=

+1, +1, +1
+1, -1, -1
-1, +1, -1
-1, -1, +1
Hexahedron 6
4 sides
8 12

L=2 units
a=24 units2
a=6L2
v=L3

cr=
mr=√2
ir=

+1, +1, +1
+1, +1, -1
+1, -1, +1
+1, -1, -1
-1, +1, +1
-1, +1, -1
-1, -1, +1
-1, -1, -1
Octahedron 8
3 sides
6 12

L=2 units
a=4√3
a=24 units2
a=2√3L2
v=(√2)/3L3

cr=
mr=1/√2
ir=

+1,  0,  0
-1, 0, 0
 0, +1,  0
0, -1, 0
 0,  0, +1
0, 0, -1
Dodecahedron 12
5 sides
20 30

L=2φ units
a=60 √(φ/√5)
a=√(360(5-√5))
a=15 √(5 φ + 10 )

a=3√(25+10√5)L2
v=(15+7√5)/4L3

cr=√3
mr=φ
ir=

0, +φ, +Φ
0, +φ, -Φ
0, -φ, +Φ
0, -φ, -Φ
+Φ, 0, +φ
+Φ, 0, -φ
-Φ, 0, +φ
-Φ, 0, -φ
+φ, +Φ, 0
+φ, -Φ, 0
-φ, +Φ, 0
-φ, -Φ, 0
+1, +1, +1
+1, +1, -1
+1, -1, +1
+1, -1, -1
-1, +1, +1
-1, +1, -1
-1, -1, +1
-1, -1, -1
Icosahedron 20
3 sides
12 30

L=2 units
a=20√3
v=20( 1 + φ)/3
v=20 Φ2/3
v=10 (3 + √5)/3

cr=
mr=(1+φ)/√3
ir=

0, +φ, +1;
0, +φ, -1;
0, -φ, +1;
0, -φ, -1;
+1, 0, +φ;
+1, 0, -φ;
-1, 0, +φ;
-1, 0, -φ;
+Φ, 0, +φ
+Φ, 0, -φ
-Φ, 0, +φ
-Φ, 0, -φ
+φ, +Φ, 0
+φ, -Φ, 0
-φ, +Φ, 0
-φ, -Φ, 0
+1, +1, +1
+1, +1, -1
+1, -1, +1
+1, -1, -1
-1, +1, +1
-1, +1, -1
-1, -1, +1
-1, -1, -1


There are five Platonic Solids

1 The Tetrahedron
Four faces of regular triangles
2 The Octahedron
Eight faces of regular triangles
3 The Hexahedron (a cube)
Six faces of regular squares
4 The Dodecahedron
Twelve faces of regular pentagons
5 The Icosahedron
Twenty faces of regular triangles
Platonic Solids, Theaetetus's Theorem, Euler's Polyhedron Theorem
A platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex. A polyhedron is a solid figure bounded by plane polygons or faces.
The Greek philosopher Plato described the solids in detail in his book "Timaeus" and assigned the items to the Platonic conception of the world, hence today they are well-known under the name "Platonic Solids."
Euler's polyhedron theorem: F + V = E + 2, where F, V, E are the number of faces, vertices, and edges in the polyhedron.


See also

Astro-Logix.com
"Astro-logix is a simple yet ingenious construction toy, for building 3D frameworks, including DNA helixes, polyhedra, 3d stars, and spaceframes."

References

  1. Pierce, Rod. "Platonic Solids" Math Is Fun. Ed. Rod Pierce. 20 Jun 2009. 7 May 2010 <http://www.mathsisfun.com/platonic_solids.html>