Can anyone explain TOCID symbols?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Can anyone explain TOCID symbols?

Postby ubersketch » Fri Feb 16, 2018 4:14 pm

I haven't found a guide anywhere.
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Re: Can anyone explain TOCID symbols?

Postby Klitzing » Fri Feb 16, 2018 5:59 pm

TOCID is just the abbreviation for Tetrahedron, Octahedron, Cube, Icosahedron, and Dodecahedron.

Norman Johnson at Wed, 07 Jun 2006 provided a detailed listing of these symbols and their usage into a private polyhedron list archive.
There he wrote:
Code: Select all
                    UNIFORM POLYHEDRA [AND DUALS]


The table below lists all uniform polyhedra, classified by type.
For each polyhedron is given its figure number in Polyhedron Models,
its modified Wythoff symbol, its TOCID symbol, its name, and [in
brackets] the name of the dual co-uniform polyhedron, if any.  Some
polyhedra can be obtained in more than one way and so appear more
than once in the table.


Regular polyhedra:  Vertex type  p^q

   1     {2 3}(3)        T  = Tetrahedron
                                [Tetrahedron]
   2     {2 3}(4)        O  = Octahedron
                                [Cube]
   3     {2 4}(3)        C  = Cube
                                [Octahedron]
   4     {2 3}(5)        I  = Icosahedron
                                [Dodecahedron]
   5     {2 5}(3)        D  = Dodecahedron
                                [Icosahedron]
  20    {2 5/2}(5)       D* = Small stellated dodecahedron
                                [Great dodecahedron]
  21    {2 5}(5/2)       E  = Great dodecahedron
                                [Small stellated dodecahedron]
  22    {2 5/2}(3)       E* = Great stellated dodecahedron
                                [Great icosahedron]
  41    {2 3}(5/2)       J  = Great icosahedron
                                [Great stellated dodecahedron]


Quasi-regular polyhedra:  Vertex type  (p.q)^r

   2     {3 3}(2)       TT  = Tetratetrahedron = octahedron
                                [Rhombic hexahedron = cube]
  11     {3 4}(2)       CO  = Cuboctahedron
                                [Rhombic dodecahedron]
  12     {3 5}(2)       ID  = Icosidodecahedron
                                [Rhombic triacontahedron]
  73    {5/2 5}(2)      ED* = Dodecadodecahedron
                                [Midly rhombic triacontahedron]
  94    {3 5/2}(2)      JE* = Great icosidodecahedron
                                [Great rhombic triacontahedron]
  70    {3 5/2}(3)      ID* = Small ditrigonary icosidodecahedron
                                [Small triambic icosahedron]
  80    {5/3 5}(3)      DE* = Ditrigonary dodecadodecahedron
                                [Midly triambic icosahedron]
  87    {3 5}(3/2)      JE  = Great ditrigonary icosidodecahedron
                                [Great triambic icosahedron]


Versi-regular polyhedra:  Vertex type  q.h.q.h

  67    [2]{3/2 3}     T|T  = Tetrahemihexahedron
                                [no dual]
  78    [3]{4/3 4}     C|O  = Cubohemioctahedron
                                [no dual]
  68    [3]{3/2 3}     O|C  = Octahemioctahedron
                                [no dual]
  91    [5]{5/4 5}     D|I  = Small dodecahemidodecahedron
                                [no dual]
  89    [5]{3/2 3}     I|D  = Small icosahemidodecahedron
                                [no dual]
102    [3]{5/4 5}     E|D* = Small dodecahemiicosahedron
                                [no dual]
100   [3]{5/3 5/2}   D*|E  = Great dodecahemiicosahedron
                                [no dual]
106   [5/3]{3/2 3}    J|E* = Great icosahemidodecahedron
                                [no dual]
107  [5/3]{5/3 5/2}  E*|J  = Great dodecahemidodecahedron
                                [no dual]


Tomo-regular polyhedra:  Vertex type  q.2p.2p

   6     [3]{3 2}       tT  = Truncated tetrahedron
                                [Triakis tetrahedron]
   7     [3]{4 2}       tO  = Truncated octahedron
                                [Tetrakis hexahedron]
   8     [4]{3 2}       tC  = Truncated cube
                                [Triakis octahedron]
  92    [4/3]{3 2}      tC* = Stellatruncated cube
                                [Great triakis octahedron]
   9     [3]{5 2}       tI  = Truncated icosahedron
                                [Pentakis dodecahedron]
  10     [5]{3 2}       tD  = Truncated dodecahedron
                                [Triakis icosahedron]
  97    [5/3]{5 2}      tD* = Small stellatruncated dodecahedron
                                [Great pentakis dodecahedron]
  75    [5]{5/2 2}      tE  = Great truncated dodecahedron
                                [Small astropentakis dodecahedron]
104    [5/3]{3 2}      tE* = Great stellatruncated dodecahedron
                                [Great triakis icosahedron]
  95    [3]{5/2 2}      tJ  = Great truncated icosahedron
                                [Great astropentakis dodecahedron]


