# Polyhedra Coordinates

In a note posted on "synergetics-l@teleport.com", Gerald de Jong showed that the vertices of the Tetrahedron, Octahedron, A quantum module, B quantum module, and Vector Equilibrium could all be given integer (x, y, z) coordinates. He also showed that the Icosahedron and Regular Dodecahedron vertex could be given in terms of integers and the number p=(1 + sqrt(5)) /2, the Golden ratio.

Following de Jong's lead, I re-calculated all the vertex coordinates for all the major polyhedra in Fuller's Synergetics books and found that almost all the polyhedra could be given integer vertex coordinates or coordinates in terms of the number p. These results are given as part of the polyhedra data. The original vertex coordinates to these polyhedra which I calculated back in 1992, which were almost all irrational numbers, can thankfully be forgotten.

The scale for the vertex coordinates listed was pick to ensure that all the rational coordinates would in fact be integer coordinates. This means that the edge length of the cube is 6 coordinate units in length, not one. (The edge length of the inscribed tetrahedron can be defined as unit length, but the coordinate scale does not give the edge length a value of one.)

The vertex coordinates give the orientation of the polyhdera as they occur in the IVM. Note that there are other orientations of the polyhedra in the IVM. I have listed only one such orientation.

For the corrdinates listed, most of the polyhedra center of volume are not at (0,0,0). Therefore, if you intend to use these coordinates as part of a computer program which rotates the polyhedra you will need to translate the polyhedra first so that the center of volume is at (0,0,0).

## Integer Coordinates

Tetrahedron
VERTEX X Y Z
A 0 0 0
B 0 6 6
C 6 0 6
D 6 6 0

1/4 Tetrahedron
VERTEX X Y Z
A 0 0 0
B 6 0 6
C 6 6 0
D 3 3 3

A+ Quantum Module
VERTEX X Y Z
A 0 0 0
B 3 3 0
C 4 2 2
D 3 3 3

A- Quantum Module
VERTEX X Y Z
A 6 6 6
B 3 3 0
C 4 2 2
D 3 3 3

Octahedron
VERTEX X Y Z
A 0 0 0
B 6 -6 0
C 12 0 0
D 6 6 0
E 6 0 6
F 6 0 -6
Octahedron edge map: (A,B)(A,D)(A,E)(A,F)(B,C)(B,E)(B,F)(C,D) (C,E)(C,F)(D,E)(D,F)

1/8 Octahedron
VERTEX X Y Z
A 0 0 0
B 6 0 6
C 6 0 0
D 6 6 0

Iceberg
VERTEX X Y Z
A 0 0 0
B 12 0 0
C 6 6 0
D 6 0 6

B+ Quantum Module
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 6 0 3
D 3 3 0

B- Quantum Module
VERTEX X Y Z
A 6 6 0
B 6 0 0
C 6 0 3
D 3 3 0
C Quantum Module
VERTEX X Y Z
A 0 0 0
B 3 3 0
C 2 4 4
D 1 5 5

D Quantum Module
VERTEX X Y Z
A 0 0 0
B 3 3 0
C 1 5 5
D 0 6 6

nth Quantum Module
(n integer)
VERTEX X Y Z
A 0 0 0
B 3 3 0
Cn 4-n 2+n 2+n
Dn 3-n 3+n 3+n

MITE
VERTEX X Y Z
A 0 0 0
B 3 3 0
C 6 0 0
D 3 3 3

Syte: Bite
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 0 6 0
D 3 3 3

Syte: Rite
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 3 3 3
D 3 3 -3

Syte: Lite
VERTEX X Y Z
A 0 0 0
B 3 3 0
C 6 0 0
D 3 3 3
E 3 0 3

Syte: Lite edge map: (A,B)(A,C)(A,E)(A,D)(B,C)(B,D)(C,D)(C,E)(D,E)

Kite: Kate
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 0 6 0
D 3 3 3
E 3 3 -3

Kite: Kate edge map: (A,B)(A,C)(A,D)(A,E)(B,D)(B,E)(C,D)(C,E)(D,E)

Kite: Kat
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 0 6 0
D 6 6 0
E 3 3 3

