Site Map Polytopes Dynkin Diagrams Vertex Figures, etc. Incidence Matrices Index

Temporary potpourri of incmats files, which still are to be shifted at a better place ...


Convex polyhedra with equal-sized edges but (some) non-regular faces   (up)

File Name Remarks
pacop pacop – partially contracted octagonal prism polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
mono-lower-sirco mono lowered small rhombicuboctahedron polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
para-bi-lower-tic para bi lowered truncated cube polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
mono-lower-tic mono lowered truncated cube polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
tet-lower-sirco patex sirco – partially tetrahedrally-expanded small rhombicuboctahedron,
tetrahedrally lowered small rhombicuboctahedron 		polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
4fold-contr-tic pactic – partially contracted truncated cube polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
4fold-elong-co pexco – partially elongated cuboctahedron polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
4fold-elong-rhombohedron ebauco – elongated biaugmented cuboctahedron polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
and rhombs {(r,R)2}, r = 60°, R = 120°
pextoe pextoe – partially expanded truncated octahedron polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pacgirco pac girco – partially contracted great rhomicuboctahedron polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
tet-trunc-cube chat (alt.: patex cube) – chamfered tetrahedron, tetrahedrally truncated cube,
partially tetrahedrally-expanded cube 		polyhedron with {(h,H,H)2} hexagons, h = 90°, H = 135°
oct-trunc-rad chic – chamfered cube, octahedrally truncated rad polyhedron with {(h,H,H)2} hexagons, h = 109.471221°, H = 125.264390°
cube-trunc-rad choct – chamfered oct, cubically truncated rad polyhedron with {(h,H,H)2} hexagons, h = 70.528779°, H = 144.735610°.
ike-trunc-rhote chado – chamfered doe, icosahedrally truncated rhote polyhedron with {(h,H,H)2} hexagons, h = 116.565051°, H = 121.717474°.
doe-trunc-rhote chike – chamfered ike, dodecahedrally truncated rhote polyhedron with {(h,H,H)2} hexagons, h = 63.434949°, H = 148.282526°.
bauco bi-augmented cuboctahedron polyhedron with {(r,R)2} rhombs, r = 60°, R = 120°
t32s6h12 expanded octa-augmented truncated-octehedral variant polyhedron (T.Dorozinski) with {(H,h,h)2} hexagons, h = 109.47°, H = 141.06°
12aug-sirco dodeca-augmented rhombicuboctahedron polyhedron (T.Dorozinski) with {(h,H)3} hexagons, h = 104.48°, H = 135.52°
30aug-srid triaconta-augmented rhombicosidodecahedron polyhedron (Klitzing & Dorozinski) with {(h,H)3} hexagons, h = 115.28°, H = 124.72°
t12s12p12h8 octahedrally expanded icosidodecahedron polyhedron (T.Dorozinski) with {(h,H)3} hexagons, h = 82.24°, H = 157.76°
ex12aug-girco expanded dodeca-augmented great rhombicuboctahedron polyhedron (T.Dorozinski) with {(d,d,D,D)3} dodecagons, d = 142.24°, D = 157.76°
t8r24 rhombi-propello-octahedron relaxed polyhedron (J.McNeill) with rhombs (angles?)
t20r60 rhombi-propello-icosahedron relaxed polyhedron (J.McNeill) with rhombs (angles?)
t24s6r12 rhombical octa-augmented truncated-octehedral variant polyhedron (C.Piché) with rhombs, r = 38.94°, R = 141.06°
t8s30r12 expanded rhombical dodecahedron polyhedron with rhombs, r = 70.53°, R = 109.47°
t20s60r30p12 expanded rhombical triacontahedron polyhedron with rhombs, r = 63.43°, R = 116.57°
t16r12 contracted t16s12r12h4 polyhedron (Klitzing & Dorozinski) with rhombs, r = 67.11°, R = 112.89°
t16s12r12h4 expanded t16r12 polyhedron (Klitzing & Dorozinski) with rhombs, r = 67.11°, R = 112.89°
t24s6r24 contracted t32s24r24o6 polyhedron (T. Dorozinski) with rhombs, r = 62.80°, R = 117.20°
t32s24r24o6 expanded t24s6r24 polyhedron (T Dorozinski) with rhombs, r = 62.80°, R = 117.20°
t60r60p12 contracted t80s60r60d12 polyhedron (Kaufman & McNeill) with rhombs, r = 60.96°, R = 119.04°
t80s60r60d12 expanded t60r60p12 polyhedron (Kaufman & McNeill) with rhombs, r = 60.96°, R = 119.04°
rhode (new: rad) rhombical dodecahedron (dual of co) polyhedron with rhombs, r = 70.53°, R = 109.47°
rhote (old: rattic) rhombical triacontahedron (dual of id) polyhedron with rhombs, r = 63.43°, R = 116.57°
r30+r60 rhombical enneacontahedron polyhedron with rhombs, r = 70.53°, R = 109.47°
resp. r' = 41.81°, R' = 138.19°
t20r60h30 icosa-expanded rhombical enneacontahedron polyhedron with rhombs, r = 70.53°, R = 109.47°,
and (h,h,H)2 hexagons, h = 110.91°, H = 138.19°
s60r60p12h30 ... polyhedron (T.Dorozinski) with rhombs, r = 70.53°, R = 109.47°,
and (h,H,H)2 hexagons, h = 41.81°, H = 159.09°
p12h4 unit edge variant of truncation of tut dual polyhedron (T.Dorozinski) with {(P,p,P0,p,P)} pentagons and {(h,H)3} hexagons
xofo5ofox_xt pentagonal rhombic barrel, xofo5ofox&#xt,
stack of segments from id (polar) and doe (equatorial) 		polyhedron (T.Dorozinski) with rhombs, r = 72°, R = 108°
phexdo partially hexa-expanded doe polyhedron (T.Dorozinski) with rhombs, r = 55.11°, R = 124.89°
phexik partially hexa-expanded ike polyhedron (T.Dorozinski) with {(h,H)3} hexagons, h = 97.76°, H = 142.24°

