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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]