# Rhombic triacontahedron
The rhombic triacontahedron is a convex polyhedron with 30 congruent rhombic faces, 60 edges of equal length, and 32 vertices; it is the dual of the icosidodecahedron and one of the 13 Catalan solids.[1](https://dmccooey.com/polyhedra/RhombicTriacontahedron.html) • [2](https://mathshistory.st-andrews.ac.uk/Extras/Catalan_solids/) [Johannes Kepler](https://www.edgepedia.org/johannes-kepler) discovered it, publishing drawings in Book II of *Harmonices mundi* (1619), and it has since become a standard building block in the geometry of quasicrystals.[3](https://old.maa.org/press/periodicals/convergence/kepler-and-the-rhombic-dodecahedron-conclusion-kepler-s-other-rhombic-polyhedron-discovery) • [4](https://www.xray.cz/aperiodic/files/125.htm)
| Key fact | Detail |
|---|---|
| Faces, edges, vertices | 30 rhombic faces, 60 equal-length edges, 32 vertices (20 where 3 faces meet, 12 where 5 meet)[1](https://dmccooey.com/polyhedra/RhombicTriacontahedron.html) |
| Face shape | Each face is a golden rhombus: the ratio of the long diagonal to the short diagonal is the golden ratio φ = (√5 + 1)/2 ≈ 1.618[5](https://www.sciencedirect.com/topics/mathematics/rhombus) |
| Dual | The icosidodecahedron; as its dual, the triacontahedron is one of the 13 Catalan solids[2](https://mathshistory.st-andrews.ac.uk/Extras/Catalan_solids/) |
| Symmetry | Full icosahedral symmetry (Ih); dihedral angle 144°[1](https://dmccooey.com/polyhedra/RhombicTriacontahedron.html) |
| Dissection | It can be assembled from ten oblate and ten prolate golden rhombohedra, a decomposition shown by Kowalewski[5](https://www.sciencedirect.com/topics/mathematics/rhombus) |
| Distinctive symmetry property | It is edge-transitive but not vertex-transitive, one of only two polyhedra in 3D with that combination (the other is the rhombic dodecahedron)[6](https://arxiv.org/abs/2005.14285) |
| Discovery | Kepler, with drawings in *Harmonices mundi* Book II (1619); one history dates the discovery itself to 1611[3](https://old.maa.org/press/periodicals/convergence/kepler-and-the-rhombic-dodecahedron-conclusion-kepler-s-other-rhombic-polyhedron-discovery) |
## Geometry of the faces and vertices
Every face is a golden rhombus, a rhombus whose diagonals are in the golden ratio. The 30 faces meet 5 at a time at twelve vertices and 3 at a time at twenty vertices, giving the vertex counts 12 and 20 that echo the dodecahedron and icosahedron.[1](https://dmccooey.com/polyhedra/RhombicTriacontahedron.html) This is no accident: the long diagonals of the rhombi trace out the edges of an icosahedron, and the short diagonals trace out the edges of a dodecahedron, so the triacontahedron contains the vertices of both Platonic solids at once.[7](https://paulbourke.org/geometry/Rhombic_triacontahedron/index.html) • [8](https://sacred-geometry.es/index.php/en/phi-in/phi-in-sacred-solids)
The solid carries full icosahedral symmetry, and its dihedral angle, the angle between adjacent faces, is exactly 144°.[1](https://dmccooey.com/polyhedra/RhombicTriacontahedron.html) Its edge graph is the planar dual of the icosidodecahedral graph, since duality exchanges vertices and faces while pairing corresponding edges.[2](https://mathshistory.st-andrews.ac.uk/Extras/Catalan_solids/)
## Place among the rhombic polyhedra
Kepler described two rhombic polyhedra: one contained by twelve rhombi, the rhombic dodecahedron, and one contained by thirty, the triacontahedron.[3](https://old.maa.org/press/periodicals/convergence/kepler-and-the-rhombic-dodecahedron-conclusion-kepler-s-other-rhombic-polyhedron-discovery) A later classification by S. Bilinski (1960) showed that only a short list of polyhedra have congruent rhombic faces: the family of rhombic hexahedra obtained by stretching a cube, the rhombic dodecahedron, the Bilinski dodecahedron, the rhombic icosahedron, and the rhombic triacontahedron.[6](https://arxiv.org/abs/2005.14285) The triacontahedron and the rhombic dodecahedron stand out within this family as the only members that are edge-transitive without being vertex-transitive; a 2026 classification proves that in four dimensions and higher, edge-transitivity of a convex polytope forces vertex-transitivity, so no higher-dimensional analog of this exceptional behavior exists.[6](https://arxiv.org/abs/2005.14285)