Convex polychora with equal-sized edges but (some) non-regular faces   (up)

File Name Remarks
trip=gybef trip || gybef polychoron either with corealmic cells or otherwise
using rhombs {(r,R)2}, r = 60°, R = 120°
abx3ooo3ooc4odo_zx edge-beveled hex polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
and rhombs {(r,R)2}, r = 60°, R = 120°
abo3ooo3ooc4odo_zx terminally edge-beveled hex polychoron with rhombs {(r,R)2}, r = 60°, R = 120°
pexrit partially Stott expanded rit polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pabexrit partially Stott bi-expanded rit,
partially Stott bi-contracted tat 		polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pactat partially Stott contracted tat polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pexrico partially Stott expanded rico polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pabexrico partially Stott bi-expanded rico,
partially Stott bi-contracted proh 		polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pacproh partially Stott contracted proh polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pextah partially Stott expanded tah polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pabextah partially Stott bi-expanded tah,
partially Stott bi-contracted grit 		polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pacgrit partially Stott contracted grit polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pextico partially Stott expanded tico polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pabextico partially Stott bi-expanded tico,
partially Stott bi-contracted gidpith 		polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°
pac gidpith partially Stott contracted gidpith polychoron with {(h,H,H)2} hexagons, h = 90°, H = 135°

Honeycombs with equal-sized edges and convex cells but (some) non-regular faces   (up)

File Name Remarks
radh rhombic-dodecahedral honeycomb honeycomb with rhombs, r = 70.53°, R = 109.47°
extoh expanded octahedral-tetrahedral honeycomb honeycomb with rhombs, r = 70.53°, R = 109.47°
atich alternatedly truncated cubical honeycomb honeycomb with non-regular hexagons {(h,H,H)2}, h = 90°, H = 135°
octet-wise-trunc-radh alternated-cubically truncated rhombidodecahedral honeycomb honeycomb with non-regular hexagons {(h,H,H)2}, h = 109.471221°, H = 125.264390°
chon-wise-trunc-radh cubically truncated rhombidodecahedral honeycomb honeycomb with non-regular hexagons {(h,H,H)2}, h = 70.528779°, H = 144.735610°

