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Polyhedra Notes

written by Rene K. Mueller, Copyright (c) 2007, last updated Wed, January 9, 2008

Square Pyramid


Square Pyramid
  • Johnson Solid: J1
  • Vertices: 5
  • Faces: 5
  • Edges: 8
  • V: s3 / 6 * √2
  • A: s2 * (1 + √3)



General Pyramid
The general pyramid:

  • q = √((s/2)2 + h2)
  • l = √((s/2)2 + q2)
  • A = 2 s q + s2
  • Awall/roof = 2 s q
  • V = s2 h / 3


Historic Pyramids


The Pyramids of Giza (courtesy Bruno Girin/DHD Multimedia Gallery )
Most obvious association with pyramids comes from the egyptian pyramids in Giza and those of the mayans in south america. It is also apparent those pyramids have been built with enormous effort and skills which is now honored by the durability of thousands of years.

The largest pyramid in Giza is the Khufu Pyramid, with following values:

  • s = 231m (original) (now: 230.4m)
  • h = 146.6m (original) (now: 138.8m)

thereby

  • q = 186.632m
  • l = 219.478m

Interesting ratio:

  • s/h = 1.57571.. ~ π/2 ~ 11/7

Read more at Great Pyramid of Giza .


Pentagonal Pyramid


Pentagonal Pyramid
  • Johnson Solid: J2
  • Vertices: 6
  • Faces: 6
  • Edges: 10
  • V: s3 / 24 * (5 + √5)
  • A: s2 / 2 * √(5/2 * (10 + √10 + √(75+30*√5))

The roof isn't very steep, unless those roof struts would be made longer - it would be a very simplistic habitat.


Triangular Cupola


Triangular Cupola
  • Johnson Solid: J3
  • Vertices: 9
  • Faces: 8
  • Edges: 15
  • V: s3 * (5/(5*√2))
  • A: s2 * (3+5/2*√3)

Square Cupola


Square Cupola
  • Johnson Solid: J4
  • Vertices: 12
  • Faces: 10
  • Edges: 20

Also cap of a Rhombicuboctahedron.


Pentagonal Cupola


Pentagonal Cupola
  • Johnson Solid: J5
  • Vertices: 15
  • Faces: 12
  • Edges: 25
  • V: s3 / 6 * (5+4*√5)
  • A: s2 / 4 * (20+√(10*(80+31*√5+√(2175+950*√5))))

Also cap of a Rhombicosidodecahedron.


Pentagonal Rotunda


Pentagonal Rotunda
  • Johnson Solid: J6
  • Vertices: 20
  • Faces: 17
  • Edges: 35
  • V: s3 / 12 * (45+17*√5)
  • A: s2 * 5/2 * (√3+√(26+(58/√5)))
  • h: s * √(1/5*(5+2*√5))
Also 1/2 of a Icosidodecahedron.

Elongated Triangular Pyramid


Elongated Triangular Pyramid
  • Johnson Solid: J7
  • Vertices: 7
  • Faces: 7
  • Edges: 12

Elongated Square Pyramid


Elongated Square Pyramid
  • Johnson Solid: J8
  • Vertices: 9
  • Faces: 9
  • Edges: 16

Elongated Pentagonal Pyramid


Elongated Pentagonal Pyramid
  • Johnson Solid: J9
  • Vertices: 11
  • Faces: 11
  • Edges: 20

Gyroelongated Square Pyramid


Gyroelongated Square Pyramid
  • Johnson Solid: J10
  • Vertices: 9
  • Faces: 13
  • Edges: 20

Gyroelongated Pentagonal Pyramid


Gyroelongated Pentagonal Pyramid
  • Johnson Solid: J11
  • Vertices: 11
  • Faces: 16
  • Edges: 25

Triangular Dipyramid


Triangular Dipyramid
  • Johnson Solid: J12
  • Vertices: 5
  • Faces: 6
  • Edges: 9

Pentagonal Dipyramid


Pentagonal Dipyramid
  • Johnson Solid: J13
  • Vertices: 7
  • Faces: 10
  • Edges: 15
  • V: s3 / 12 * (5 + √5)
  • A: s2 * 5/12 * √3
  • router: s / 10 * √(50+10*√5)
  • h: s / 10 * √(50-10*√5)

Elongated Triangular Dipyramid


Elongated Triangular Dipyramid
  • Johnson Solid: J14
  • Vertices: 8
  • Faces: 9
  • Edges: 15

Elongated Square Dipyramid


Elongated Square Dipyramid
  • Johnson Solid: J15
  • Vertices: 10
  • Faces: 12
  • Edges: 20

Elongated Pentagonal Dipyramid


Elongated Pentagonal Dipyramid
  • Johnson Solid: J16
  • Vertices: 12
  • Faces: 15
  • Edges: 25

Gyroelongated Square Dipyramid


Gyroelongated Square Dipyramid
  • Johnson Solid: J17
  • Vertices: 10
  • Faces: 16
  • Edges: 24
  • V: s3 * 2(1/4) / 4 * (1 + √2 + 2(1/4) )
  • A: s2 * 4 * √3

Elongated Triangular Cupola


Elongated Triangular Cupola
  • Johnson Solid: J18
  • Vertices: 15
  • Faces: 14
  • Edges: 27

Elongated Square Cupola


Elongated Square Cupola
  • Johnson Solid: J19
  • Vertices: 20
  • Faces: 18
  • Edges: 36

Elongated Pentagonal Cupola


Elongated Pentagonal Cupola
  • Johnson Solid: J20
  • Vertices: 25
  • Faces: 22
  • Edges: 45

Elongated Pentagonal Rotunda


Elongated Pentagonal Rotunda
  • Johnson Solid: J21
  • Vertices: 30
  • Faces: 27
  • Edges: 55
Very suitable for a habitat, the top smaller pentagon as skylight, and eventually the side pentagons further triangulated each.

Gyroelongated Triangular Cupola


Gyroelongated Triangular Cupola
  • Johnson Solid: J22
  • Vertices: 15
  • Faces: 20
  • Edges: 33

Gyroelongated Square Cupola


Gyroelongated Square Cupola
  • Johnson Solid: J23
  • Vertices: 20
  • Faces: 26
  • Edges: 44

Gyroelongated Pentagonal Cupola


Gyroelongated Pentagonal Cupola
  • Johnson Solid: J24
  • Vertices: 25
  • Faces: 32
  • Edges: 55


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