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Small stellated dodecahedron

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Small stellated dodecahedron
TypeKepler–Poinsot polyhedron
Faces12
Edges30
Vertices12
Schläfli symbol[1]
Dual polyhedrongreat dodecahedron
3D model of a small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {⁠5/2⁠,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.

It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure.

It is the second of four stellations of the dodecahedron (including the original dodecahedron itself).

The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect.

Construction and properties

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The small stellated dodecahedron is constructed by attaching twelve pentagonal pyramids onto a regular dodecahedron's faces.[1] Suppose the pentagrammic faces are considered as five triangular faces. In that case, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar. The critical angle is atan(2) above the dodecahedron face.

Regarding the small stellated dodecahedron has 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, one can compute its genus using Euler's formula and conclude that the small stellated dodecahedron has genus 4.[2] This observation, made by Louis Poinsot, was initially confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. This Riemann surface, called Bring's curve, has the greatest number of symmetries of any Riemann surface of genus 4: the symmetric group acts as automorphisms.[3]

The dual polyhedron of a small stellated dodecahedron is the great dodecahedron which shares the same number of vertices, edges, and faces.[4]

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Small stellated dodecahedron (bottom left) in Johannes Kepler's Harmonices Mundi
Floor mosaic by Paolo Uccello

Small stellated dodecahedra can be seen in antiquity, as in Johannes Kepler's Harmonices Mundi, and a floor mosaic in Paolo Uccello's St Mark's Basilica circa 1430.[5] The same shape is central to two lithographs by M. C. Escher: Contrast (Order and Chaos) (1950) and Gravitation (1952).[6]

Formulas

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For a small stellated dodecahedron with edge length ,

  • Inradius =
  • Midradius =
  • Circumradius =
  • Area =
  • Volume =
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Animated truncation sequence from {⁠5/2⁠,5} to {5,⁠5/2⁠}.[7]

Its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron; the compound with both is the great complex icosidodecahedron.

There are four related uniform polyhedra, constructed as degrees of truncation. The dual is a great dodecahedron. The dodecadodecahedron is a rectification, where edges are truncated down to points.

The truncated small stellated dodecahedron can be considered a degenerate uniform polyhedron since edges and vertices coincide, but it is included for completeness. Visually, it looks like a regular dodecahedron on the surface, but it has 24 faces in overlapping pairs. The spikes are truncated until they reach the plane of the pentagram beneath them. The 24 faces are 12 pentagons from the truncated vertices and 12 decagons taking the form of doubly-wound pentagons overlapping the first 12 pentagons. The latter faces are formed by truncating the original pentagrams. When an {⁠n/d⁠}-gon is truncated, it becomes a {⁠2n/d⁠}-gon. For example, a truncated pentagon {⁠5/1⁠} becomes a decagon {⁠10/1⁠}, so truncating a pentagram {⁠5/2⁠} becomes a doubly-wound pentagon {⁠10/2⁠} (the common factor between 10 and 2 mean we visit each vertex twice to complete the polygon).

Stellations of the dodecahedron
Platonic solid Kepler–Poinsot solids
Dodecahedron Small stellated dodecahedron Great dodecahedron Great stellated dodecahedron
Name Small stellated dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated
great
dodecahedron 		Great
dodecahedron
Coxeter–Dynkin
diagram
Picture

See also

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References

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  1. ^ a b Kappraff, Jay (2001). Connections: The Geometric Bridge Between Art and Science (2nd ed.). World Scientific. p. 309. ISBN 981-02-4585-8.
  2. ^ Vince, Andrew. "Maps". In Gross, Jonathan L.; Yellen, Jay (eds.). Handbook of Graph Theory. CRC Press. p. 710.
  3. ^ Weber, Matthias (2005). "Kepler's small stellated dodecahedron as a Riemann surface". Pacific Journal of Mathematics. 220: 167–182. doi:10.2140/pjm.2005.220.167.
  4. ^ Conrad, J.; Chamberland, C.; Breuckmann, N. P.; Terhal, B. M. (2018). "The small stellated dodecahedron code and friends". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 376 (2123): 20170323. arXiv:1712.07666. Bibcode:2018RSPTA.37670323C. doi:10.1098/rsta.2017.0323. ISSN 1364-503X. PMC 5990658. PMID 29807900.
  5. ^ Coxeter, H. S. M. (2013). "Regular and semiregular polyhedra". In Senechal, Marjorie (ed.). Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination (2nd ed.). Springer. pp. 41–52. doi:10.1007/978-0-387-92714-5_3. ISBN 978-0-387-92713-8. See in particular p. 42.
  6. ^ Barnes, John (2012). Gems of Geometry (2nd ed.). Springer. p. 46.
  7. ^ Grünbaum, Branko (2007). "Graphs of polyhedra; polyhedra as graphs". Discrete Mathematics. 307 (3–5): 445–463. doi:10.1016/j.disc.2005.09.037.

Further reading

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