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One of the lost icosahedra
 

Guy's polyhedra pages

Essays, current research and random resources

Contents


A spacefilling hendecahedron
 

What's new

For regular visitors, here is a handy reference to what has changed over the last year or so.

11 June 2014. Ditela, polytopes and dyads
A new name for closed line segments completes the pantheon of names for polytopes - updated with minor corrections.

27 March 2011. It's a long way to the stars
or, The sorry state of polyhedron theory today - updated with some more and better answers, and a sneak preview of my topological journey.

30 October 2010. Flextegrity: a new structural paradigm
A review of Flextegrity: equilibriated polyhedral structures by Sam Lanahan.

4 Mar 2008. Stellation and facetting - a brief history
General update: 2 new entries, 2 new references, other bits and pieces.

16 Feb 2008. Filling polytopes
Updated but still unfinished. In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete.

2 Feb 2008. Polytopes - abstract and real
By Norman Johnson, edited by me. Real polytopes are a consistent mathematical formulation of the true geometric figures that we instinctively think of as "polygons", "polyhedra" and so on, in contrast to such things as abstract polytopes on the one hand and the woolly ideas we have inherited down the ages from Euclid and his successors on the other.

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Space-filling polyhedra

Illustrated essays, published and unpublished. Most have links to printable "nets" for making card models.

Five space-filling polyhedra
As in The Mathematical Gazette 80, November 1996, p.p. 466-475.
My most popular polyhedron page. The text is reproduced by kind permission, with some revision and additions.

The Archimedean honeycomb duals
As in The Mathematical Gazette 81, July 1997, p.p. 213-219.
A remarkable family of 14 polyhedra. The text is reproduced by kind permission.

A 3-D quasicrystal structure?
A possible candidate for a 3-dimensional, aperiodic crystal structure.

Weaire-Phelan Bubbles
The closest to ideal bubbles yet found.

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Stellation and facetting

Star polyhedra include some of the most beautiful mathematical shapes around (see for example my icosahedron pages). Their mathematical theory remains surprisingly primitive and incomplete.

It's a long way to the stars
or, The sorry state of polyhedron theory today.

Stellation and facetting - a brief history
Title says it all, really.

Facetting diagrams
The facetting diagram may be used to find facettings of a polyhedron, whereby pieces are cut away to reveal new faces and edges, but not new vertices.

The regular star (Kepler-Poinsot) polyhedra:

A.-L. Cauchy, Researches on polyhedra, Part I
Paper deriving Poinsot's regular star polyhedra by stellating the regular convex solids, and proving that the set is complete. English translation from: Recherches sur les polyèdres, Prèmiere partie, Journal de l' École Polytechnique, 16 (1813). Includes PDF file for downloading/printing.

J. Bertrand, Note on the theory of regular polyhedra
Paper deriving Poinsot's regular star polyhedra by facetting the regular convex solids. English translation from: Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences 46 (1858). Includes PDF file for downloading/printing.

Stellating the icosahedron and facetting the dodecahedron - index page
Introduction and links to the rest. The main pages are listed below:

Some lost stellations of the icosahedron
This page brings together many of the lost stellations of the icosahedron that I have come across so far, including two discovered, probably by H.T. Flather, before publication of the famous fifty-nine.
In search of the lost icosahedra
As in The Mathematical Gazette 86, July 2002, p.p. 208-215.
Towards stellating the icosahedron and faceting the dodecahedron
As in Symmetry: Culture and Science, Vol. 11, 1-4, 2000, p.p. 269-291.
Tidy dodecahedra and icosahedra
Unpublished. Includes a discussion of Bridge's 1974 paper on Facetting the dodecahedron.
Icosahedral precursors
Unfinished. Waiting for me to develop some very fundamental theory about polyhedra.

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General theory of polytopes and polyhedra

Yikes! The serious interest department.

It's a long way to the stars
or, The sorry state of polyhedron theory today.

Ditela, polytopes and dyads
A new name for closed line segments completes the pantheon of names for polytopes in any number of dimensions.

Polytopes - abstract and real
By Norman Johnson, edited by me. Real polytopes are a consistent mathematical formulation of the true geometric figures that we instinctively think of as "polygons", "polyhedra" and so on, in contrast to such things as abstract polytopes on the one hand and the woolly ideas we have inherited down the ages from Euclid and his successors on the other.

Vertex figures
Mathematicians have used different definitions of the vertex figure for different purposes. Different types are examined and classified leading, via the idea of the complete vertex, to a more general definition.

Filling polytopes
In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete. This essay is incomplete too.

Polytopes, duality and precursors
A new definition of polytopes as set pairs is presented, based on the inclusion of vertex figures. Dual or reciprocal polytopes are understood as differing manifestations of the same precursor. A notation suited to the simpler and more regular cases is described. Unfinished.

Polytopes: degeneracy and tidiness
Geometrical untidiness is distinguished from from topological degeneracy. Some types of untidiness and degeneracy are discussed. Features located at infinity can have two opposing images. Unfinished.

Trimethoric (and trisynaptic) polyhedra
As in Mathematics and Informatics Quarterly 2/2001, Vol. 11, p.p. 71-74.
Trimethoric and trisynaptic polyhedra represent two previously unrecognised classes.

Is there a self-dual hendecahedron?
Thanks to those who to told me yes, there is one called the canonical form. But are there also any "non-canonical" solutions?

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Other resources

Random stuff I thought you might like.

30 October 2010. Flextegrity: a new structural paradigm
A review of Flextegrity: equilibriated polyhedral structures by Sam Lanahan.

Wenninger's Polyhedron Models - ERRATA. List of known errors in this ever-popular book. Endorsed by Wenninger.

Stardust. Polyhedron kits and puzzles.

Links to other polyhedron websites (now a bit out of date).

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