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° |
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 = 36.26°, R = 143.74° |
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 |
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,
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 |
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 |
Further polyhedra with different edge sizes occur naturally either within the investigation of Catalan solids or as cells of their 4D counterparts.
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 |
Further polychora with different edge sizes occur naturally within the investigation of the 4D counterparts of the Catalan solids.
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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