User:Tomruen/Uniform operator conversion
Table expanded a bit from [1], trying to sort out the Bowers naming system.
General (selective)[edit]
| Prefix | Coxeter-Dynkin diagram {p,q,r,s,t} |
Bowers adjectives | Johnson adjectives |
|---|---|---|---|
| t0 |       
|
regular | |
| t1 |
|
rectified | |
| t2 |
|
bi-rectified | |
| t3 |
|
tri-rectified | |
| t4 |
|
quadri-rectified | |
| t5 |
|
quinti-rectified | |
| t6 | 
|
sexti-rectified | |
| t7 | 
|
septi-rectified | |
| t8 | 
|
octi-rectified | |
| t9 | 
|
noni-rectified | |
| t10 | 
|
deci-rectified | |
| t11 | 
|
unideci-rectified | |
| t0,1 |
|
truncated | |
| t0,2 |
|
(small) rhombated | cantellated / canti- |
| t0,3 |
|
(small) prismated / prismato- | runcinated / runci- |
| t0,4 |
|
(small) cellated / celli- | stericated / steri- |
| t0,5 |
|
(small) terated / tera- | [pentellated / penti-] |
| t0,6 |
|
(small) petated / peta- | [hexicated / hexi-] |
| t0,7 |
|
(small) exated / exa- | [heptellated / hepti-] |
| t0,8 |
|
(small) zettated / zetta- | [octellated / octi-] |
| t0,9 |
|
(small) yottated / yotta- | [ennecated / enni-] |
| t1,2 |
|
bi-truncated | |
| t1,3 |
|
(small) bi-rhombated | bi-cantellated |
| t1,4 |
|
(small) bi-prismated | bi-runcinated |
| t1,5 |
|
(small) bi-cellated | bi-stericated |
| t2,3 |
|
tri-truncated | |
| t2,4 |
|
(small) tri-rhombated | tri-cantellated |
| t2,5 |
|
(small) tri-prismated | tri-runcinated |
| t3,4 |
|
quadri-truncated | |
| t3,5 |
|
(small) quadri-rhombated | quadri-cantellated |
| t4,5 |
|
quinti-truncated | |
| t0,1,2 |
|
great rhombated | cantitruncated |
| t0,1,3 |
|
prismatotruncated | runcitruncated |
| t0,1,4 |
|
cellitruncated | steritruncated |
| t0,1,5 |
|
teratruncated | pentitruncated |
| t0,2,3 |
|
prismatorhombated | runcicantellated |
| t0,2,4 |
|
(small) cellirhombated | stericantellated |
| t0,2,5 |
|
(small) terarhombated | penticantellated |
| t0,3,4 |
|
celliprismated | steriruncinated |
| t0,3,5 |
|
teraprismated | pentiruncinated |
| t0,4,5 |
|
teracellated | pentistericated |
| t1,2,3 |
|
great bi-rhombated | bi-cantitruncated |
| t1,2,4 |
|
bi-prismatotruncated | bi-runcitruncated |
| t1,2,5 |
|
bi-cellitruncated | bi-steritruncated |
| t1,3,4 |
|
bi-prismatorhombated | bi-runcicantellated |
| t1,3,5 |
|
bi-cellirhombated | bi-stericantellated |
| t1,4,5 |
|
bi-celliprismated | bi-steriruncinated |
| t2,3,4 |
|
great tri-rhombated | tri-cantitruncated |
| t2,3,5 |
|
tri-prismatotruncated | tri-runcitruncated |
| t2,4,5 |
|
tri-prismatorhombated | tri-runcicantellated |
| t3,4,5 |
|
great quadri-rhombated | quadri-cantitruncated |
| t0,1,2,3 |
|
great prismated | runcicantitruncated |
| t0,1,2,4 |
|
great cellirhombated | stericantitruncated |
| t0,1,2,5 |
|
great terarhombated | penticantitruncated |
| t0,1,3,4 | ...
