---
title: "The Fifty-Nine Icosahedra"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T08:44:41.601047+00:00"
instance: "kb-cron"
---

mainline stellations are the stellations  whose name consist of a single capital letterA till (and including)  H
fully supported stellations: stellations where there are no overhangs, and all visible parts of a face can be seen from the same side. these are those stellations where in the faces column there is no face that has  an apostrophe, for example 3'.
The only Kepler–Poinsot polyhedron in the list is the great icosahedron G.
The Great stellated dodecahedron is an edge-stellated icosahedron but only face-stellated icosahedra are in the list and it is therefore not included in the list.

Some images illustrate the mirror-image icosahedron with the f1 rather than the f1 cell.

== Subsequent debate ==
There has subsequently been some debate on Miller's rules, with some writers questioning them. Flather himself made and exhibited models which were "non-Miller". Bridge (1974) obtained stellations of the icosahedron by dualising facettings of the dodecahedron, noting the significance of internal structure in distinguishing between stellations which Miller's rules treat as identical. Hudson and Kingston (1988) adopted a reduced rule set. Inchbald noted two non-Miller stellations, and went on to discuss various related issues.

== See also ==
List of Wenninger polyhedron models – Wenninger's book Polyhedron models included 21 of these stellations.
Solids with icosahedral symmetry

== Notes ==

== References ==
Brückner, Max (1900). Vielecke und Vielflache: Theorie und Geschichte [Polygons and Polyhedra: Theory and History]. Leipzig: B.G. Treubner. (in German). OCLC 25080888. (Plate scans at the Internet Archive, at bulatov.org. Reprinted 2004: Michigan Publishing. ISBN 978-1-4181-6590-1.)
H. S. M. Coxeter, P. Du Val, H. T. Flather, J. F. Petrie (1938) The Fifty-nine Icosahedra, University of Toronto studies, mathematical series 6: 1–26.
Third edition (1999) Tarquin ISBN 978-1-899618-32-3 MR 0676126
Cromwell, Peter R. (1997). "Stars, Stellations and Skeletons". Polyhedra . Cambridge University Press. Ch. 7, pp. 249–287. ISBN 0-521-55432-2.
Wenninger, Magnus J. (1983) Polyhedron models; Cambridge University Press, Paperback edition (2003). ISBN 978-0-521-09859-5.
Wheeler, Albert Harry (1924) "Certain forms of the icosahedron and a method for deriving and designating higher polyhedra." Archived 2017-12-01 at the Wayback Machine In Proceedings of the International Mathematical Congress, Toronto, vol. 1, pp. 701–708.

== External links ==
Vladimir Bulatov (2001). "An Interactive Creation of Polyhedra Stellations with Various Symmetries" VisMath. Vol. 3, No. 2.
The fifty nine stellations of the regular icosahedron
Weisstein, Eric W. "Fifty nine icosahedron stellations". MathWorld.
Weisstein, Eric W. "Echidnahedron". MathWorld.
Stellations of the icosahedron
George Hart, 59 Stellations of the Icosahedron - VRML 3D files.
John L. Hudson (1987). "Set of 70 polyhedron models showing the icosahedron and its 58 stellations". Science Museum, London.
A. Harry Wheeler. Models at the National Museum of American History. Collection search for "Harry Wheeler icosahedron"