Table of Altitudes
Common Parameters:  Formula Value:  Approx Value:  Relationship:       
R  (1 + sqrt(5)) / 2  1.61803398874989  R^{2} = R + 1       
A   1.53884176858763  A = a + r       
r   0.85065080835204  r = RH       
a   0.688190960235587  2a = R^{2}H       
H  H = 1 / sqrt (R + 2)  0.525731112119134  H = 2a  r       

Lower Symm. Polyhedron:  Height between faces:  Value:  Approx:  Height between faces:  Value:  Approx:    
Q_{5} = pentagonal cupola  h(5,10,Q_{5}) =  H = 2ar  0.525731112119134       
S_{5} = pentagonal antiprism  h(5,5,S_{5}) =  RH = r  0.85065080835204  h(3,3,S_{5}) =  (R + 1)sqrt(3) / 3  1.51152262815234    
R_{5} = pentagonal rotunda  h(5,10,R_{5}) =  R^{2}H = 2a  1.37638192047117       

Icosahedral Polyhedron:  Fivefold Axis  Value:  Approx:  Threefold Axis  Value:  Approx:  Approx:  Value:  Approx: 
D_{5} = dodecahedron  h(5,5,D_{5}) =  R^{3}H = 2a + r  2.22703272882321  diameter =  (3R)sqrt(3)/3  2.80251707688815  edgetoedge  R + 1  2.61803398874989 
I_{5} = icosahedron  diameter =  (R + 2)H  1.90211303259031  h(3,3,I_{5}) =  (R + 1)sqrt(3) / 3  1.51152262815234  edgetoedge  R  1.61803398874989 
B_{5} = icosidodecahedron  h(5,5,B_{5}) =  2R^{2}H = 4a  2.75276384094235  h(3,3,B_{5}) =  2(R + 1)sqrt(3) / 3  3.02304525630468  diameter =  2R  3.23606797749979 
C_{5} = truncated icosahedron  h(5,5,C_{5}) =  (3R^{2}+1)H = 8a  r  4.65487687353265  h(6,6,C_{5}) =  (R + 1)sqrt(3)  4.53456788445702  edgetoedge  3R  4.85410196624968 
T_{5} = truncated dodecahedron  h(10,10,T_{5}) =  (3R^{2}+R)H = 6a + r  4.97979656976556  h(3,3,T_{5}) =  (5R + 2)sqrt(3) / 3  5.82556233319283  edgetoedge  3R + 1  5.85410196624968 
E_{5} = rhombicosidodecahedron  h(5,5,E_{5}) =  3R^{2}H = 6a  4.12914576141352  h(3,3,E_{5}) =  (4R + 1)sqrt(3) / 3  4.31403970504049  h(4,4,E_{5}) =  2R + 1  4.23606797749979 
K_{5} = truncated icosidodecahedron  h(10,10,K_{5}) =  5R^{2}H = 10a  6.88190960235587  h(6,6,K_{5}) =  (2R + 1)sqrt(3)  7.33708496134517  h(4,4,K_{5}) =  4R + 1  7.47213595499958 

Lower Symm. Polyhedron:  Height between faces:  Value:  Approx:       
Q_{3} = triangular cupola  h(3,6,Q_{3}) =  sqrt(6) / 3  0.816496580927726       
Q_{4} = square cupola  h(4,8,Q_{4}) =  sqrt(2) / 2  0.707106781186548       

