User:Tomruen/Uniform operator conversion

Table expanded a bit from, trying to sort out the Bowers naming system.

2-polytopes
(*) unneeded - can be made by reverse construction

Johnson
http://www.mathconsult.ch/lists/cgi/private/polyhedron/2006b/msg00273.html Higher Wythoffian operators (Johnson) a.. From: "Norman Johnson" b.. Subject: Re: [Polyhedron] Higher Wythoffian operators c.. Date: Mon, 31 Jul 2006 12:25:03 -0400 d.. To: "Polyhedron Discussion List" 

My names for the operations corresponding to ringing various nodes of a Coxeter diagram for a reflection group, thereby converting it into a Wythoff diagram for a uniform polytope or honeycomb, are as follows:

t_0 original t_1 rectified t_2 birectified t_3 trirectified t_4 quadrirectified t_5 quintirectified . . .       t_0,1  truncated t_1,2 bitruncated t_2,3 tritruncated . . .       t_0,2  cantellated t_1,3 bicantellated t_2,4 tricantellated . . .       t_0,3  runcinated t_1,4 biruncinated t_2,5 triruncinated . . .       t_0,4  stericated t_1,5 bistericated t_2,6 tristericated . . .     t_0,1,2  cantitruncated t_1,2,3 bicantitruncated t_2,3,4 tricantitruncated . . .     t_0,1,3  runcitruncated t_1,2,4 biruncitruncated t_2,3,5 triruncitruncated . . .     t_0,1,4  steritruncated t_1,2,5 bisteritruncated t_2,3,6 tristeritruncated . . .     t_0,2,3  runcicantellated t_1,3,4 biruncicantellated t_2,4,5 triruncicantellated . . .   t_0,1,2,3  runcicantitruncated t_1,2,3,4 biruncicantitruncated t_2,3,4,5 triruncicantitruncated

If only the end nodes 0 and n are ringed, the term "expanded" can be used; when all nodes are ringed, the term is "omnitruncated." It should be borne in mind that these operations apply to regular figures and others whose Wythoff diagrams have their nodes numbered from left to right.

I have not invented TOCID equivalents for n-polytopes or (n-1)- honeycombs with n > 4.

Norman

Bowers
http://www.mathconsult.ch/lists/cgi/private/polyhedron/2006b/msg00270.html Subject: Re: [Polyhedron] Higher Wythoffian operators? nth order rectification and truncation? Date: Mon, 31 Jul 2006 17:48:11 +0200 Cc: polyhedron@lists.mathconsult.ch Tom asked for a rule based naming according to Wythoff's kaleidoscopical construction, i.e. a naming scheme which translates the decoration of Dynkin diagrams into names. This was given twice within the archive. One by Jonathan Bowers, one by Norman Johnson (the ndiffering names in parantheses). In what follows this system is given with respect to 6-dimensional linear Dynkin diagrams. Obviously, the higher the dimension, the more additional operators will be needed. [+pentellated/penti]

000001 - regular 000010 - rectated 000011 - truncated 000100 - birectated 000101 - small rhombated (cantellated) 000110 - bitruncated 000111 - great rhombated (cantitruncated) 001001 - small prismated (runcinated) 001010 - small birhombated (bicantellated) 001011 - prismatotruncated (runcitruncated) 001100 - tritruncated 001101 - prismatorhombated (runcicantellated) 001110 - great birhombated (bicantitruncated) 001111 - great prismated (runcicantitruncated) 010001 - small cellated (stericated) 010010 - small biprismated (biruncinated) 010011 - cellitruncated (steritruncated) 010101 - small cellirhombated (stericantellated) 011010 - biprismatorhombated (biruncitruncated) 010111 - great cellirhombated (stericantitruncated) 011001 - celliprismated (steriruncinated) 011011 - celliprismatotruncated (steriruncitruncated) 011101 - celliprismatorhombated (steriruncicantellated) 011110 - great biprismated (biruncicantitruncated) 011111 - great cellated (steriruncicantitruncated) 100001 - small terated [pentellated] 110001 - teracellated [pentitruncated] 101001 - small teraprismated [penticantellated] 111001 - teracelliprismated [penticantitruncated] 110101 - small teracellirhombated [pentiruncitruncated] 101101 - teraprismatorhombated [pentiruncicantellated] 101111 - great teraprismated [pentiruncicantitruncated] 110011 - teracellitruncated [pentisteritruncated] 110111 - great teracellirhombated (pentistericantitruncated] 111111 - great terated [pentisteriruncicantitruncated = omnitruncated]