List of noble polyhedra

A visual index of noble polyhedra sorted by convex hull types. The purple lines connect dual pairs, the purple polyhedra are self-dual, the magenta polyhedron is dual to its enantiomorph. The tetragonal disphenoid and the rhombic disphenoid are variants of the regular tetrahedron. The bottom row represents the stephanoid infinite family.

Besides the tetragonal disphenoid, rhombic disphenoid and the infinite family of crown polyhedra, there are 94 known non-exotic noble polyhedra, 9 of which are regular and 2 of which are fissary:

Tetrahedral symmetry

There exists only one noble polyhedron with tetrahedral symmetry: the tetrahedron. There do not exist noble polyhedra with chiral tetrahedral or pyritohedral symmetry.

Name	Convex hull	Faces	Edges	Vertices	Dual	Minimal ratio	Notes
Tetrahedron	Tetrahedron	4 triangles	6	4	Tetrahedron	1:1	Regular, self-dual

Octahedral symmetry

Name	Convex hull	Faces	Edges	Vertices	Dual	Minimal ratio	Notes
Cube	Cube	6 squares	12	8	Octahedron	1:1	Regular
Octahedron	Octahedron	8 triangles	12	6	Cube	1:1	Regular
Noble faceting of the truncated cube/ Noble pentagrammic icositetrahedron	Truncated cube	24 mirror-symmetric pentagrams	60	24	Noble piscoidal icositetrahedron	1:1.30656
Noble faceting of the small rhombicuboctahedron/ Noble piscoidal icositetrahedron	Small rhombicuboctahedron	24 mirror-symmetric pentagons	60	24	Noble pentagrammic icositetrahedron	1:1.84776
Noble tetragonal tetracontoctahedron	Nonuniform truncated octahedron	48 irregular tetragons	96	24	Noble octagrammic icositetrahedron	1:1.57021
Noble octagrammic icositetrahedron	Nonuniform great rhombicuboctahedron	24 mirror-symmetric octagrams	96	48	Noble tetragonal tetracontoctahedron	1:1.19166
Noble pentagonal tetracontoctahedron	Nonuniform great rhombicuboctahedron	48 irregular pentagons	120	48	Noble pentagrammic tetracontoctahedron	1:4.66313
Noble pentagrammic tetracontoctahedron	Nonuniform great rhombicuboctahedron	48 irregular pentagrams	120	48	Noble pentagonal tetracontoctahedron	1:1.22064

Chiral octahedral symmetry

Name	Convex hull	Faces	Edges	Vertices	Dual	Minimal ratio	Notes
First noble kipiscoidal icositetrahedron	Snub cube	24 irregular pentagons	60	24	First noble kipiscoidal icositetrahedron	1:1.68502	Self-dual
First noble kisombreroidal icositetrahedron	Nonuniform snub cube	24 irregular pentagons	60	24	First noble kisombreroidal icositetrahedron	1:2.41421	Self-dual
Noble kipentagrammic icositetrahedron	Nonuniform snub cube	24 irregular pentagrams	60	24	Noble kipentagrammic icositetrahedron	1:1.25108	Self-dual
Second kipiscoidal icositetrahedron	Nonuniform snub cube	24 irregular pentagons	60	24	Second noble kisombreroidal icositetrahedron	1:2.68320
Second noble kisombreroidal icositetrahedron	Nonuniform snub cube	24 irregular pentagons	60	24	Second kipiscoidal icositetrahedron	1:2.28455
Third noble kisombreroidal icositetrahedron	Nonuniform snub cube	24 irregular pentagons	60	24	Fourth noble kisombreroidal icositetrahedron	1:3.30901
Fourth noble kisombreroidal icositetrahedron	Nonuniform snub cube	24 irregular pentagons	60	24	Third noble kisombreroidal icositetrahedron	1:1.60002

