Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
There are 2 opposite paths which lead to closed finite flat complexes, i.e. adjoins of polytopes, which then finally are closed with respect to the dihedrality rule, thereby completely remaining within euclidean space of component dimension: On the one hand those are produced bottomup as continually growing complexes, which finally allow for a closure by means of a not completely coincident second such. Esp. (but not necessary) whenever that second one happens to be a single larger polytope. On the other hand those are derived as degenerate cases of higher dimensional polytopes, when the height between consecutive layers evaluates to zero.
The former sense can be found in the extraction of finite patches of euclidean tesselations.
The second pathway e.g. is used by degenerate segmentotopes.
Esp. in the former sense, there come in some unusual additions: besides of locally finite euclidean reflection groups there are also locally dense ones. Those are usually rejected right for that reason. But here, when using only finite patches (which somehow can be closed dihedrally for sure), all those groups come back into play! Just because of using only finite complexes therefrom, and therefore this displeasing density does not show up here already.
With respect to the dihedral angles we get the restriction, that either (in the interior) those of the incident components will add directly to 360° (as for tesselations), or (at the boarder) a subset of those component angles would add to some value, which by the angles of the complemental subset would be exhausted again. In cases of a complete decomposition the choice of that former subset is obvious.
There is neither a firm definition nor a completed research to this subject. This page rather serves as an attempt of a temporary collection.
 2D 
For locally 2D polytopal manifolds oPoQoR*a the restriction to euclidean geometry amounts to 1/P + 1/Q + 1/R = 1. For R = 2 this then just resolves to Q = 2P/(P2). Therefore we might consider all the following (linear symbol) groups:
o3o6o (locally finite) o4o4o (locally finite) o5o10/3o (locally dense) o5/2o10o (locally dense) o6o3o (locally finite) o7o14/5o (locally dense) o7/2o14/3o (locally dense) o7/3o14o (locally dense) o8o8/3o (locally dense) o9o18/7o (locally dense) o9/2o18/5o (locally dense) o9/4o18o (locally dense) ...
from locally finite groups  
o6o  o3/2o 

pt  {6} = ox6oo&#x = hippy {6}  {12} = ox6xx&#x = hicu pt  pseudo {6}  {12} = oxx6oox&#xt {6}  pseudo {12}  dual {6} = xxo6oxx&#xt 
{3/2}ap = ss3/2ss&#x = trirp 
 3D 
Several of the Johnson solids in fact are a stack (or, more general: an external blend) of elementary components. Accordingly those can serve here as a corresponding double cover too: at the one hand just that composition of components, on the other one right the resulting solid. – Similarily also some uniform or even regular ones can be decomposed using Johnson solids as (some of their) elements.
2x tet = tridpy (J12) 2x squippy (J1) = oct (regular) 2x peppy (J2) = pedpy (J13) 2x tricu (J3) = co (uniform) 2x tricu (J3) = tobcu (J27) 2x pero (J6) = id (uniform) 2x pero (J6) = pobro (J34) tet + trip = etripy (J7) squippy + cube = esquipy (J8) peppy + pip = epeppy (J9) squippy + squap = gyesp (J10) peppy + pap = gyepip (J11) peppy + mibdi (J62) = gyepip (J11) 2x peppy + teddi (J63) = gyepip (J11) 2x squippy + squap = gyesqidpy (J17) 2x peppy + pap = ike (regular) 2x peppy + mibdi (J62) = ike (regular) 3x peppy + teddi (J63) = ike (regular) 3x squippy + trip = tautip (J51) 2x squacu (J4) + op = sirco (uniform) 2x squacu (J4) + op = esquigybcu (J37) ... 
© 
A related research, digging tunnels through larger polyhedra, was published as "Adventures Among the Toroids" by Stewart. There he restricts to larger convex polyhedra, being tunneled by smaller polyhedra. Further excluding thereby intersecting faces.
(In fact, reintroducing the missing cells, which were defining all these tunnels, then would provide corresponding closed finite flat complexes again.)
A nice collection of those toroids is outlined by McNeill at this link. According VRMLs also are linked by courtesy of him. – A further one was found at this link written up by Doskey. (On that page he further classifies into linear, planar, and spherical dissections when refering to Johnsonians which can be seen as lace towers, to prisms of the 2D decompositions, resp. to the ones defined by toroids.)
In the context of toroids for historical reasons some special abbreviations are in use for building components, which (except of Steward's finds, G3 and Z4) generally are known otherwise:
© 
© 
G3  Z4 
B4 (co) 
D5 (doe) 
E4 (sirco) E5 (srid) 
G3