Some non-self-intersecting, though concave polyhedra with regular faces (nsiCvRF)   (up)

File Name Remarks
xfo3foo5oxf_zx 20 doe-dimples + 12 id-dimples dihedral angle between {5a} and {5a}:   360°-arccos(-1/sqrt(5)) = 243.434949°
dihedral angle between {3} and {5b}:   360°-arccos(-sqrt[(5+2 sqrt(5))/15]) = 217.377°
xfo3foo5xuFx_zx 20 doe-dimples + 12 ti-dimples dihedral angle between {5a} and {5a}:   360°-arccos(-1/sqrt(5)) = 243.434949°
dihedral angle between {6} and {6}:   360°-arccos(-sqrt(5)/3) = 221.810315°
dihedral angle between {5b} and {6}:   360°-arccos(-sqrt[(5+2 sqrt(5))/15]) = 217.377368°
xfoE3fooo5xuFx_zxt s30-p72-h60 tetraform (E = 2f-x = sqrt(5), F = f+x = ff)
36t triangle triacontahexahedron triform and isohedral
120t triangle hecatonicosahedron triform and isohedral

Some non-self-intersecting, though concave polychora with regular faces (nsiCvRF)   (up)

File Name Remarks
{3} || gybef bistratic stack of traf and trippy,
featuring gybefs 		dihedral angle at {3} between tet and tet:   360°-arccos(-7/8) = 208.955024°
pautpen penta-augmented truncated pentachoron dihedral angle at {6} between tricu and tricu:   360°-arccos(-11/16) = 226.567463°
spysp small pyramidic swirlprism dihedral angle at {5} between peppy and peppy:   216°
hi-120ikadoes   hi dimpled in by 120 ikadoes dihedral angle at {3} between ike and tet:   360°-arccos[-sqrt(5/8)] = 217.761244°,
one type of those at {3} between tet and tet:   360°-arccos[-(1+3 sqrt(5))/8] = 195.522488°

Some self-intersecting concave polyhedra with regular faces   (up)

File Name Remarks
sidtid-0-3-3-3 reduced( ofx3/2oxx&#xt, by x3/2x ) sidtid edge faceting
tuhip 2-hip-blend toroidal

Some self-intersecting concave polychora with regular faces   (up)

File Name Remarks
reduced_ofx32oxx3ooo_xt reduced( ofx3/2oxx3ooo&#xt, by 2tet ) orbiform
reduced_ofx32oxx3xxx_xt reduced( ofx3/2oxx3xxx&#xt, by 4x {6/2} ) orbiform
reduced_ox32xx4oo_x cuboctahedral retro-cuploid cuploid
hocucup "coord-planes squares star" || "coord-planes squares star" alterprism
hossdap narrower pseudo sissid || pseudo sissid alterprism

Some non-self-intersecting, though concave polyhedra with (some) non-regular faces   (up)

File Name Remarks
xfoa3fooo5oxfo_zx 41st stellation of ti polyhedron with rhombs, r = 60°, R = 120°
(where a = sqrt(5) = 2.236068)

Some convex polyhedra with different edge sizes   (up)

