List of noble polyhedra
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Besides the tetragonal disphenoid, rhombic disphenoid and the infinite family of crown polyhedra, there are 146 known non-exotic noble polyhedra, 9 of which are regular and 2 of which are fissary:
Tetrahedral symmetry[edit | edit source]
There exists only one noble polyhedron with tetrahedral symmetry: the tetrahedron. There do not exist noble polyhedra with chiral tetrahedral or pyritohedral symmetry.
| Name | Image | Convex hull | Faces | Edges | Vertices | Dual | Schläfli type | Minimal ratio | Notes |
|---|---|---|---|---|---|---|---|---|---|
| Tetrahedron |  |
Tetrahedron | 4 triangles | 6 | 4 | Self-dual | {3,3} | 1:1 | Regular |
Octahedral symmetry[edit | edit source]
| Name | Image | Convex hull | Faces | Edges | Vertices | Dual | Schläfli type | Minimal ratio | Notes | Discoverer, year |
|---|---|---|---|---|---|---|---|---|---|---|
| Cube |  |
Cube | 6 squares | 12 | 8 | Octahedron | {4,3} | 1:1 | Regular | |
| Octahedron |  |
Oct | 8 triangles | 12 | 6 | Cube | {3,4} | 1:1 | Regular | |
| Noble faceting of the truncated cube / |  |
Tic | 24 mirror-symmetric pentagrams | 60 | 24 | Noble piscoidal icositetrahedron | {5,5} | 1:1.30656 | Edmund Hess, 1877 | |
| Noble faceting of the small rhombicuboctahedron / |  |
Sirco | 24 mirror-symmetric pentagons | 60 | 24 | Noble pentagrammic icositetrahedron | {5,5} | 1:1.84776 | Hess, 1877 | |
| Noble tetragonal tetracontoctahedron |  |
Semi-uniform Toe | 48 irregular tetragons | 96 | 24 | Noble octagrammic icositetrahedron | {4,8} | 1:1.57021 | Plasmath, 2022 | |
| Noble octagrammic icositetrahedron |  |
Semi-uniform Girco 1 | 24 mirror-symmetric octagrams | 96 | 48 | Noble tetragonal tetracontoctahedron | {8,4} | 1:1.19166 | Plasmath, 2023 | |
| Noble pentagonal tetracontoctahedron |  |
Semi-uniform Girco 2 | 48 irregular pentagons | 120 | 48 | Noble pentagrammic tetracontoctahedron | {5,5} | 1:4.66313 | Plasmath, 2023 | |
| Noble pentagrammic tetracontoctahedron |  |
Semi-uniform Girco 3 | 48 irregular pentagrams | 120 | 48 | Noble pentagonal tetracontoctahedron | {5,5} | 1:1.22064 | Plasmath, 2023 |
Chiral octahedral symmetry[edit | edit source]
| Name | Image | Convex hull | Faces | Edges | Vertices | Dual | Schläfli type | Minimal ratio | Notes | Discoverer, year |
|---|---|---|---|---|---|---|---|---|---|---|
| First kipiscoidal icositetrahedron |  |
Snic | 24 irregular pentagons | 60 | 24 | Self-dual | {5,5} | 1:1.68502 | Robert Webb, 2008 | |
| First kisombreroidal icositetrahedron |  |
Non-uniform Snic 3 | 24 irregular pentagons | 60 | 24 | Self-dual | {5,5} | 1:2.41421 | Max Brückner, 1906 | |
| Noble kipentagrammic icositetrahedron |  |
Non-uniform Snic 1 | 24 irregular pentagrams | 60 | 24 | Self-dual | {5,5} | 1:1.25108 | Plasmath, 2023 | |
| Second kipiscoidal icositetrahedron |  |
Non-uniform Snic 1 | 24 irregular pentagons | 60 | 24 | Second kisombreroidal icositetrahedron | {5,5} | 1:2.68320 | Plasmath, 2023 | |
| Second kisombreroidal icositetrahedron |  |
Non-uniform Snic 2 | 24 irregular pentagons | 60 | 24 | Second kipiscoidal icositetrahedron | {5,5} | 1:2.28455 | Plasmath, 2023 | |
| Third kisombreroidal icositetrahedron |  |
Non-uniform Snic 4 | 24 irregular pentagons | 60 | 24 | Fourth kisombreroidal icositetrahedron | {5,5} | 1:3.30901 | Plasmath, 2023 | |
| Fourth kisombreroidal icositetrahedron |  |
Non-uniform Snic 5 | 24 irregular pentagons | 60 | 24 | Third kisombreroidal icositetrahedron | {5,5} | 1:1.60002 | Plasmath, 2023 |
Icosahedral symmetry[edit | edit source]
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Chiral icosahedral symmetry[edit | edit source]
Fissary nobles[edit | edit source]
Two non-exotic fissary nobles are known. Both of them have dodecahedral symmetry.