|
celliprismatotruncated | steriruncitruncated |
| t0,1,3,5 |
|
teraprismatotruncated | pentiruncitruncated |
| t0,1,4,5 |
|
teracellitruncated | pentisteritruncated |
| t0,2,3,4 |
|
celliprismatorhombated | steriruncicantellated |
| t0,2,3,5 |
|
teraprismatorhombated | pentiruncicantellated |
| t0,2,4,5 |
|
(small) teracellirhombated | pentistericantellated |
| t1,2,3,4 |
|
great bi-prismated | bi-runcicantitruncated |
| t1,2,3,5 |
|
great bi-cellirhombated | bi-stericantitruncated |
| t1,2,4,5 |
|
bi-celliprismatotruncated | bi-steriruncitruncated |
| t1,3,4,5 |
|
bi-celliprismatorhombated | bi-steriruncicantellated |
| t2,3,4,5 |
|
great tri-prismated | tri-runcicantitruncated |
| t0,1,2,3,4 |
|
great cellated | steriruncicantitruncated |
| t0,1,2,3,5 |
|
great teraprismated | pentiruncicantitruncated |
| t0,1,2,4,5 |
|
great teracellirhombated | pentistericantitruncated |
| t0,1,3,4,5 |
|
teracelliprismatotruncated | pentisteriruncitruncated |
| t0,2,3,4,5 |
|
teracelliprismatorhombated | pentisteriruncicantellated |
| t1,2,3,4,5 |
|
great bi-cellated | bi-steriruncicantitruncated |
| t0,1,2,3,4,5 |
|
great terated | pentisteriruncicantitruncated |
| t0,last |
|
expanded | |
| tall |
|
omnitruncated | |
5-polytopes[edit]
| Prefix | Coxeter-Dynkin diagram {p,q,r,s} |
Bowers adjectives | Johnson adjectives |
|---|---|---|---|
| t0 |
|
regular | |
| t1 |
|
rectified | |
| t2 |
|
bi-rectified | |
| (*) t3 |
|
tri-rectified | |
| (*) t4 |
|
quadri-rectified | |
| t0,1 |
|
truncated | |
| t0,2 |
|
(small) rhombated | cantellated / canti- |
| t0,3 |
|
(small) prismated / prismato- | runcinated / runci- |
| t0,4 |
|
(small) cellated / celli- | stericated / steri- |
| t1,2 |
|
bi-truncated | |
| t1,3 |
|
(small) bi-rhombated | bi-cantellated |
| (*) t1,4 |
|
(small) bi-prismated | bi-runcinated |
| (*) t2,3 |
|
tri-truncated | |
| (*) t2,4 |
|
(small) tri-rhombated | tri-cantellated |
| (*) t3,4 |
|
quadri-truncated | |
| t0,1,2 |
|
great rhombated | cantitruncated |
| t0,1,3 |
|
prismatotruncated | runcitruncated |
| t0,1,4 |
|
cellitruncated | steritruncated |
| t0,2,3 |
|
prismatorhombated | runcicantellated |
| t0,2,4 |
|
(small) cellirhombated | stericantellated |
| (*) t0,3,4 |
|
celliprismated | steriruncinated |
| t1,2,3 |
|
great bi-rhombated | bi-cantitruncated |
| (*) t1,2,4 |
|
bi-prismatotruncated | bi-runcitruncated |
| (*) t1,3,4 |
|
bi-prismatorhombated | bi-runcicantellated |
| (*) t2,3,4 |
|
great tri-rhombated | tri-cantitruncated |
| t0,1,2,3 |
|
great prismated | runcicantitruncated |
| t0,1,2,4 |
|
great cellirhombated | stericantitruncated |
| (*) t0,1,3,4 |
|
celliprismatotruncated | steriruncitruncated |
| (*) t0,2,3,4 |
|
celliprismatorhombated | steriruncicantellated |
| (*) t1,2,3,4 |
|
great bi-prismated | bi-runcicantitruncated |
| t0,1,2,3,4 |
|
great cellated | steriruncicantitruncated |
| tall |
|
omnitruncated | |
4-polytopes[edit]
| Prefix | Coxeter-Dynkin diagram {p,q,r} |
Bowers adjectives | Johnson adjectives |
|---|---|---|---|
| t0 |
|
regular | |
| t1 |
|
rectified | |
| (*) t2 |
|
bi-rectified | |
| (*) t3 |
|
tri-rectified | |
| t0,1 |
|
truncated | |
| t0,2 |
|
