Site Map | Polytopes | Dynkin Diagrams | Vertex Figures, etc. | Incidence Matrices | Index |
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 |
patex cube – partially tetrahedrally-expanded cube, tetrahedrally truncated cube, chamfered tetrahedron | polyhedron with {(h,H,H)^{2}} hexagons, h = 90°, H = 135° |
oct-trunc-rad | octahedrally truncated rad, chamfered cube | polyhedron with {(h,H,H)^{2}} hexagons, h = 109.471221°, H = 125.264390° |
cube-trunc-rad | cubically truncated rad, chamfered oct | polyhedron with {(h,H,H)^{2}} hexagons, h = 70.528779°, H = 144.735610°. |
ike-trunc-rhote | icosahedrally truncated rhote, chamfered doe | polyhedron with {(h,H,H)^{2}} hexagons, h = 116.565051°, H = 121.717474°. |
doe-trunc-rhote | dodecahedrally truncated rhote, chamfered ike | 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° |
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° |
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° |
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,P_{0},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° |
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 | chamfered hex |
polychoron with {(h,H,H)^{2}} hexagons, h = 90°, H = 135° and rhombs {(r,R)^{2}}, r = 60°, R = 120° |
abo3ooo3ooc4odo_zx | terminally chamfered 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° |
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° |
File | Name | Remarks |
xfo3foo5oxf_zx | 20 doe-dimples + 12 id-dimples |
dihedral angle between {5_{a}} and {5_{a}}: 360°-arccos(-1/sqrt(5)) = 243.434949° dihedral angle between {3} and {5_{b}}: 360°-arccos(-sqrt[(5+2 sqrt(5))/15]) = 217.377° |
xfo3foo5xuFx_zx | 20 doe-dimples + 12 ti-dimples |
dihedral angle between {5_{a}} and {5_{a}}: 360°-arccos(-1/sqrt(5)) = 243.434949° dihedral angle between {6} and {6}: 360°-arccos(-sqrt(5)/3) = 221.810315° dihedral angle between {5_{b}} and {6}: 360°-arccos(-sqrt[(5+2 sqrt(5))/15]) = 217.377368° |
36t | triangle triacontahexahedron | triform and isohedral |
120t | triangle hecatonicosahedron | triform and isohedral |
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° |
File | Name | Remarks |
sidtid-0-3-3-3 | reduced( ofx3/2oxx&#xt, by x3/2x ) | sidtid edge faceting |
File | Name | Remarks |
reduced_ofx32oxx3ooo_xt | reduced( ofx3/2oxx3ooo&#xt, by 2tet ) | 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 |
File | Name | Remarks |
xfoa3fooo5oxfo_zx | 41st stellation of ti |
polyhedron with rhombs, r = 60°, R = 120° (where a = sqrt(5) = 2.236068) |
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 trip, 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[(a^{2}-ac+c^{2})/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, ambiated tut | using edge sizes x and h = sqrt(3) = 1.732051 |
ambo-tic | retic, ambiated tic | using edge sizes x and k = sqrt[2+sqrt(2)] = 1.847759 |
ambo-toe | retoe, ambiated 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 ambiation) |
trunc-tic | dittec, truncated tic |
using edge sizes x, k = sqrt[2+sqrt(2)] = 1.847759, and arbitrary y (expansion parameter from ambiation) |
trunc-trip | truncated trip |
using edge sizes x, q = sqrt(2) = 1.414214, and arbitrary y (expansion parameter from ambiation) |
rectangular trapezoprism | ab ba&#zc | using edge sizes a, b, and c > |b-a|/sqrt(2) |
Further polyhedra with different edge sizes occur naturally either within the investigation of Catalan solids or as cells of their 4D counterparts. Some more could be found at the page for isogonal polytopes.
File | Name | Remarks |
o3m3o3o |
dual of rectified pentachoron, polychoron co3oo3oo3ox&#zy |
Catalan polychoron, using edge sizes x and y = 2/3 (pseudo edge size c = 2/3) |
o4m3o3o |
dual of rectified tesseract, polychoron co4oo3oo3ox&#zy |
Catalan polychoron, using edge sizes x and y = sqrt(7/18) = 0.623610 (pseudo edge size c = sqrt(2)/3 = 0.471405) |
o5m3o3o |
dual of rectified hecatonicosachoron, polychoron co5oo3oo3ox&#zy |
Catalan polychoron, using edge sizes x and y = sqrt[(9-sqrt(5))/18] = 0.613004 (pseudo edge size c = (sqrt(5)-1)/3 = 0.412023) |
o3m4o3o |
dual of rectified icositetrachoron, polychoron co3oo4oo3ox&#zy |
Catalan polychoron, using edge sizes x and y = sqrt(5)/3 = 0.745356 (pseudo edge size c = sqrt(8)/3 = 0.942809 |
ico |
dual of rectified hexadecachoron = ico, polychoron qo3oo3oo4ox&#zx |
Catalan polychoron, using edge sizes x only (Wythoffian) |
o3m3o5o |
dual of rectified hexacosachoron, polychoron co3oo3oo5ox&#zy |
Catalan polychoron, using edge sizes x and y = sqrt[(6+2 sqrt(5))/5] = 1.447214 (pseudo edge size c = (5+3 sqrt(5))/5 = 2.341641 |
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 xo3ou uo3ox&#zq | using edge sizes x, 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 xo3yb by3ox&#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 3c^{2} = 2a^{2}-6ab+6b^{2} |
spidrico | swirlprismatodiminished rectified icositetrachoron | using edge sizes x and q |
bhidtex | bi-hecatonicosidiminished truncated hexacosichoron | using edge sizes x and u |
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.
File | Name | Remarks |
tithah | truncation of thah |
polyhedron with (x,q)-bowties as well as x4q |
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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