Deltahedra and deltahedral surfaces

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.
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
• 2019-07-31Add Mobius deltahedron
• 2019-07-30Initial release

B

Bellows deltahedra

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

Biform deltahedra

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".

Boerdijk–Coxeter helix

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).

C

Convex biform deltahedra

Five convex deltahedra are biform.

Convex deltahedra

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

Conway's toroidal deltahedron

A smallest deltahedron with a hole.

Coplanar deltahedra

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.

Corpuscles

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

Concave Cupola

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.

Cylindrical deltahedra

Composed of antiprismatic rings.

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

D

Diamond lattice

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

F

Fractal crystal

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.

G

Geraldine (Endo-pentakis icosi-dodecahedron)

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

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

Goldberg's bistable icosahedron (Siamese dipyramid)

Composed of two adjoined pentagonal dipyramids. Multistable polyhedron.

H

Helical deltahedron

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

Hexaflexagon

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

Honeycomb (space-filling tessellations)

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

I

Isohedral deltahedra

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

M

Mobius deltahedra

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.

N

Nets of regular convex 4-polytopes

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

P

Paper pentasia

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.

R

Realization of cubic polyhedral graphs

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.

S

Skew apeirohedron (infinite Polyhedra)

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

Spiral deltahedra

Composed of connected strips of equilateral triangles. Tetrahelix.

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

Stewart toroids

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

T

Tetrahelix

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.

Twisted dome

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