(small) rhombated | cantellated / canti- |
| t0,3 |
|
(small) prismated / prismato- | runcinated / runci- |
| t1,2 |
|
bi-truncated | |
| (*) t1,3 |
|
(small) bi-rhombated | bi-cantellated |
| (*) t2,3 |
|
tri-truncated | |
| t0,1,2 |
|
great rhombated | cantitruncated |
| t0,1,3 |
|
prismatotruncated | runcitruncated |
| (*) t0,2,3 |
|
prismatorhombated | runcicantellated |
| (*) t1,2,3 |
|
great bi-rhombated | bi-cantitruncated |
| t0,1,2,3 |
|
great prismated | runcicantitruncated |
| tall |
|
omnitruncated | |
3-polytopes[edit]
| Prefix | Coxeter-Dynkin diagram {p,q} |
Bowers adjectives | Johnson adjectives |
|---|---|---|---|
| t0 |
|
regular | |
| t1 |
|
rectified | |
| (*) t2 |
|
bi-rectified | |
| t0,1 |
|
truncated | |
| t0,2 |
|
(small) rhombated | cantellated / canti- |
| (*) t1,2 |
|
bi-truncated | |
| t0,1,2 |
|
great rhombated | cantitruncated |
| tall |
|
omnitruncated | |
2-polytopes[edit]
| Prefix | Coxeter-Dynkin diagram {p} |
Bowers adjectives | Johnson adjectives |
|---|---|---|---|
| t0 |
|
regular | |
| (*) t1 |
|
rectified | |
| t0,1 |
|
truncated | |
| tall |
|
omnitruncated | |
(*) unneeded - can be made by reverse construction
4-polytopes explicit[edit]
| Prefix | t0 | t1 | t2 | t3 | t0,1 | t0,2 | t0,3 | t1,2 | t1,3 | t2,3 | t0,1,2 | t0,1,3 | t0,2,3 | t1,2,3 | t0,1,2,3 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Coxeter-Dynkin diagram {p,q,r} |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Johnson prefix | regular | rectified | bi-rectified | tri-rectified | truncated | cantellated | runcinated | bi-truncated | bi-cantellated | tri-truncated | cantitruncated | runcitruncated | runcicantellated | bi-cantitruncated | runcicantitruncated (omnitruncated) |
| Bowers prefix | regular | rectified | bi-rectified | tri-rectified | truncated | rhombated | prismated | bi-truncated | bi-rhombated | tri-truncated | great rhombated | prismatotruncated | prismatorhombated | great bi-rhombated | great prismated |
| {3,3,3} | 
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Johnson | 5-cell | rectified 5-cell | bi-rectified 5-cell | tri-rectified 5-cell | truncated 5-cell | cantellated 5-cell | runcinated 5-cell | bi-truncated 5-cell | bi-cantellated 5-cell | tri-truncated 5-cell | cantitruncated 5-cell | runcitruncated 5-cell | runcicantellated 5-cell | bi-cantitruncated 5-cell | omnitruncated 5-cell |
| Bowers | pentachoron (pen) |
rectified pentachoron (rap) |
bi-rectified pentachoron | tri-rectified pentachoron | truncated pentachoron (tip) |
rhombated pentachoron (srip) |
prismodecachoron (spid) |
decachoron (deca) |
bi-rhombated pentachoron | tri-truncated pentachoron | great rhombated pentachoron (grip) |
prismatotruncated pentachoron | prismatorhombated pentachoron (prip) |
great bi-rhombated pentachoron | great prismatodecachoron (gippid) |
| {4,3,3} | 
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Johnson | 8-cell | rectified 8-cell | bi-rectified 8-cell | tri-rectified 8-cell | truncated 8-cell | cantellated 8-cell | runcinated 8-cell | bi-truncated 8-cell | bi-cantellated 8-cell | tri-truncated 8-cell | cantitruncated 8-cell | runcitruncated 8-cell | runcicantellated 8-cell | bi-cantitruncated 8-cell | omnitruncated 8-cell |