File Name Remarks
m m3o co oo3ox&#zy, tridpy variant,
pen derived,
dual of trip 		using edge sizes x and y = 2/3
cell of o3m3o3o
co2oo3ox_zy-o4m3o3o co oo3ox&#zy, tridpy variant,
tes derived 		using edge sizes x and y = sqrt(7/18) = 0.623610
cell of o4m3o3o
co2oo3ox_zy-o5m3o3o co oo3ox&#zy, tridpy variant,
hi derived 		using edge sizes x and y = sqrt[(9-sqrt(5))/18] = 0.613004
cell of o5m3o3o
co2oo3ox_zy-o3m4o3o co oo3ox&#zy, tridpy variant,
ico derived 		using edge sizes x and y = sqrt(5)/3 = 0.745356
cell of o3m4o3o
oct qo oo4ox&#zx,
hex derived 		using edge sizes x only (Wythoffian)
cell of o3m3o4o
co2oo5ox_zy-o3m3o5o co oo3ox&#zy, pedpy variant,
ex derived 		using edge sizes x and y = sqrt[(6+2 sqrt(5))/5] = 1.447214
cell of o3m3o5o
oqo3coc_xt oqo3coc&#xt, rectified/ambified trip (retrip),
truncated tridpy variant,
o2o3o symmetric co relative 		using edge sizes x and c = 1/sqrt(2) = 0.707107,
relates to pen edges
obo3coc_xt-ico oqo3coc&#xt, truncated tridpy variant,
o2o3o symmetric co relative 		using edge sizes x and c = b/2 = 1/sqrt(3) = 0.577350,
relates to ico edges
obo3coc_xt-tes oqo3coc&#xt, truncated tridpy variant,
o2o3o symmetric co relative 		using edge sizes x and c = b/2 = sqrt(2/3) = 0.816497,
relates to tes edges
obo3coc_xt-hi oqo3coc&#xt, truncated tridpy variant,
o2o3o symmetric co relative 		using edge sizes x and c = b/2 = sqrt[(5+sqrt(5))/10] = 0.850651,
relates to hi edges
co oqo4xox&#xt using edge sizes x only (Wythoffian),
relates to hex edges
oqo5coc_xt oqo5coc&#xt, truncated pedpy variant,
o2o5o symmetric co relative 		using edge sizes x and c = (1+sqrt(5))/sqrt(8) = 1.144123,
relates to ex edges
vov3ofx_xt vov3ofx&#xt, axially trigonal variant of pentagonal rotunda using edge sizes x and v = (sqrt(5)-1)/2 = 0.618034
tet-dim-doe tetrahedrally-diminished dodecahedron using edge sizes x and f = (1+sqrt(5))/2 = 1.618034
cube-dim-doe oxF xFo Fox&#zf, cubically-diminished dodecahedron,
pyritohedrally symmetric variant of ike 		using edge sizes x and f = (1+sqrt(5))/2 = 1.618034
cao2aoc2oca_zd cao aoc oca&#zd, pyritohedral ike variant
(a < c results in c being pseudo) 		using edge sizes a and d = sqrt[(a2-ac+c2)/2]
oxqxo8ooooo&#qt octagonal Leonardo style "polyhedron of renaissance" using edge sizes x and q = sqrt(2) = 1.414214
24t6s8n xA3Bo4oC&#zx, near miss Johnson solid with enneagons using edge sizes x and C = 1.049668
pystid VooFxfu oVofFxu ooVxfFu&#z(x,v),
pyritohedrally symmetric convex hull of id and u-cube 		using edge sizes x and v = (sqrt(5)-1)/2 = 0.618034
oqo3ooq_xt cubera, Dan Moore's self-dual cube faceting using edge sizes x and q = sqrt(2) = 1.414214
hiktut xo3xo3oy&#z, hexakis truncatet tetrahedron using edge sizes x and z = sqrt[11-sqrt(33)]/2 = 1.146237
tepdid xfFo3oxox&#(x,f)t, tripentadiminished icosidodecahedron using edge sizes x and f = (1+sqrt(5))/2 = 1.618034
xo3ox2qo3oq_zx cyclo-hexagonally diminished ico using edge sizes x and q = sqrt(2) = 1.414214
ambo-tut retut, ambified tut using edge sizes x and h = sqrt(3) = 1.732051
ambo-tic retic, ambified tic using edge sizes x and k = sqrt[2+sqrt(2)] = 1.847759
ambo-toe retoe, ambified toe using edge sizes x and b = sqrt(3/2) = 1.224745
trunc-tut dittet, truncated tut using edge sizes x, h = sqrt(3) = 1.732051,
and arbitrary y (expansion parameter from ambification)
trunc-tic dittec, truncated tic using edge sizes x, k = sqrt[2+sqrt(2)] = 1.847759,
and arbitrary y (expansion parameter from ambification)
trunc-trip truncated trip using edge sizes x, q = sqrt(2) = 1.414214,
and arbitrary y (expansion parameter from ambification)
rectangular trapezoprism ab ba&#zc using edge sizes a, b, and c > |b-a|/sqrt(2)
id-faceting faces: x2f, f5/2o, and retrograde x3o using edge sizes x, f
uoo2oux_zqhq line atop dual square using edge sizes x, q, h
dohany Dohány or Tabakgasse polyhedron using edge sizes x and y = sqrt(2/3) = 0.816497