Kite: Kat edge map: (A,B)(B,C)(C,D)(D,A)(A,E)(B,E)(C,E)(D,E)

Octet
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 6 6 0
D 6 0 6
E 3 3 3

Octet edge map: (A,B)(A,C)(A,D)(A,E)(B,C)(B,D)(C,D)(C,E)(D,E)

Coupler
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 0 6 0
D 3 3 3
E 6 6 6
F 3 3 -3

Coupler edge map: (A,B)(A,C)(A,D)(A,F)(B,D)(B,E)(B,F) (C,D)(C,E)(C,F)(D,E)(E,F)

Cube
VERTEX X Y Z
A 0 0 0
B 6 0 0
C 6 6 0
D 0 6 0
E 0 0 6
F 6 0 6
G 6 6 6
H 0 6 6

edge map: (A,B)(A,D)(A,E)(B,C)(B,F)(C,D)(C,G) (D,H)(E,F)(E,H)(F,G)(G,H)

Vector Equilibrium
VERTEX X Y Z
V1 6 0 6
V2 0 6 6
V3 -6 0 6
V4 0 -6 6
V5 6 6 0
V6 -6 6 0
V7 -6 -6 0
V8 6 -6 0
V9 6 0 -6
V10 0 6 -6
V11 -6 0 -6
V12 0 -6 -6

Vector Equilibrium edge map: (V1,V2)(V1,V4)(V1,V5)(V1,V8)(V2,V3)(V2,V5)(V2,V6) (V3,V4)(V3,V6)(V3,V7)(V4,V7)(V4,V8)(V5,V9)(V5,V10) (V6,V10)(V6,V11)(V7,V11)(V7,V12)(V8,V12)(V8,V9) (V9,V10)(V9,V12)(V10,V11)(V11,V12)

Rhombic Dodecahedron
VERTEX X Y Z
V1 6 0 0
V2 0 6 0
V3 -6 0 0
V4 0 -6 0
V5 0 0 6
V6 0 0 -6
V7 3 3 3
V8 -3 3 3
V9 -3 -3 3
V10 3 -3 3
V11 3 3 -3
V12 -3 3 -3
V13 -3 -3 -3
V14 3 -3 -3

Tetrakaidecahedron edge map: (V1,V2)(V2,V3)(V3,V4)(V1,V4)(V1,V5)(V2,V6)(V3,V7)(V4,V8) (V5,V9)(V5,V16)(V6,V10)(V6,V11)(V7,V12)(V7,V13)(V8,V14)(V8,V15) (V9,V10)(V11,V12)(V13,V14)(V15,V16)(V17,V9)(V17,V16) (V18,V10)(V18,V11)(V19,V12)(V19,V13)(V20,V14)(V20,V15) (V17,V21)(V18,V22)(V19,V23)(V20,V24)(V21,V22)(V22,V23) (V23,V24)(V24,V21)

### FIVE FOLD SYMMETRY BASED POLYHEDRA

Let p = (1 + sqrt(5)) /2

Tetrakaidecahedron (Lord Kelvin's Solid)
VERTEX X Y Z
V1 2 0 4
V2 0 2 4
V3 -2 0 4
V4 -2 -2 4
V5 4 0 2
V6 0 4 2
V7 -4 0 2
V8 0 -4 2
V9 4 2 0
V10 2 4 0
V11 -2 4 0
V12 -4 2 0
V13 -4 -2 0
V14 -2 -4 0
V15 2 -4 0
V16 4 -2 0
V17 4 0 -2
V18 0 4 -2
V19 -4 0 -2
V20 0 -4 -2
V21 2 0 -4
V22 0 2 -4
V23 -2 0 -4
V24 -2 -2 -4
Icosahedron
VERTEX X Y Z
V1 1 0 p
V2 -1 0 p
V3 1 0 -p
V4 -1 0 -p
V5 0 p 1
V6 0 -p 1
V7 0 p -1
V8 0 -p -1
V9 p 1 0
V10 -p 1 0
V11 p -1 0
V12 -p -1 0

edge map: (V1,V5)(V1,V2)(V1,V6)(V1,V9)(V1,V11)(V2,V5)(V2,V6)(V2,V10)(V2,V12) (V3,V4)(V3,V7)(V3,V8)(V3,V9)(V3,V11)(V4,V7)(V4,V8)(V4,V10)(V4,V12) (V5,V7)(V5,V9)(V5,V10)(V6,V8)(V6,V11)(V6,V12)(V7,V9)(V7,V10) (V8,V11)(V8,V12)(V9,V11)(V10,V12)