| Bowers prefix | tesseract (tes) |
rectified tesseract (rit) |
bi-rectified tesseract rectified hexadecachoron [24-cell] |
tri-rectified tesseract hexadecachoron (hex) |
truncated tesseract (tat) |
rhombated tesseract (srit) |
prismatotesseractihexadecachoron (sidpith) |
tesseractihexadecachoron (tah) |
bi-rhombated tesseract rhombated hexadecachoron [r.24-cell] |
tri-truncated tesseract truncated hexadecachoron (thex) |
great rhombated tesseract (grit) |
prismatotruncated tesseract prismatorhombated hexadecachoron (proh) |
prismatorhombated tesseract (prit) |
great bi-rhombated tesseract great rhombated hexadecachoron [tr.24-cell] |
great prismatotesseractihexadecachoron (gidpith) |
| {3,4,3} |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Johnson | 24-cell | rectified 24-cell | bi-rectified 24-cell | tri-rectified 24-cell | truncated 24-cell | cantellated 24-cell | runcinated 24-cell | bi-truncated 24-cell | bi-cantellated 24-cell | tri-truncated 24-cell | cantitruncated 24-cell | runcitruncated 24-cell | runcicantellated 24-cell | bi-cantitruncated 24-cell | omnitruncated 24-cell |
| Bowers | icositetrachoron (ico) |
rectified icositetrachoron (rico) |
bi-rectified icositetrachoron | tri-rectified icositetrachoron | truncated icositetrachoron (tico) |
rhombated icositetrachoron (srico) |
prismotetracontaoctachoron (spic) |
tetracontaoctachoron (cont) |
bi-rhombated icositetrachoron | tri-truncated icositetrachoron | great rhombated icositetrachoron (grico) |
prismatotruncated icositetrachoron | prismatorhombated icositetrachoron (prico) |
great bi-rhombated icositetrachoron | great prismatotetracontaoctachoron (gippic) |
| {5,3,3} | 
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Johnson | 120-cell | rectified 120-cell | bi-rectified 120-cell | tri-rectified 120-cell | truncated 120-cell | cantellated 120-cell | runcinated 120-cell | bi-truncated 120-cell | bi-cantellated 120-cell | tri-truncated 120-cell | cantitruncated 120-cell | runcitruncated 120-cell | runcicantellated 120-cell | bi-cantitruncated 120-cell | omnitruncated 120-cell |
| Bowers prefix | hecatonicosachoron (hi) |
rectified hecatonicosachoron (rahi) |
bi-rectified hecatonicosachoron rectified hecatonicosachoron (rox) |
tri-rectified hecatonicosachoron hexacosichoron (ex) |
truncated hecatonicosachoron (thi) |
rhombated hecatonicosachoron (srahi) |
prismatohexacosihecatonicosachoron (sidpixhi) |
hexacosihecatonicosachoron (xhi) |
bi-rhombated hecatonicosachoron rhombated hexacosichoron (srix) |
tri-truncated hecatonicosachoron truncated hexacosichoron (tex) |
great rhombated hecatonicosachoron (grahi) |
prismatotruncated hecatonicosachoron prismatorhombated hexacosichoron (prix) |
prismatorhombated hecatonicosachoron (prahi) |
great bi-rhombated hecatonicosachoron great rhombated hexacosichoron (grix) |
great prismatohexacosihecatonicosachoron (gidpixhi) |
Johnson[edit]
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" <polyhedron@lists.mathconsult.ch>
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[edit]
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]