This page aims to collect all types of deltahedron and polyhedral surfaces composed of congruent
equilateral triangles. If you know of one that is not listed here, please let me know. @tsurutana

Numerical method is required to construct some deltahedra. Here's a link to my add-on for regularizing triangle mesh with Blender.

Numerical method is required to construct some deltahedra. Here's a link to my add-on for regularizing triangle mesh with Blender.

このページは様々な形のデルタ多面体と正三角形集合で表現される曲面（Deltahedral surface）を収集しています。下記のリストに載っていない形を見つけたら@tsurutana までご連絡ください。

History:

- 2020-01-31Add goldberg's icosahedron
- 2019-12-03Add Fractal crystal
- 2019-08-28Add link to Blender Add-on
- 2019-07-31Add Mobius deltahedron
- 2019-07-30Initial release

Flexible deltahedra made by jointing cut-opened dipyramids. The word "Bellows" comes from *Bellows
theorem*. See Robert Dawson's website for
details.

Deltahedra which have only two forms of vertices. See Roger Kaufman's website for details.

- Cundy, H. M. "Deltahedra." Math. Gaz. 36, 263-266, 1952.
- Olshevsky G. "Polytopics #28: Breaking Cundy's Deltahedra Record".

See wikipedia for details. Tetrahelix. Helical deltahedra.

- A.H. Boerdijk, Philips Res. Rep. 7, 303 (1952).
- H.S.M. Coxeter, Can. Math. Bull. 28, 385 (1985).

Five convex deltahedra are biform.

There are only eight convex deltahedra. See Deltahedron at MathWorld.

- Freudenthal, H. and van der Waerden, B. L. "On an Assertion of Euclid." Simon Stevin 25, 115-121, 1947.
- Weisstein, Eric W. "Deltahedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Deltahedron.html

A smallest deltahedron with a hole.

A polyhedron whose faces are equilateral triangles and polygons which can be divided into a set of coplanar equilateral triangles. See Roger Kaufman's web and wikipedia for details.

Compact geometric structures formed by regular triangles with edges functioning like hinges. e.g. Goldberg's icosahedron.

- Wohlleben, E., & Liebermeister, W. (2012). Elastic Deformations in Polyhedral Rings Formed by Corpuscle Elements. link

Dome-like surfaces made of equilateral triangles.

- Misic, S., Obradovic, M., Dukanovic, G. (2015). Composite concave cupolae as geometric and architectural forms. Journal for Geometry and Graphics. 19. 79-91.

Composed of antiprismatic rings.

- Roelofs, Rinus. (2013). The Discovery of a New Series of Uniform Polyhedra. Bridges Enschede, Conference Proceedings 2013, 369-376

Deltahedral lattice made by augmentating icosahedrons and octahedrons. Relation between this lattice and hyperbolic plane is discussed on David A. Richter's page.

Fractal crystal comprised of Tetrahedra or Octahedra can be realized as deltahedron (with coplanar faces).

- Robert W. Fathauer, A Fractal Crystal Comprised of Cubes and Some Related Fractal Arrangements of other Platonic Solids, in Proceedings of the Bridges Leeuwarden, edited by Reza Sarhangi and Carlo Sequin, pp. 289-296, 2008.

Geraldine and its transformations are discussed in the following Knoll's papar.

- Knoll, Eva. (2000). Decomposing Deltahedra. ISAMA Conference Proceedings, Albany, NY. link

Composed of two adjoined pentagonal dipyramids. Multistable polyhedron.

- Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165-170, 1978.
- Weisstein, Eric W. "Multistable Polyhedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MultistablePolyhedron.html

Constructed from side-by-side connected strips of equilateral triangles.

- Roelofs, Rinus. (2013). The Discovery of a New Series of Uniform Polyhedra. Bridges Enschede, Conference Proceedings 2013, 369-376

A flexagon made up of nineteen triangles folded from a strip of paper. See Hexaflexagon at MathWorld.

- Gardner, M. "Hexaflexagons." Ch. 1 in Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games. New York: Simon and Schuster, pp. 1-14, 1959.
- Weisstein, Eric W. "Hexaflexagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Hexaflexagon.html

Some of honeycomb patterns use deltahedron. e.g. Octet Truss

Face-transive and may includes intersecting faces. See Jim McNeill's website for details.

- Shephard G.C. (1999) Isohedral Deltahedra, Periodica Mathematica Hungarica Vol. 39 (1-3), 83-106

Five deltahedra which have special symmetry properties. Mobius deltahedra have the same topology as
*Kleetopes*. See Melinda Green's
website and Roger Kaufman's
website for details.

Nets of 5-cell, 16-cell, 24-cell and 600-cell form deltahedron.

Equilateral triangle surface based on Penrose tiling.

- Robert J. Lang and Barry Hayes, "Paper Pentasia: an Aperiodic Surface in Modular Origami", The Mathematical Intelligencer, December 2013, Volume 35, Issue 4, pp 61-74.

Examples of deltahedral forms reconstructed from cubic polyhedral graphs with up to 10 vertices. List of reconstrcuted forms is here.

- Tsuruta, N., Mitani J., Kanamori Y., Fukui, Y., "Random Realization of Polyhedral Graphs as Deltahedra", Journal of Geometry Graphics, 19(2), pp.227-236, 2016.

Some of regular skew apeirohedron and Gott's regular pseudopolyhedrons have deltahedral surface. See the page on wikipedia.

Composed of connected strips of equilateral triangles. Tetrahelix.

- Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57

A toroidal deltahedron by augmentation of eight octahedron is one of Stewart toroids. See Jim McNeill's web for details.

A Helical deltahedron composed of face bond regular tetrahedra. See R. W. Gray's page for details.

- Fuller, R.Buckminster (1975). Applewhite, E.J. (ed.). Synergetics. Macmillan.

Deltahedra made by subdivision, twist and triangulation (regularization of irregular triangles).

- Gailiunas, Paul. (2014). Twisted Domes. Bridges: Mathematical Connections in Art, Music, and Science, 45-52