K3 (toe)
K4 (girco) 
P3 (trip)
P4 (cube) P6 (hip) 
Q3 (tricu)
Q4 (squacu) Q5 (pecu) 
R5 (pero) 
S3 (oct)
S5 (pap) 
T3 (tut)
T4 (tic) T5 (tid) 
Z4 
In cases of more complex interior structures the central element then is given in parantheses and counts are used as numeric prefixes for lacing structures.
Q3P6 \ P3Q3 = xxx3oxx&#xt \ xxx3oox&#xt Q3Q3 \ S3S3 = xxx3oxo&#xt \ xox3oxo&#xt E4 \ B4P4 = xxxx4oxxo&#xt \ xxox4ooqo&#xt T4 \ Q4P4Q4 = xwwx4xoox&#xt \ xxxx4xoox&#xt (esp.: tic = ext. blend of 1 cube + 8 tets + 6 squacues) Q4Q4 \ B4 = xxx4oxo&#xt \ xox4oqo&#xt K3 \ Q3T3 = xuxx3xxux&#xt \ xoox3xxux&#xt R5 \ S5Q5 = xox5ofx&#xt \ xox5oxx&#xt Q4T4Q4 \ B4P4B4 = xxooxx4oxwwxo&#xt \ xoxxox4oqooqo&#xt K3 \ 4Q3(S3) (esp.: toe = ext. blend of 1 oct + 8 tricues + 6 squippies) J91 \ Z4 K4 \ 12P4(E4) K4 \ 6Q4(E4) K4 \ 8Q3(E4) (esp.: girco = ext. blend of 1 sirco + 8 tricues + 12 cubes + 6 squacues) (more general: K4 \ aQ4 bQ3 cP4(E4)  518 members) K4 \ 2P4(Q4),2P4(Q4) K4 \ 12P4(E4 \ B4P4) E5 \ 6J91(P4) = 8G3,12Q5 (esp.: srid = ext. blend of 1 cube + 8 peppies + 12 bilbiros + 6 G3s T5 \ 12Q5S5(D5) T5 \ 6Q5S5 R5(D5) (more general: T5 \ aQ5S5 bR5(D5)) ...
o3o3o *b4o (locally finite) o5o5/2o *b3o (locally dense) ...
from locally finite groups  from locally dense groups  
o3o3o  o3o4o  o o6o  o5o5/2o  ... 

pt  co = ox3oo3ox&#x = copy oct  toe = ox3xx3ox&#x = octatoe tet  inv tut = ox3ox3xo&#x †) = tetaltut pt  oho = oβ3oo3ox&#x †) = ohopy 
pt  co = oo3ox4oo&#x = copy cube  tic = oo3ox4xx&#x = cubatic oct  toe = xx3ox4oo&#x = octatoe sirco  girco = xx3ox4xx&#x = sircoagirco 
line  hip = xx ox6oo&#x †) = hippyp hip  twip = xx ox6xx&#x †) = hicupe 
pt  did = oo5ox5/2oo&#x = didpy gad  tigid = xx5ox5/2oo&#x = gaddatigid 