S Quantum Module
VERTEX X Y Z
A 1 0 p
B 0 0 p
C 0 p 1
D 0 0 p^2

Regular Dodecahedron
VERTEX X Y Z
V1 0 p p^3
V2 0 -p p^3
V3 p^2 p^2 p^2
V4 -p^2 p^2 p^2
V5 -p^2 -p^2 p^2
V6 p^2 -p^2 p^2
V7 p^3 0 p
V8 -p^3 0 p
V9 p p^3 0
V10 -p p^3 0
V11 -p -p^3 0
V12 p -p^3 0
V13 p^3 0 -p
V14 -p^3 0 -p
V15 p^2 p^2 -p^2
V16 -p^2 p^2 -p^2
V17 -p^2 -p^2 -p^2
V18 p^2 -p^2 -p^2
V19 0 p -p^3
V20 0 -p -p^3

edge map: (V1,V2)(V1,V3)(V1,V4)(V2,V5)(V2,V6)(V3,V7)(V3,V9) (V4,V8)(V4,V10)(V5,V8)(V5,V11)(V6,V12)(V6,V7) (V7,V13)(V8,V14)(V9,V10)(V9,V15)(V10,V16)(V11,V12) (V11,V17)(V12,V18)(V13,V15)(V13,V18)(V14,V16)(V14,V17) (V15,V19)(V16,V19)(V17,V20)(V18,V20)(V19,V20)

Rhombic Triacontahedron
VERTEX X Y Z
V1 -p^2 0.0 p^3
V2 0.0 p p^3
V3 -p^2 p^2 p^2
V4 -p^3 0.0 p
V5 -p^2 -p^2 p^2
V6 0.0 -p p^3
V7 p^2 0.0 p^3
V8 0.0 p^3 p^2
V9 -p^3 p^2 0.0
V10 -p^3 -p^2 0.0
V11 0.0 -p^3 p^2
V12 p^2 p^2 p^2
V13 -p p^3 0.0
V14 -p^3 0.0 -p
V15 -p -p^3 0.0
V16 p^2 -p^2 p^2
V17 p^3 0.0 p
V18 p p^3 0.0
V19 -p^2 p^2 -p^2
V20 -p^2 -p^2 -p^2
V21 p -p^3 0.0
V22 p^3 p^2 0.0
V23 0.0 p^3 -p^2
V24 -p^2 0.0 -p^3
V25 0.0 -p^3 -p^2
V26 p^3 -p^2 0.0
V27 p^3 0.0 -p
V28 p^2 p^2 -p^2
V29 0.0 p -p^3
V30 0.0 -p -p^3
V31 p^2 -p^2 -p^2
V32 p^2 0.0 -p^3

edge map: (V1,V2)(V1,V3)(V1,V4)(V1,V5)(V1,V6)(V2,V7) (V2,V8)(V3,V8)(V3,V9)(V4,V9)(V4,V10)(V5,V10)(V5,V11) (V6,V11)(V6,V7)(V7,V12)(V7,V16)(V7,V17)(V8,V12)(V8,V13) (V8,V18)(V9,V13)(V9,V14)(V9,V19)(V10,V14)(V10,V15)(V10,V20) (V11,V15)(V11,V16)(V11,V21)(V12,V22)(V13,V23)(V14,V24)(V15,V25) (V16,V26)(V17,V26)(V17,V22)(V18,V22)(V18,V23)(V19,V23) (V19,V24)(V20,V24)(V20,V25)(V21,V25)(V21,V26)(V22,V27) (V22,V28)(V23,V28)(V23,V29)(V24,V29)(V24,V30)(V25,V30) (V25,V31)(V26,V31)(V26,V27)(V27,V32)(V28,V32)(V29,V32) (V30,V32)(V31,V32)

## To Do

I have not found an orientation for the following polyhedra which result in simple coordinates for their vertices. I will give the decimal value for the (x, y, z) coordinates which I calculated some years ago. The exact expressions are too difficult to express here at this time. I hope to give the exact expressions in later updates to this web page.

