One of the lost icosahedra |
Guy's polyhedra pagesEssays, current research and random resourcesContents |
A spacefilling hendecahedron |
What's new
For regular visitors, here is a handy reference to significant changes over the last couple of years.
30 Oct 2018. Ditela, polytopes and dyads.
Update following Johnson's Geometries and Transformations. Note on complex configurations.
29 Sept 2018. Folding Space-Filling Bisymmetric Hendecahedron for a Large-Scale Art Installation, Wu, Jiangmei and Inchbald, Guy; Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, Pages 483–486. (External link to PDF)
8 July 2018. Morphic polytopes.
Towards a new definition of polyhedra and higher polytopes, seeking to reconcile abstract and more traditional topological approaches. (Blocki post)
15 Aug 2017. Egyptian Pyramid Puzzle.
Print and assemble the pieces, then solve the puzzle. The first of my old dissection puzzles to be made a free PDF download. (Stardust kit)
14 Dec 2016. Icosahedral precursors
Updated with the outline of a foundation in abstract and morphic theories, revised criteria for stellations and facettings.
23 Nov 2016. Polytopes, Duality and Precursors.
Deleted because the idea already exists and is called abstract polytopes.
4 Oct 2016. It's a long way to the stars
or, The sorry state of polyhedron theory today - corrected and updated. (Blocki post)
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Illustrated essays, published and unpublished.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.
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, why stellation theory is in such a mess. (Blocki post)
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:
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.
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:
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.
As in The Mathematical Gazette 86, July 2002, p.p. 208-215.
As in Symmetry: Culture and Science, Vol. 11, 1-4, 2000, p.p. 269-291.
Unpublished. Includes a discussion of Bridge's 1974 paper on Facetting the dodecahedron.
Unfinished and subject to revision. New rules for stellation, founded in abstract and morphic theories.
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. (Blocki post)
Morphic polytopes.
Towards a new definition of polyhedra and higher polytopes, seeking to reconcile abstract and more traditional topological approaches. (Blocki post)
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: 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?
Other resources
Random stuff I thought you might like.
Flextegrity: a new structural paradigm. A review of Flextegrity: equilibriated polyhedral structures by Sam Lanahan. (Blocki post)
Wenninger's Polyhedron Models - ERRATA.
List of known errors in this ever-popular book. Endorsed by Wenninger.
Stardust.
Polyhedron kits and puzzles.
Updated 30 Sept 2018