Chapter 19 - Tables of Altitudes

  • This chapter includes altitude information on some of the more commonly used polyhedra, and how some commonly occurring heights led to the combinations throughout the book. I don't have much for this chapter, but I did arrange the table of altitudes into an actual table, for easy reference. Also, with approximate values, a researcher may be able to more quickly deduce a relationship, and then verify a particular value.
    Model.Stel GenusProperties Pages with Diagram Pages with DescriptionNotes
    Pentagon-DecagonRelationship.StelNANA219  

    Table of Altitudes
    Common Parameters:Formula Value:Approx Value:Relationship:      
    R(1 + sqrt(5)) / 21.61803398874989R2 = R + 1      
    A 1.53884176858763A = a + r      
    r 0.85065080835204r = RH      
    a 0.6881909602355872a = R2H      
    HH = 1 / sqrt (R + 2)0.525731112119134H = 2a - r      
     
    Lower Symm. Polyhedron:Height between faces:Value:Approx:Height between faces:Value:Approx:   
    Q5 = pentagonal cupolah(5,10,Q5) = H = 2a-r0.525731112119134      
    S5 = pentagonal antiprismh(5,5,S5) = RH = r0.85065080835204h(3,3,S5) = (R + 1)sqrt(3) / 31.51152262815234   
    R5 = pentagonal rotundah(5,10,R5) = R2H = 2a1.37638192047117      
     
    Icosahedral Polyhedron:Fivefold AxisValue:Approx:Threefold AxisValue:Approx:Approx:Value:Approx:
    D5 = dodecahedronh(5,5,D5) = R3H = 2a + r2.22703272882321diameter = (3R)sqrt(3)/32.80251707688815edge-to-edgeR + 12.61803398874989
    I5 = icosahedrondiameter = (R + 2)H1.90211303259031h(3,3,I5) = (R + 1)sqrt(3) / 31.51152262815234edge-to-edgeR1.61803398874989
    B5 = icosidodecahedronh(5,5,B5) = 2R2H = 4a2.75276384094235h(3,3,B5) = 2(R + 1)sqrt(3) / 33.02304525630468diameter = 2R3.23606797749979
    C5 = truncated icosahedronh(5,5,C5) = (3R2+1)H = 8a - r4.65487687353265h(6,6,C5) = (R + 1)sqrt(3)4.53456788445702edge-to-edge3R4.85410196624968
    T5 = truncated dodecahedronh(10,10,T5) = (3R2+R)H = 6a + r4.97979656976556h(3,3,T5) = (5R + 2)sqrt(3) / 35.82556233319283edge-to-edge3R + 15.85410196624968
    E5 = rhombicosidodecahedronh(5,5,E5) = 3R2H = 6a4.12914576141352h(3,3,E5) = (4R + 1)sqrt(3) / 34.31403970504049h(4,4,E5) = 2R + 14.23606797749979
    K5 = truncated icosidodecahedronh(10,10,K5) = 5R2H = 10a6.88190960235587h(6,6,K5) = (2R + 1)sqrt(3)7.33708496134517h(4,4,K5) = 4R + 17.47213595499958
     
    Lower Symm. Polyhedron:Height between faces:Value:Approx:      
    Q3 = triangular cupolah(3,6,Q3) = sqrt(6) / 30.816496580927726      
    Q4 = square cupolah(4,8,Q4) = sqrt(2) / 20.707106781186548      
     
    Octahedral Polyhedron:Fourfold AxisValue:Approx:Threefold AxisValue:Approx:Approx:Value:Approx:
    P4 = Cubeh(4,4,P4) = 11diameter = sqrt(3)1.73205080756888edge-to-edgesqrt(2)1.4142135623731
    S3 = Octahedrondiameter = sqrt(2)1.4142135623731h(3,3,S3) = sqrt(6) / 30.816496580927726edge-to-edge11
    B4 = Cuboctahedronh(4,4,B4) = sqrt(2)1.4142135623731h(3,3,B4) = 2*sqrt(6) / 31.63299316185545diameter = 22
    K3 = Truncated Octahedronh(4,4,K3) = 2*sqrt(2)2.82842712474619h(6,6,K3) = sqrt(6)2.44948974278318edge-to-edge33
    T4 = Truncated Cubeh(8,8,T4) = 1+sqrt(2)2.41421356237309h(3,3,T4) = sqrt(3) + 2*sqrt(6) / 33.36504396942433edge-to-edge2+sqrt(2)3.41421356237309
    E4 = Rhombicuboctahedronh(4,4,E4) = 1+sqrt(2)2.41421356237309h(3,3,E4) = sqrt(3) + sqrt(6) / 32.5485473884966h(4,4,E4) = 1+sqrt(2)2.41421356237309
    K4 = Truncated Cuboctahedronh(8,8,K4) = 1+2*sqrt(2)3.82842712474619h(6,6,K4) = sqrt(3) + sqrt(6)4.18154055035206h(4,4,K4) = 3+sqrt(2)4.41421356237309
    sB4 = snub cuboctahedronh(4,4,sB4) = 2t2.28522701785h(3,3,sB4) =  2.42671160004378edge-to-edge 2.49444633598728
    note: t is the solution to the equation 32t6 - 32t4 - 12t2 - 1 = 0
     
    Tetrahedral Polyhedron:Threefold AxisValue:Approx:Twofold AxisValue:Approx:   
    Y3 = Tetrahedronaltitude = sqrt(6) / 30.816496580927726edge-to-edgesqrt(2) / 20.707106781186548   
    T3 = Truncated Tetrahedronh(3,6,T3) = 2*sqrt(6) / 31.63299316185545edge-to-edgesqrt(2) + sqrt(2) / 22.12132034355964   
     
    Lower Symm. Polyhedron:Height between faces:Value:Approx:      
    Pm = m-gonal prismh(m,m,Pm) = 11h(4,4,P2k) = cot(90°/k)    
    Sm = m-gonal antiprismh(m,m,Sm) = (1/2)*sqrt(3-tan2(90°/m)) h(3,3,Sm) = h(m,m,Sm) * sqrt(3) / 3*tan(90°/m)    
     
    hm2 = hn2 + 4an2 - 4am2hm2 = hn2 + 4an2 - 4am2pg. 225    
    E5 = rhombicosidodecahedron((4R + 1)sqrt(3) / 3)2 = (2R + 1)2 + 4*(1/2)2 - 4*(sqrt(3)/6)2pg. 228    
    K4 = truncated cuboctahedron(3+sqrt(2))2 = (sqrt(3) + sqrt(6))2 + 4*(sqrt(3)/2)2 - 4*(1/2)2pg. 228    
    K5 = truncated icosidodecahedron(4R + 1)2 = ((2R + 1)sqrt(3))2 + 4*(sqrt(3)/2)2 - 4*(1/2)2pg. 228    

    Prev - Chapter 18 - Knotted (R)(A) Toroids
    Next - Chapter 20 - Supplements
    Back to Adventures Among the Toroids - Index

    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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