Simo-regular polyhedra:  Vertex type  p.3.p.3.3  or  p.3.p.3.q/2.3

   4     {3}|4 2|       sO  = Snub octahedron
                                   = (small) icosahedron
                                [Petaloid dodecahedron
                                   = (small) dodecahedron]
  41    {3/2}|4 2|     s*O  = Retrosnub octahedron
                                   = great icosahedron
                                [Astroid dodecahedron
                                   = great stellated dodecahedron]
110   {\3\}||5 2||    ssI  = Holosnub icosahedron
                                   = snub disicosidodecahedron
                                [no dual]
118  {\3/2\}||5 2||  ss*I  = Retroholosnub icosahedron
                                   = retrosnub disicosidodecahedron
                                [no dual]


Quasi-quasi-regular polyhedra:  Vertex type  p.2r.q.2r  or  p.2s.q.2s

  11     [2]{3 3}      rTT  = Rhombitetratetrahedron
                                   = cuboctahedron
                                [Lanceal dihexahedron
                                   = rhombic dodecahedron]
  68    [3]{3/2 3}     aTT  = Allelotetratetrahedron
                                   = octahemioctahedron
                                [no dual]
  13     [2]{3 4}      rCO  = (Small) rhombicuboctahedron
                                [(Small) lanceal disdodecahedron]
  69    [4]{3/2 4}     bCO  = Small cubicuboctahedron
                                [Small sagittal disdodecahedron]
  77    [4/3]{3 4}     cOC* = Great cubicuboctahedron
                                [Great lanceal disdodecahedron]
  85    [2]{3/2 4}     rOC* = Great rhombicuboctahedron
                                [Great sagittal disdodecahedron]
  14     [2]{3 5}      rID  = (Small) rhombicosidodecahedron
                                [(Small) lanceal ditriacontahedron]
  72    [5]{3/2 5}     dID  = Small dodekicosidodecahedron
                                [Small sagittal ditriacontahedron]
  71    [3]{3 5/2}     iID* = Small icosified icosidodecahedron
                                [Small lanceal trisicosahedron]
  82    [5]{3 5/3}     dID* = Small dodekified icosidodecahedron
                                [Small sagittal trisicosahedron]
  76    [2]{5/2 5}     rED* = Rhombidodecadodecahedron
                                [Midly lanceal ditriacontahedron]
  83    [3]{5/3 5}     iED* = Icosified dodecadodecahedron
                                [Midly sagittal ditriacontahedron]
  81    [5/3]{3 5}     eJE  = Great dodekified icosidodecahedron
                                [Great lanceal trisicosahedron]
  88    [3]{3/2 5}     iJE  = Great icosified icosidodecahedron
                                [Great sagittal trisicosahedron]
  99   [5/3]{3 5/2}    eJE* = Great dodekicosidodecahedron
                                [Great lanceal ditriacontahedron]
105    [2]{3 5/3}     rJE* = Great rhombicosidodecahedron
                                [Great sagittal ditriacontahedron]


Versi-quasi-regular polyhedra:  Vertex type  2r.2s.2r.2s

  78   3/2[2 3]3/2   ra|TT  = Rhomballelohedron
                                   = cubohemioctahedron
                                [no dual]
  86   3/2[2 4]4/2   rb|CO  = Small rhombicube
                                [Small dipteral disdodecahedron]
103  3/2[2 4/3]4/2  rc|OC* = Great rhombicube
                                [Great dipteral disdodecahedron]
  74   3/2[2 5]5/2   rd|ID  = Small rhombidodecahedron
                                [Small dipteral ditriacontahedron]
  90   3/2[3 5]5/4   di|ID* = Small dodekicosahedron
                                [Small dipteral trisicosahedron]
  96   5/4[2 3]5/2   ri|ED* = Rhombicosahedron
                                [Midly dipteral ditriacontahedron]
101  3/2[3 5/3]5/2  ei|JE  = Great dodekicosahedron
                                [Great dipteral trisicosahedron]
109  3/2[2 5/3]5/4  re|JE* = Great rhombidodecahedron
                                [Great dipteral ditriacontahedron]