Further polyhedra with different edge sizes occur naturally either within the investigation of Waterman polyhedra, Catalan solids, or as cells of their 4D counterparts. Some more could be found at the page for isogonal polytopes.


Some convex polychora with different edge sizes   (up)

File Name Remarks
oq3oo3qo3oc_zx pentachoron-derived Gévay polychoron oq3oo3qo3oc&#zx perfect polychoron, using edge sizes x and c = 1/sqrt(2) = 0.707107
rico hexadecachoron-derived Gévay polychoron oq3oo3qo4ox&#zx perfect polychoron, using edge size x only (Wythoffian)
oq3oo3qo5oc_zx hexacosachoron-derived Gévay polychoron oq3oo3qo5oc&#zx perfect polychoron, using edge sizes x and c = (1+sqrt(5))/sqrt(8) = 1.144123
oa3oo4bo3oc_zx icositetrachoron-derived Gévay polychoron oa3oo4bo3oc&#zx perfect polychoron, using edge sizes x and c = 1/sqrt(3) = 0.577350
oa4oo3bo3oc_zx tesseract-derived Gévay polychoron oa4oo3bo3oc&#zx perfect polychoron, using edge sizes x and c = sqrt(2/3) = 0.816497
oa5oo3bo3oc_zx hecatonicosachoron-derived Gévay polychoron oa4oo3bo3oc&#zx perfect polychoron, using edge sizes x and c = sqrt[(5+sqrt(5))/10] = 0.850651
xuo3uoo3oou3oux_zqqh decachoron-derived Gévay polychoron xuo3uoo3oou3oux&#z(q,q,h) perfect polychoron, using edge sizes x, q = sqrt(2) = 1.414214, and h = sqrt(3) = 1.732051
aco3boo4oob3oca_zxxd tetracontoctachoron-derived Gévay polychoron aco3boo4oob3oca&#z(x,x,d) perfect polychoron, using edge sizes x, a = (sqrt(8)-1)/sqrt(3) = 1.055643, and d = sqrt[(6-sqrt(2))/3] = 1.236364
doe-rico rectified icositetrachoron-derived polychoron with dodecahedra using edge sizes x and v = (sqrt(5)-1)/2 = 0.618034
ooxf3xfox3oxFx&#xt - using edge sizes x and f = (1+sqrt(5))/2 = 1.618034
lace tower with tetrahedral across symmetry,
using f sized edges only in the final layer
4d-corner-hypercubera Dan Moore's self-dual tes faceting using edge sizes x, q = sqrt(2) = 1.414214, and h = sqrt(3) = 1.732051
rect-deca redeca, rectified decachoron xo3od3do3ox&#zh using edge sizes x and h = sqrt(3) = 1.732051
rect-spid respid, rectified small prismated decachoron uo3ox3xo3ou&#zq using edge sizes x and q = sqrt(2) = 1.414214
rect-cont recont, rectified tetracontoctachoron xo3oK4Ko3ox&#zk using edge sizes x and k = sqrt[2+sqrt(2)] = 1.847759
rect-spic respic, rectified small prismated tetracontoctachoron uo3ox4xo3ou&#zq using edge sizes x and q = sqrt(2) = 1.414214
retriddip retdip, rectified triddip uo3ox xo3ou&#zq
 using edge sizes x and q = sqrt(2) = 1.414214
retepe retepe, rectified/ambified tepe uo3ox3oo ou&#zq
 using edge sizes x and q = sqrt(2) = 1.414214
trunc-deca tadeca, truncated deca xo3yb3by3ox&#zh
(using pseudo edge b=y+3) 		using edge sizes x, h = sqrt(3) = 1.732051,
and arbitrary y (expansion parameter from rectification)
trunc-cont ticont, truncated cont xo3yb4by3ox&#zk
(using pseudo edge b=y+2+sqrt(2)) 		using edge sizes x, k = sqrt[2+sqrt(2)] = 1.847759,
and arbitrary y (expansion parameter from rectification)
trunc-spid tispid, truncated spid by3ox3xo3yb&#zq
(using pseudo edge b=y+2) 		using edge sizes x, q = sqrt(2) = 1.414214,
and arbitrary y (expansion parameter from rectification)
trunc-spic tispic, truncated spic by3ox4xo3yb&#zq
(using pseudo edge b=y+2) 		using edge sizes x, q = sqrt(2) = 1.414214,
and arbitrary y (expansion parameter from rectification)
trunc-triddip tatriddip, truncated triddip by3ox xo3yb&#zq
(using pseudo edge b=y+2) 		using edge sizes x, q = sqrt(2) = 1.414214,
and arbitrary y (expansion parameter from rectification)
titepe titepe, truncated tepe by3ox3oo yb&#zq
(using pseudo edge b=y+2) 		using edge sizes x, q = sqrt(2) = 1.414214,
and arbitrary y (expansion parameter from rectification)
biambo-deca triangle based biambodecachoron oo3xo3ox3oo&#zy using edge sizes x and y = sqrt(2/5) = 0.632456
biambo-cont triangle based biambotetracontoctachoron oo3xo4ox3oo&#zy using edge sizes x and y = 2-sqrt(2) = 0.585786
ab3oo2ba3oo_zc triangular ditetragoltriate ab3oo ba3oo&#zc using edge sizes a, b, and c = (b-a) sqrt(2/3)
ab3ob2ba3bo_zc triangular duoexpandoprism ab3ob ba3bo&#zc using edge sizes a, b, and 3c2 = 2a2-6ab+6b2
spidrico swirlprismatodiminished rectified icositetrachoron using edge sizes x and q = sqrt(2) = 1.414214
bhidtex bi-hecatonicosidiminished truncated hexacosichoron using edge sizes x and u = 2
xo3uo3ou3ox_zh tegum sum of 2 mut. inv. (x,u)-tips using edge sizes x, h = sqrt(3) = 1.732051, and u = 2
tedope tetra(hedrally) diminished ope using edge sizes x and q = sqrt(2) = 1.414214