†)  The ones marked such not only are degenerate segmentotopes themselves, but would ask for subdimensional degeneracies as well.
 4D 
o3o3o3o3o3*a (locally finite) o4o3o3o4o (locally finite) o3o3o4o3o (locally finite) o3o3o *b3o4o (locally finite) o5o3o3o5/2o (locally dense) o5/2o5o3o5/2o (locally dense) o5o5/2o5o5/2o (locally dense) o5o3o5/2o5o (locally dense) o3o5o5/2o3o (locally dense) o3o3o3o3o5/2*c (locally dense) o3o3o3o3/2o5*c (locally dense) ...
from locally finite groups  
o3o3o3o  o3o3o4o  o3o4o3o  

pt  spid = ox3oo3oo3ox&#x = spidpy rap  prip = ox3xx3oo3ox&#x deca  gippid = ox3xx3xx3ox&#x rap  inv tip = ox3ox3xo3oo&#x †) tip  inv prip = ox3ox3xo3xx&#x †) pen  inv srip = xo3ox3oo3ox&#x †) srip  inv grip = xo3ox3xx3ox&#x †) spid  deca = xo3ox3ox3xo&#x †) 
pt  tes = oo3oo3oo4ox&#x = tespy rit  tat = oo3oo3xx4ox&#x ico  srit = oo3xx3oo4ox&#x hex  sidpith = xx3oo3oo4ox&#x tah  grit = oo3xx3xx4ox&#x rico  proh = xx3oo3xx4ox&#x thex  prit = xx3xx3oo4ox&#x tico  gidpith = xx3xx3xx4ox&#x pt  ico = oo3ox3oo4oo&#x = icopy tes  srit = oo3ox3oo4xx&#x rit  tah = oo3ox3xx4oo&#x hex  thex = xx3ox3oo4oo&#x tat  grit = oo3ox3xx4xx&#x sidpith  prit = xx3ox3oo4xx&#x rico  tico = xx3ox3xx4oo&#x proh  gidpith = xx3ox3xx4xx&#x hex  rit = xo3oo3ox4oo&#x †) sidpith  tat = xo3oo3ox4xx&#x †) thex  tah = xo3xx3ox4oo&#x †) prit  grit = xo3xx3ox4xx&#x †) rit  thex = ox3ox3xo4oo&#x †) tat  prit = ox3ox3xo4xx&#x †) ico  rico = ox3xo3ox4oo&#x srit  proh = ox3xo3ox4xx&#x tes  sadi = os3os3os4xo&#x 
pt  ico = ox3oo4oo3oo&#x = icopy ico  spic = ox3oo4oo3xx&#x rico  gyro srico = ox3oo4xx3oo&#x rico  tico = ox3xx4oo3oo&#x tico  gyro prico = ox3oo4xx3xx&#x srico  prico = ox3xx4oo3xx&#x cont  grico = ox3xx4xx3oo&#x grico  gippic = ox3xx4xx3xx&#xx ico  rico = xo3ox4oo3oo&#x spic  srico = xo3ox4oo3xx&#x srico  cont = xo3ox4xx3oo&#x prico  gyro grico = xo3ox4xx3xx&#x ico  sadi = os3os4oo3xo&#x 

o3o3o *b3o  o o3o4o  o4o o4o  
pt  ico = oo3ox3oo *b3oo&#x = icopy hex  thex = oo3ox3oo *b3xx&#x rit  tah = oo3ox3xx *b3xx&#x rico  tico = xx3ox3xx *b3xx&#x hex  rit = ox3oo3ox *b3xo&#x †) thex  tah = ox3xx3ox *b3xo&#x †) rit  thex = xo3ox3xo *b3ox&#x †) ico  rico = ox3xo3ox *b3ox&#x 
pt  tes = ox oo3oo4ox&#x = tespy oct  sircope = ox xx3oo4ox&#x co  ticcup = ox oo3xx4ox&#x toe  gircope = ox xx3xx4ox&#x line  perp cube = xo oo3oo4ox&#x = cubasc ope  sirco = xo xx3oo4ox&#x cope  tic = xo oo3xx4ox&#x tope  girco = xo xx3xx4ox&#x 
pt  tes = ox4oo ox4oo&#x = tespy {4}  sodip = ox4oo ox4xx&#x tes  odip = ox4xx ox4xx&#x {4}  perp {4} = ox4oo xo4oo&#x {8}  tes = ox4oo xo4xx&#x sodip  ortho sodip = ox4xx xo4xx&#x 