| Name | Image | Convex hull | Faces | Edges | Vertices | Dual | Schläfli type | Minimal ratio | Notes | Discoverer, year |
|---|---|---|---|---|---|---|---|---|---|---|
| Eleventh kipiscoidal hecatonicosahedron | .png) |
Semi-uniform Ti 8 | 120 irregular pentagons | 300 | 60 | Tenth kipentagrammic hecatonicosahedron | {5,5} | 1:2.94665 | Fissary | Plasmath, 2022 |
| Twelth kipiscoidal hecatonicosahedron | .png) |
Semi-uniform Srid 8 | 120 irregular pentagons | 300 | 60 | Eleventh kipentagrammic hecatonicosahedron | {5,5} | 1:2.41571 | Fissary | Plasmath, 2022 |
Nonregular nonprismatic noble polyhedra sorted by Schläfli types[edit | edit source]
Noble polyhedra with exotic faces[edit | edit source]
There also exists a number of noble polyhedra which have faces that can be interpreted either as singular polygons with some of their vertices coinciding or as compounds of polygons. Therefore the polyhedra themselves can be interpreted either as singular noble polyhedra or as polyhedral compounds.
| Name | Image | Convex hull | Faces | Edges | Vertices | Dual | Schläfli type | Minimal ratio | Notes | Discoverer, year |
|---|---|---|---|---|---|---|---|---|---|---|
| Exotic-combo-faced noble triangular hexecontahedron
(ECF faceting of dodecahedron #1) |
|
Doe | 60
(20*3) scalene triangles |
90 | 20 | 
|
{3,9} | 1:2.61803 | Can be interpreted | |
| Exotic-combo-faced noble enneagrammic icosahedron
(ECF faceting of dodecahedron #2) |
|
Doe | 20 exotic irregular | 90 | 60
(20*3) |
|
{9,3} | 1:2.61803 | Fissary, can be interpreted | |
| Exotic-combo-faced noble crossed trapezoidal hexecontahedron
(ECF faceting of dodecahedron #3) |
|
Doe | 60
(20*3) butterflies |
120 | 20 | 
|
{4,12} | 1:2.61803 | Can be interpreted | |
| Exotic-combo-faced dodecagonal icosahedron
(ECF faceting of dodecahedron #4) |
|
Doe | 20 exotic irregular dodecagrams | 120 | 60
(20*3) |
|
{12,4} | 1:2.61803 | Fissary, can be interpreted | |
| First exotic-combo-faced ditrapezoidal icositetrahedron | 
|
Sirco | 24 exotic mirror-symmetric hexagons | 72 | 24 | 
|
{6,6} | 1:1.84776 | Can be interpreted
as a compound of 12 rhombic disphenoids |
Brückner, 1906 |
| Second exotic-combo-faced ditrapezoidal icositetrahedron |
|
Tic | 24 exotic mirror-symmetric hexagons | 72 | 24 |
|
{6,6} | 1:1.30656 | Can be interpreted
as a compound of 12 rhombic disphenoids |
Brückner, 1906 |
| Third exotic-combo-faced ditrapezoidal icositetrahedron | 
|
Semi-uniform sirco | 24 exotic mirror-symmetric hexagons | 72 | 24 | Self-dual | {6,6} | 1:1.93185 | Can be interpreted
as a compound of 12 rhombic disphenoids |
Brückner, 1906 |
| Exotic-combo-faced kignathogrammic icositetrahedron | 
|
Non-uniform snic 3
(Same hull as first kisombreroidal icositetrahedron) |
24 exotic irregular hexagons | 72 | 24 | Self-dual | {6,6} | 1:2.41421 | Can be interpreted
as a compound of 12 rhombic disphenoids |
Brückner, 1906 |
Bibliography[edit | edit source]
- Hess, Edmund (1876). Über die Zugleich Gleicheckigen und Gleichflächigen Polyeder. Kassel.
- Brückner, Max (1906). Über die Gleicheckig-Gleichflächigen Diskontinuierlichen und Nichtkonvexen Polyeder. Halle.
- Wenninger, Magnus (1971). Polyhedron Models. Cambridge University Press. ISBN 978-0-511-56974-6.
- Grünbaum, Branko (1993). "Polyhedra With Hollow Faces (Proc. NATO - ASI Conference)". In Bisztriczky, T.; et al. (eds.). POLYTOPES: Abstract, Convex and Computational. Toronto: Kluwer Acad. Publ. pp. 43–70. ISBN 978-94-010-4398-4.
- Grünbaum, Branko (2003). Are Your Polyhedra the Same as My Polyhedra. Discrete and Computational Geometry: The Goodman-Pollack Festschrift. New York: Springer. pp. 461–488.
- Brückner, Max; Stratton, R.; Mikloweit, Ulrich (2009). Concerning the Isogonal-Isohedral, Discontinuous and Nonconvex Polyhedra. Colorado Springs. ISBN 978-0-692-00323-7. (Translation of Max Brückner's work)
- Webb, Robert (2008). "Noble Faceting of Snub Cube". Stella.
- Mikloweit, Ulrich (2020). "Exploring Noble Polyhedra With the Program Stella4D" (PDF). Bridges 2020 Conference Proceedings. Helsinki and Espoo, Finland. 25: 257–264. ISBN 9781938664366.