**Decomposition into rhombohedra.** Kowalewski showed that the triacontahedron can be built from one acute and one obtuse rhombohedron, ten of each; in quasicrystal terminology these are called the oblate and prolate rhombohedra.[5](https://www.sciencedirect.com/topics/mathematics/rhombus) Hands-on teaching materials use exactly this construction, assembling 3D-printed oblate and prolate golden rhombohedra into a triacontahedron, and the same pieces can fill space in aperiodic ensembles.[9](https://journals.iucr.org/j/issues/2020/06/00/gj5252/gj5252fig9.html) The two rhombohedra have the property that they fill space without translational symmetry, a fact probably already known to Kepler.[5](https://www.sciencedirect.com/topics/mathematics/rhombus)
## Role in quasicrystals
The triacontahedron's fivefold-compatible geometry made it central to the study of quasicrystals, the intermetallic phases discovered in the 1980s that show diffraction symmetries forbidden to periodic crystals, such as fivefold axes.[10](https://doi.org/10.1107/S1600576720011772) A projection of the triacontahedron along a fivefold axis gives a pattern equivalent to a [Penrose tiling](https://www.edgepedia.org/penrose-tiling), as A. L. Mackay showed.[5](https://www.sciencedirect.com/topics/mathematics/rhombus)
In structure models of Tsai-type icosahedral quasicrystals, such as the binary YbCd5.7 and ScZn7.33 phases, the solid appears as an atomic cluster: the rhombic triacontahedron (RTH) unit sits at the 12-fold vertices of a three-dimensional Penrose tiling, while acute and obtuse rhombohedra decorated with atoms fill the gaps between clusters.[4](https://www.xray.cz/aperiodic/files/125.htm) The Tsai-type cluster itself is built of five concentric shells, an innermost tetrahedron, a dodecahedron, an icosahedron, an icosidodecahedron, and an outermost triacontahedron.[11](https://journals.iucr.org/m/issues/2016/04/00/hi0151/hi0151.pdf) Real crystals show the shape directly: solution-grown single grains of the binary quasicrystal Sc12Zn88 form faceted morphologies of pentagonal dodecahedra and rhombic triacontahedra under only marginally different growth conditions, and dendrites of the stable icosahedral Al–Cu–Li phase have the faceting of the triacontahedron.[12](https://arxiv.org/abs/0910.2507) • [13](https://ufn.ru/ufn10/ufn10_6/ufn106a.pdf)
## Stellations and related solids
Extending the 30 face planes of the triacontahedron produces a family of stellations. One of them, the rhombic hexecontahedron, is a 60-faced polyhedron obtained by placing a plane along each edge perpendicular to the symmetry plane containing that edge and taking the bounded solid.[14](https://mathworld.wolfram.com/RhombicHexecontahedron.html) Removing a band of faces from the triacontahedron and closing the gap yields the rhombic icosahedron, a 20-faced convex relative.[15](https://www.orchidpalms.com/polyhedra/rhombic/RTC/RTC.htm)
## References
1. [Rhombic Triacontahedron, David McCooey polyhedron pages](https://dmccooey.com/polyhedra/RhombicTriacontahedron.html)
2. [Catalan solids, MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/Extras/Catalan_solids/)
3. [Kepler and the Rhombic Dodecahedron: Conclusion, Convergence, Mathematical Association of America](https://old.maa.org/press/periodicals/convergence/kepler-and-the-rhombic-dodecahedron-conclusion-kepler-s-other-rhombic-polyhedron-discovery)
4. [Structure models of icosahedral quasicrystals, H. Takakura, Hokkaido University](https://www.xray.cz/aperiodic/files/125.htm)
5. [The Language of Shape (Hyde, Hyde & Lidin, 1997), section 2.9 Quasicrystals, ScienceDirect Topics: Rhombus](https://www.sciencedirect.com/topics/mathematics/rhombus)
6. [The Edge-transitive Polytopes that are not Vertex-transitive, arXiv:2005.14285](https://arxiv.org/abs/2005.14285)
7. [Rhombic Triacontahedron, Paul Bourke](https://paulbourke.org/geometry/Rhombic_triacontahedron/index.html)
8. [Phi in Sacred Solids, Sacred Geometry](https://sacred-geometry.es/index.php/en/phi-in/phi-in-sacred-solids)
9. [Teaching periodicity and aperiodicity using 3D-printed tiles and polyhedra, Journal of Applied Crystallography 53 (2020) 1583–1592](https://journals.iucr.org/j/issues/2020/06/00/gj5252/gj5252fig9.html)
10. [L. Casas (2020), Teaching periodicity and aperiodicity using 3D-printed tiles and polyhedra, J. Appl. Cryst. 53, doi:10.1107/S1600576720011772](https://doi.org/10.1107/S1600576720011772)
11. [Mysteries of icosahedral quasicrystals: how are the atoms arranged?, IUCrJ commentary (2016)](https://journals.iucr.org/m/issues/2016/04/00/hi0151/hi0151.pdf)
12. [Discovery of a binary icosahedral quasicrystal in Sc12Zn88, arXiv:0910.2507](https://arxiv.org/abs/0910.2507)
13. [Quasicrystals, Physics-Uspekhi review](https://ufn.ru/ufn10/ufn10_6/ufn106a.pdf)
14. [Rhombic Hexecontahedron, Wolfram MathWorld](https://mathworld.wolfram.com/RhombicHexecontahedron.html)
15. [Convex Polyhedra containing Golden Rhombi, Orchid Palms](https://www.orchidpalms.com/polyhedra/rhombic/RTC/RTC.htm)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Elementary and Euclidean geometry › Solid geometry*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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