T Quantum Module
VERTEX X Y Z
A -0.231643001 0.047721083 0.124935417
B 0.077214334 -0.143163248 0.124935417
C 0.077214334 0.047721083 0.124935417
D 0.077214334 0.047721083 -0.374806250

E Quantum Module
VERTEX X Y Z
A -0.231762746 0.047745751 0.125
B 0.077254249 -0.143237254 0.125
C 0.077254249 0.047745751 0.125
D 0.077254249 0.047745751 -0.375

Truncated Icosahedron
(Buckminsterfullerene)
VERTEX X Y Z
V1 0.0 -0.850650808 2.327438437
V2 0.809016994 -0.262865556 2.327438437
V3 0.5 0.688190960 2.327438437
V4 -0.5 0.688190960 2.327438437
V5 -0.809016994 -0.262865556 2.327438437
V6 0.0 -1.701301617 1.801707325
V7 1.618033989 -0.525731112 1.801707325
V8 1.0 1.376381920 1.801707325
V9 -1.0 1.376381920 1.801707325
V10 -1.618033989 -0.525731112 1.801707325
V11 0.809016994 -1.964167173 1.275976213
V12 1.618033989 -1.376381920 1.275976213
V13 2.118033989 0.162459848 1.275976213
V14 1.809016994 1.113516364 1.275976213
V15 0.5 2.064572881 1.275976213
V16 -0.5 2.064572881 1.275976213
V17 -1.809016994 1.113516364 1.275976213
V18 -2118033989 0.162459848 1.275976213
V19 -1.618033989 -1.376381920 1.275976213
V20 -0.809016994 -1.964167173 1.275976213
V21 0.5 -2.389492577 0.425325404
V22 2.118033989 -1.213922072 0.425325404
V23 2.427050983 -0.262865556 0.425325404
V24 1.809016994 1.639247477 0.425325404
V25 1.0 2.227032729 0.425325404
V26 -1.0 2.227032729 0.425325404
V27 -1.809016994 1.639247477 0.425325404
V28 -2.427050983 -0.262865556 0.425325404
V29 -2.118033989 -1.213922072 0.425325404
V30 -0.5 -2.389492577 0.425325404
V31 1.0 -2.227032729 -0.425325404
V32 1.809016994 -1.639247477 -0.425325404
V33 2.427050983 0.262865556 -0.425325404
V34 2.118033989 1.213922072 -0.425325404
V35 0.5 2.389492577 -0.425325404
V36 -0.5 2.389492577 -.0425325404
V37 -2.118033989 1.213922072 -0.425325404
V38 -2.427050983 0.262865556 -0.425325404
V39 -1.809016994 -1.639247477 -0.425325404
V40 -1.0 -2.227032723 -0.425325404
V41 0.5 -2.064572881 -1.275976213
V42 1.809016994 -1.113516364 -1.275976213
V43 2.118033989 -0.162459848 -1.275976213
V44 1.618033989 1.376382920 -1.275976213
V45 0.809016994 1.964167173 -1.275976213
V46 -0.809016994 1.964167173 -1.275976213
V47 -1.618033989 1.376381920 -1.275976213
V48 -2.118033989 -0.162459848 -1.275976213
V49 -1.809016994 -1.113516364 -1.275976213
V50 -0.5 -2.064572881 -1.275976213
V51 1.0 -1.376381920 -1.801707325
V52 1.618033989 0.525731112 -1.801707325
V53 0.0 1.701301617 -1.801707325
V54 -1.618033989 0.525731112 -1.891707325
V55 -1.0 -1.376381920 -1.801707325
V56 0.5 -0.688190960 -2.327438437
V57 0.809016994 0.262865556 -2.327438437
V58 0.0 0.850650808 -2.327438437
V59 -0.809016994 0.262865556 -2.327438437
V60 -0.5 -0.688190960 -2.327438437

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