Icosahedral symmetry

Name	Convex hull	Faces	Edges	Vertices	Dual	Notes
Dodecahedron	Dodecahedron	12 pentagons	30	20	Icosahedron	Regular
Icosahedron	Icosahedron	20 triangles	30	12	Dodecahedron	Regular
Small stellated dodecahedron	Icosahedron	12 pentagrams	30	12	Great dodecahedron	Regular
Great dodecahedron	Icosahedron	12 pentagons	30	12	Small stellated dodecahedron	Regular
Great stellated dodecahedron	Dodecahedron	12 pentagrams	30	20	Great icosahedron	Regular
Great icosahedron	Icosahedron	20 triangles	30	12	Great stellated dodecahedron	Regular, stellation of the icosahedron
Ditrigonal icosahedron	Dodecahedron	20 propeller tripods	60	20	Ditrigonal icosahedron	Self-dual, semi-uniform, icosahedron stellation, dodecahedron faceting
Small noble triangular hexecontahedron	Dodecahedron	60 isosceles triangles	90	20	Crennell number 4 stellation of the icosahedron
Great noble triangular hexecontahedron	Dodecahedron	60 isosceles triangles	90	20	Final stellation of the icosahedron
Crennell number 4 stellation of the icosahedron	Rhombicosidodecahedron	20 irregular enneagons	90	60	Small noble triangular hexecontahedron	Stellation of the icosahedron
Final stellation of the icosahedron	Non-uniform truncated icosahedron (hull of qrid)	20 irregular great enneagrams	90	60	Great noble triangular hexecontahedron	Stellation of the icosahedron
Noble faceting of icosidodecahedron 1	Icosidodecahedron	60 butterflies	120	30	Noble stellation of rhombic triacontahedron 1
Noble faceting of icosidodecahedron 2	Icosidodecahedron	60 butterflies	120	30	Noble stellation of rhombic triacontahedron 2
Noble faceting of icosidodecahedron 3	Icosidodecahedron	60 butterflies	120	30	Noble stellation of rhombic triacontahedron 3
Noble faceting of icosidodecahedron 4	Icosidodecahedron	120 isosceles triangles	180	30	Final stellation of the rhombic triacontahedron
Noble faceting of icosidodecahedron 5	Icosidodecahedron	120 isosceles triangles	180	30	Noble stellation of rhombic triacontahedron 5
Noble faceting of icosidodecahedron 6	Icosidodecahedron	120 isosceles triangles	180	30	Noble stellation of rhombic triacontahedron 6
Noble stellation of rhombic triacontahedron 1	Non-uniform truncated icosahedron	30 irregular octagons	120	60	Noble faceting of icosidodecahedron 1
Noble stellation of rhombic triacontahedron 2	Non-uniform truncated icosahedron	30 irregular octagrams	120	60	Noble faceting of icosidodecahedron 2
Noble stellation of rhombic triacontahedron 3	Non-uniform truncated icosahedron	30 irregular octagrams	120	60	Noble faceting of icosidodecahedron 3
Final stellation of the rhombic triacontahedron	Great rhombicosidodecahedron	30 irregular dodecagrams	180	120	Noble faceting of icosidodecahedron 4
Noble stellation of rhombic triacontahedron 5	Non-uniform great rhombicosidodecahedron	30 irregular dodecagrams	180	120	Noble faceting of icosidodecahedron 5
Noble stellation of rhombic triacontahedron 6	Non-uniform great rhombicosidodecahedron	30 irregular dodecagrams	180	120	Noble faceting of icosidodecahedron 6
Noble pentagonal hexecontahedron	Semi-uniform small rhombicosidodecahedron 3	60 mirror-symmetric convex pentagons	150	60	Noble pentagrammic hexecontahedron
Noble pentagrammic hexecontahedron	Semi-uniform small rhombicosidodecahedron 4	60 mirror-symmetric pentagrams	150	60	Noble pentagonal hexecontahedron