from locally dense groups  
o3o3o5o  o5o5/2o3o  o5/2o5o3o  
hi  rahi = oo3oo3ox5xo&#x srahi  xhi = oo3xx3ox5xo&#x sidpixhi  srix = xx3oo3ox5xo&#x prahi  grix = xx3xx3ox5xo&#x rox  hi = oo3xo3oo5ox&#x xhi  thi = oo3xo3xx5ox&#x tex  sidpixhi = xx3xo3oo5ox&#x grix  prix = xx3xo3xx5ox&#x rahi  sidpixhi = ox3oo3xo5ox&#x xhi  prahi = ox3xx3xo5ox&#x rahi  tex = ox3ox3xo5oo&#x †) thi  prahi = ox3ox3xo5xx&#x †) srix  srahi = xo3ox3xo5ox&#x 
pt  gaghi = ox5oo5/2oo3oo&#x = gahipy gofix  quipdohi = ox5oo5/2oo3xx&#x rigfix  sirgaghi = ox5oo5/2xx3oo&#x ragaghi  tigaghi = ox5xx5/2oo3oo&#x tigfix  pirgaghi = ox5oo5/2xx3xx&#x gaghi  gofix = xo5oo5/2oo3ox&#x sirgaghi  tigfix = xo5oo5/2xx3ox&#x rigfix  quipdohi = ox5oo5/2xo3ox&#x 
pt  sishi = ox5/2oo5oo3oo&#x = sishipy fix  padohi = ox5/2oo5oo3xx&#x rofix  sirsashi = ox5/2oo5xx3oo&#x tiffix  pirshi = ox5/2oo5xx3xx&#x sishi  fix = xo5/2oo5oo3ox&#x sirsashi  tiffix = xo5/2oo5xx3ox&#x padohi  rofix = xo5/2oo5ox3xo&#x 

o3o3o3o5/2*b  o3o3o3/2o5*b  o o3o5o  
pt  sidtixhi = ox3oo3oo3oo5/2*b&#x = sidtixhipy rissidtixhi  tissidtixhi = ox3xx3oo3oo5/2*b&#x swavixady  sphixhi = ox3xx3xx3oo5/2*b&#x tissidtixhi  swavixady = xo3xx3ox3oo5/2*b&#x sidtixhi  rissidtixhi = xo3ox3oo3oo5/2*b&#x 
pt  gidtixhi = ox3oo3oo3/2oo5*b&#x = gidtixhipy giddatady  sadtef pixady = ox3oo3oo3/2xx5*b&#x riggidtixhi  tiggidtixhi = ox3xx3oo3/2oo5*b&#x grawvixady  tiggidtixhi = ox3xx3xo3/2oo5*b&#x gidtixhi  riggidtixhi = xo3ox3oo3/2oo5*b&#x 
doe  ipe = ox ox3oo5xo&#x tid  tipe = ox ox3xx5xo&#x id  sriddip = ox ox3xo5ox&#x ike  dope = ox xo3oo5ox&#x ti  tiddip = ox xo3xx5ox&#x iddip  srid = ox xo3ox5xo&#x 

o o5o5/2o  o5o o5o  o5o o5/2o  
sissid  gaddip = ox ox5oo5/2xo&#x sissiddip  gad = ox xo5oo5/2ox&#x 
{5}  gyro pedip = ox5oo ox5xo&#x pedip  gyro padedip = ox5xx ox5xo&#x 
pt  starpedip = ox5oo ox5/2oo&#x = starpedippy {5}  stardedip = ox5xx ox5/2oo&#x {5}  perp {5/2} = ox5oo xo5/2oo&#x {10}  starpedip = ox5xx xo5/2oo&#x 
†)  The ones marked such not only are degenerate segmentotopes themselves, but would ask for subdimensional degeneracies as well.
© 20042018  top of page 