Octahedral Polyhedron:  Fourfold Axis  Value:  Approx:  Threefold Axis  Value:  Approx:  Approx:  Value:  Approx: 
P_{4} = Cube  h(4,4,P_{4}) =  1  1  diameter =  sqrt(3)  1.73205080756888  edgetoedge  sqrt(2)  1.4142135623731 
S_{3} = Octahedron  diameter =  sqrt(2)  1.4142135623731  h(3,3,S_{3}) =  sqrt(6) / 3  0.816496580927726  edgetoedge  1  1 
B_{4} = Cuboctahedron  h(4,4,B_{4}) =  sqrt(2)  1.4142135623731  h(3,3,B_{4}) =  2*sqrt(6) / 3  1.63299316185545  diameter =  2  2 
K_{3} = Truncated Octahedron  h(4,4,K_{3}) =  2*sqrt(2)  2.82842712474619  h(6,6,K_{3}) =  sqrt(6)  2.44948974278318  edgetoedge  3  3 
T_{4} = Truncated Cube  h(8,8,T_{4}) =  1+sqrt(2)  2.41421356237309  h(3,3,T_{4}) =  sqrt(3) + 2*sqrt(6) / 3  3.36504396942433  edgetoedge  2+sqrt(2)  3.41421356237309 
E_{4} = Rhombicuboctahedron  h(4,4,E_{4}) =  1+sqrt(2)  2.41421356237309  h(3,3,E_{4}) =  sqrt(3) + sqrt(6) / 3  2.5485473884966  h(4,4,E_{4}) =  1+sqrt(2)  2.41421356237309 
K_{4} = Truncated Cuboctahedron  h(8,8,K_{4}) =  1+2*sqrt(2)  3.82842712474619  h(6,6,K_{4}) =  sqrt(3) + sqrt(6)  4.18154055035206  h(4,4,K_{4}) =  3+sqrt(2)  4.41421356237309 
sB_{4} = snub cuboctahedron  h(4,4,sB_{4}) =  2t  2.28522701785  h(3,3,sB_{4}) =   2.42671160004378  edgetoedge   2.49444633598728 
note: t is the solution to the equation 32t^{6}  32t^{4}  12t^{2}  1 = 0 

Tetrahedral Polyhedron:  Threefold Axis  Value:  Approx:  Twofold Axis  Value:  Approx:    
Y_{3} = Tetrahedron  altitude =  sqrt(6) / 3  0.816496580927726  edgetoedge  sqrt(2) / 2  0.707106781186548    
T_{3} = Truncated Tetrahedron  h(3,6,T_{3}) =  2*sqrt(6) / 3  1.63299316185545  edgetoedge  sqrt(2) + sqrt(2) / 2  2.12132034355964    

Lower Symm. Polyhedron:  Height between faces:  Value:  Approx:       
P_{m} = mgonal prism  h(m,m,P_{m}) =  1  1  h(4,4,P_{2k}) =  cot(90°/k)     
S_{m} = mgonal antiprism  h(m,m,S_{m}) =  (1/2)*sqrt(3tan^{2}(90°/m))   h(3,3,S_{m}) =  h(m,m,S_{m}) * sqrt(3) / 3*tan(90°/m)     

h_{m}^{2} = h_{n}^{2} + 4a_{n}^{2}  4a_{m}^{2}  h_{m}^{2} =  h_{n}^{2} +  4a_{n}^{2}   4a_{m}^{2}  pg. 225     
E_{5} = rhombicosidodecahedron  ((4R + 1)sqrt(3) / 3)^{2} =  (2R + 1)^{2} +  4*(1/2)^{2}   4*(sqrt(3)/6)^{2}  pg. 228     
K_{4} = truncated cuboctahedron  (3+sqrt(2))^{2} =  (sqrt(3) + sqrt(6))^{2} +  4*(sqrt(3)/2)^{2}   4*(1/2)^{2}  pg. 228     
K_{5} = truncated icosidodecahedron  (4R + 1)^{2} =  ((2R + 1)sqrt(3))^{2} +  4*(sqrt(3)/2)^{2}   4*(1/2)^{2}  pg. 228     
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Disclaimer:
These pages are dedicated to B.M. Stewart. They are intended to be used by readers of his book to better illustrate his ideas in three dimensions. His book has tons of information that I do not intend to represent here, and I highly encourage anyone interested in this subject to buy the book. If you like my illustrations, please let me know. And if you are interested in a particular Stewart Toroid being modeled, please let me know and I will do what I can.
Thanks also to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things).
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