First noble piscoidal hexecontahedron	Semi-uniform small rhombicosidodecahedron 4	60 mirror-symmetric pentagons	150	60	First noble sombreroidal hexecontahedron
First noble sombreroidal hexecontahedron	Semi-uniform truncated icosahedron 6	60 mirror-symmetric pentagons	150	60	First noble piscoidal hexecontahedron
Second noble piscoidal hexecontahedron	Semi-uniform truncated icosahedron 5	60 mirror-symmetric pentagons	150	60	Second noble sombreroidal hexecontahedron
Second noble sombreroidal hexecontahedron	Semi-uniform truncated dodecahedron 3	60 mirror-symmetric pentagons	150	60	Second noble piscoidal hexecontahedron
Noble faceting of the truncated dodecahedron / First noble unihexagrammic hexecontahedron	Truncated dodecahedron	60 unicursal hexagrams	180	60	Second noble unihexagrammic hexecontahedron
Second noble unihexagrammic hexecontahedron	Semi-uniform small rhombicosidodecahedron 1	60 unicursal hexagrams	180	60	First noble unihexagrammic hexecontahedron
First noble faceting of the rhombicosidodecahedron / Third noble unihexagrammic hexecontahedron	Small rhombicosidodecahedron	60 unicursal hexagrams	180	60	First noble ditrapezoidal hexecontahedron
First noble ditrapezoidal hexecontahedron	Semi-uniform truncated icosahedron 1	60 mirror-symmetric hexagons	180	60	Third noble unihexagrammic hexecontahedron
Fourth noble unihexagrammic hexecontahedron	Semi-uniform truncated dodecahedron 1	60 unicursal hexagrams	180	60	First noble pterogrammic hexecontahedron
First noble pterogrammic hexecontahedron	Semi-uniform truncated icosahedron 3	60 mirror-symmetric hexagons	180	60	Fourth noble unihexagrammic hexecontahedron
Second noble pterogrammic hexecontahedron	Semi-uniform small rhombicosidodecahedron 2	60 mirror-symmetric hexagons	180	60	Fifth noble unihexagrammic hexecontahedron
Fifth noble unihexagrammic hexecontahedron	Semi-uniform truncated icosahedron 3	60 unicursal hexagrams	180	60	Second noble pterogrammic hexecontahedron
Third noble pterogrammic hexecontahedron	Semi-uniform small rhombicosidodecahedron 7	60 mirror-symmetric hexagons	180	60	Sixth noble unihexagrammic hexecontahedron
Sixth noble unihexagrammic hexecontahedron	Semi-uniform truncated icosahedron 7	60 unicursal hexagrams	180	60	Third noble pterogrammic hexecontahedron
Second noble ditrapezoidal hexecontahedron	Semi-uniform small rhombicosidodecahedron 2	60 mirror-symmetric hexagons	180	60	Fourth noble pterogrammic hexecontahedron
Fourth noble pterogrammic hexecontahedron	Semi-uniform truncated dodecahedron 1	60 mirror-symmetric hexagons	180	60	Second noble ditrapezoidal hexecontahedron
Seventh noble unihexagrammic hexecontahedron	Semi-uniform small rhombicosidodecahedron 5	60 unicursal hexagrams	180	60	Third noble ditrapezoidal hexecontahedron
Third noble ditrapezoidal hexecontahedron	Semi-uniform small rhombicosidodecahedron 6	60 mirror-symmetric hexagons	180	60	Seventh noble unihexagrammic hexecontahedron
First noble propellogrammic hexecontahedron	Semi-uniform truncated icosahedron 2	60 mirror-symmetric hexagons	180	60	Self-dual
Second noble propellogrammic hexecontahedron	Semi-uniform truncated dodecahedron 2 (hull of sirsid)	60 mirror-symmetric hexagons	180	60	Self-dual
Noble Propeller tripodal hexecontahedron	Semi-uniform truncated icosahedron 4 (hull of seside)	60 mirror-symmetric hexagons	180	60	Self-dual