Further polychora with different edge sizes occur naturally within the investigation of the 4D counterparts of the Catalan solids. Some more could be found at the page for isogonal polytopes.


Some convex polytera with different edge sizes   (up)

File Name Remarks
rescad rescad, rectified scad uo3ox3oo3xo3ou&#zq using edge sizes x and q = sqrt(2) = 1.414214
tiscad tiscad, truncated scad by3ox3oo3xoyb&#zq
(using pseudo edge b=y+2) 		using edge sizes x, q = sqrt(2) = 1.414214,
and arbitrary y (expansion parameter from rectification)

Some self-intersecting concave polyhedra with non-regular faces   (up)

File Name Remarks
tithah truncation of thah polyhedron with (x,q)-bowties
as well as x4q
12S-24R-6O faceting of quitco polyhedron with (h,w)-rectangles

Polyhedra of higher genus   (up)

File Name Remarks
4_5__19 {4,5;19}, i.e. combinatorically regular {4,5} of genus 19 polyhedral relization using rhombs of 2 sizes and trapezia
4_5__31 {4,5;31}, i.e. combinatorically regular {4,5} of genus 31 polyhedral relization using 4 types of trapezia and squares of 2 sizes
5_4__13 {5,4;13}, i.e. combinatorically regular {5,4} of genus 13 polyhedral relization using 4 types of pentagons
toroidal-tiler toroidal tiler of 3d space faces are 4 x-{4} and 8 (x-h/2-u-h/2)-trapeziums
24aab_toroid E. Pegg's 24 face equihedral toroid using edge sizes a = 1.398966 and b = 1
32aab_toroid E. Pegg's 32 face equihedral toroid using edge sizes a = 1.157493 and b = 1


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