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 having 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).

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

The smallest deltahedron with a hole.

• Weisstein, Eric W. "Toroidal Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ToroidalPolyhedron.html

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

(Triangulation of) Cowley's dodecarhombus

A polyhedron obtained by triangulating Bilinski Dodecahedron with 60 degree rhombuses.

• Izidor Hafner and Ed Pegg Jr, "1. Nets for Cowley's Dodecarhombus", http://demonstrations.wolfram.com/1NetsForCowleysDodecarhombus/, Wolfram Demonstrations Project, Published: March 2019 25

Diamond lattice

Deltahedral lattice made by augmenting icosahedrons and octahedrons. The relation between this lattice and a hyperbolic plane is discussed on David A. Richter's page.

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.

Geodesic folding of tetrahedron

A family of deltahedra that can be constructed by folding regular triangular grids. See also Twisted Dome.

• S. Nishimoto, T. Horiyama, T. Tachi, Geodesic Folding of Tetrahedron, Journal of the International Society for the Interdisciplinary Study of Symmetry, Symmetry: Art and Science, 2019, Special Issue: 11th Congress and Exhibition of SIS, Kanazawa, Japan, November 25-30, 2019

Geraldine (Endo-pentakis icosi-dodecahedron)

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

• 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.

• 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

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 polyhedron/tessellations)

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

• Weisstein, Eric W. "Space-Filling Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Space-FillingPolyhedron.html

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

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.

Nets of regular convex 4-polytopes

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

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.

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.

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 composed of eight octahedra is one of Stewart toroids. See Jim McNeill's web for details.

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.

Toroidal deltahedron

A g-holed torus, having a topological genus of 1 or greater. e.g. Conway's toroldai deltahedron. Stewart toroids.

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