Tomo-quasi-regular polyhedra:  Vertex type  2p.2q.2r

   7     [2 3 3]       tTT  = Truncated tetratetrahedron
                                   = truncated octahedron
                                [Disdyakis hexahedron
                                   = tetrakis hexahedron]
  15     [2 3 4]       tCO  = Truncated cuboctahedron
                                [(Small) disdyakis dodecahedron]
  93    [2 3 4/3]      tOC* = Stellatruncated cuboctahedron
                                [Great disdyakis dodecahedron]
  79    [3 4/3 4]     tCOC* = Cubitruncated cuboctahedron
                                [Trisdyakis octahedron]
  16     [2 3 5]       tID  = Truncated icosidodecahedron
                                [(Small) disdyakis triacontahedron]
  98    [2 5/3 5]      tED* = Stellatruncated dodecadodecahedron
                                [Midly disdyakis triacontahedron]
  84    [3 5/3 5]     tIDE* = Icositruncated dodecadodecahedron
                                [Trisdyakis icosahedron]
108    [2 3 5/3]      tJE* = Stellatruncated icosidodecahedron
                                [Great disdyakis triacontahedron]


Simo-quasi-regular polyhedra:  Vertex type  p.3.q.3.3  or  p.3.q.3.r.3

   4     {2 3 3}       sTT  = Snub tetratetrahedron
                                   = (small)icosahedron
                                [Petaloid dihexahedron
                                   = (small) dodecahedron]
  41   {2 3/2 3/2}    s*TT  = Retrosnub tetratetrahedron
                                   = great icosahedron
                                [Astroid dihexahedron
                                   = great stellated dodecahedron]
  17     {2 3 4}       sCO  = Snub cuboctahedron
                                [Petaloid disdodecahedron]
  18     {2 3 5}       sID  = Snub icosidodecahedron
                                [(Small) petaloid ditriacontahedron]
110    {3 3 5/2}     sIID* = Snub disicosidodecahedron
                                [no dual]
118  {3/2 3/2 5/2}  s*IID* = Retrosnub disicosidodecahedron
                                [no dual]
111    {2 5/2 5}      sED* = Snub dodecadodecahedron
                                [Midly petaloid ditriacontahedron]
114    {2 5/3 5}     s'ED* = Vertisnub dodecadodecahedron
                                [Midly dentoid ditriacontahedron]
112    {3 5/3 5}     sIDE* = Snub icosidodecadodecahedron
                                [Petaloidal trisicosahedron]
113    {2 3 5/2}      sJE* = Great snub icosidodecahedron
                                [Great petaloid ditriacontahedron]
116    {2 3 5/3}     s'JE* = Great vertisnub icosidodecahedron
                                [Great dentoid ditriacontahedron]
117   {2 3/2 5/3}    s*JE* = Great retrosnub icosidodecahedron
                                [Great astroid ditriacontahedron]
115   {5/3 3 5/2}   sE*JE* = Great snub dodekicosidodecahedron
                                [no dual]


Snub quasi-regular polyhedron:  Vertex type  (p.4.q.4)^2

119 {3/2 5/3 3 5/2} s(JE*)^2 = Great snub disicosidisdodecahedron
                                  [no dual]


Prisms:  Vertex type  p.4.4

       [2]{n/d 2}   (n/d)P  = (d-fold) n-gonal prism
                                [(d-fold) n-gonal fusil]
                                   (n/d > 2)
       [2 2 n/d]   t(n/d)P  = (d-fold) 2n-gonal prism
                                [(d-fold) 2n-gonal fusil]
                                   (d odd, n/d > 1)


Antiprisms:  Vertex type  p.3.3.3

       {2 2 n/d}    (n/d)Q  = (d-fold) n-gonal antiprism
                                [(d-fold) n-gonal antifusil]
                                   (n/d > 2)
     {2 2 n/(n-d)}  (n/d)R  = d-fold n-gonal crossed antiprism
                                [d-fold n-gonal concave antifusil]
                                   (2 < n/d < 3)

Therefrom I for one learned of those. But the provided formatting of that enlisting looks like he just pasted and copied that from an even older source. None the less, I'm sure at least, that those were his own creations, because he insisted on that.

--- rk
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Re: Can anyone explain TOCID symbols?

Postby Mercurial, the Spectre » Sat Feb 17, 2018 2:06 pm

Yeah, what Klitzing told you.
In fact, the Conway polyhedron operator (you can make your own polyhedron using polyhedronisme at github) utilizes these symbols and adds in operators such as t for truncate, r for rectify, d for dual, s for snub, and so on. This means that a truncated tetrahedron is tT, the snub cube is sC, and the icosahedron is dD or sT (dual of dodecahedron and snub tetrahedron, respectively).
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