There are two other noble polyhedra with this symmetry not listed in the above table. They are both in the grid army.

Chiral icosahedral symmetry

Name	Convex hull	Faces	Edges	Vertices	Dual	Notes
Second noble faceting of the small rhombicosidodecahedron / First kipiscoidal hexecontahedron	Small rhombicosidodecahedron	60 irregular pentagons	150	60	First kipentagrammic hexecontahedron
First kipentagrammic hexecontahedron	Semi-uniform truncated icosahedron 1	60 irregular pentagons	150	60	Second noble faceting of the small rhombicosidodecahedron / First kipiscoidal hexecontahedron
Third noble faceting of the small rhombicosidodecahedron / First kisombreroidal hexecontahedron	Small rhombicosidodecahedron	60 irregular pentagons	150	60	Second kisombreroidal hexecontahedron
Second kisombreroidal hexecontahedron	Semi-uniform truncated icosahedron 1	60 irregular pentagons	150	60	Third noble faceting of the small rhombicosidodecahedron / First kisombreroidal hexecontahedron
Fourth noble faceting of the small rhombicosidodecahedron / Third kisombreroidal hexecontahedron	Small rhombicosidodecahedron	60 irregular pentagons	150	60	Fourth kisombreroidal hexecontahedron
Fourth kisombreroidal hexecontahedron	Semi-uniform truncated icosahedron 1	60 irregular pentagons	150	60	Fourth noble faceting of the small rhombicosidodecahedron / Third kisombreroidal hexecontahedron
Fifth kisombreroidal hexecontahedron	Semi-uniform srid 5	60 irregular pentagons	150	60	Sixth kisombreroidal hexecontahedron
Sixth kisombreroidal hexecontahedron	Semi-uniform srid 6	60 irregular pentagons	150	60	Fifth kisombreroidal hexecontahedron
Second kipiscoidal hexecontahedron	Semi-uniform srid 7	60 irregular pentagons	150	60	Second kipentagrammic hexecontahedron
Second kipentagrammic hexecontahedron	Semi-uniform truncated icosahedron 7	60 irregular pentagons	150	60	Second kipiscoidal hexecontahedron
Kizippergrammic hexecontahedron	Semi-uniform truncated icosahedron 4 (hull of seside)	60 nonregular hexagons	180	60	Kiunihexagrammic hexecontahedron	Seside faceting
Kiunihexagrammic hexecontahedron	Semi-uniform tid 2 (hull of sirsid)	60 nonregular hexagons	180	60	Kizippergrammic hexecontahedron	Sirsid faceting
Fifth noble faceting of the small rhombicosidodecahedron / Kignathogrammic hexecontahedron	Small rhombicosidodecahedron	60 nonregular hexagons	180	60	Crossed kignathogrammic hexecontahedron
Crossed kignathogrammic hexecontahedron	Semi-uniform truncated icosahedron 1	60 nonregular hexagons	180	60	Fifth noble faceting of the small rhombicosidodecahedron / Kignathogrammic hexecontahedron
First noble snub dodecahedral hexecontahedron	Nonuniform snub dodecahedron	60 nonregular hexagons	180	60	Self-dual
Second noble snub dodecahedral hexecontahedron	Nonuniform snub dodecahedron (hull of sided)	60 nonregular hexagons	180	60	Self-dual	Sided faceting
Seventh noble kisombreroidal hexacontahedron	Nonuniform snub dodecahedron	60 nonregular pentagons	150	60	Second noble kisombreroidal hexacontahedron
Eighth noble kisombreroidal hexacontahedron	Nonuniform snub dodecahedron	60 nonregular pentagons	150	60	First noble kisombreroidal hexacontahedron

There are a further 8 noble polyhedra with this symmetry not listed in the above table.

Noble polyhedra that are nonregular and nonprismatic sorted by Schlälfi types

Bibliography

• 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). 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.