Counting deltahedra with $2n$ faces

Question

A deltahedron is, according to Wikipedia,

a polyhedron whose faces are all equilateral triangles.

There is only one deltahedron with four faces: the tetrahedron.

Likewise, there is only one deltahedron with six faces: the triangular bipyramid

There are two with eight faces:

• the octahedron
• the biaugmented tetrahedron (not convex)

And at least five with 10 faces:

• the pentagonal bipyramid
• the augmented octahedron (contains coplanar faces)
• the three triaugmented tetrahedra

I'm looking to see if anyone has computed the number of deltahedra with $2n$ faces. I did a cursory OEIS search, but didn't find anything.

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Is there a known formula or recurrence relation for the number of deltahedra that can be constructed from $2n$ equilateral triangles? If so, is there a corresponding OEIS entry?

Answer 1

Consider the Shlegel diagram of these polyhedra https://en.wikipedia.org/wiki/Schlegel_diagram

The Schlegel diagram of these object are planar graphs all of whose faces are triangles, so their duals are trivalent planar graphs. These objects are enumerated by the OEIS sequence $A000109$ https://oeis.org/A000109

Answer 2

I applied OEIS A000577 to proof the count of 2n faces deltahedra.

I used 3 4-iamonds (A000577(4)=3) to proof only one deltahedron with four face. I used 12 6-iamonds (A000577(6)=12) proof only one deltahedron with six faces. Also, I used 66 8-iamonds (A000577(8)=66) proof only two deltahedron with eight faces. In the same manners, I used 448 10-iamonds (A000577(10)=448) proof only five (5) deltahedron with ten faces. Recently, I have spent two years (year 2018 -2020) and used 3334 12-iamonds (A000577(12)=3334) to construct and proof thirteen (13) deltahedra with twelve faces.

They are D12(0,4,4,0), D12(1,3,3,1), D12(1,4,1,2), D12(2,0,6,0), D12(2,1,4,1), D12(2,2,2,2)1, D12(2,2,2,2)2, D12(2,2,3,0,1), D12(2,3,1,1,1)1, D12(2,4,0,0,2), D12(3,1,1,3), D12(3,1,2,1,1), D12(4,0,0,4). Where D12 stands for a deltahedron with 12 faces. For example, D12(0,4,4,0) denote that 12-deltahedron with 8 vertices: 4 of degree 4 and 4 of degrees 5 and none of degree 3 and 6. Since having known the count of thirteen (13) deltahedra with twelve faces, by single-face augmentation and multiple-face augmentation methods (Naoya Tsuruta, Enumeration of Deltahedral Graphs With Up to 10 Vertices, @2014 ISGG) we discovered forty-seven (47) graphs of deltahedra with fourteen faces. They are D14(0,3,6,0), D14(0,4,4,1), D14(0,5,2,2), D14(0,6,0,3), D14(1,2,5,1), D14(1,3,3,2), D14(1,4,1,3), D14(2,1,4,2)1, D14(2,1,4,2)2, D14(2,2,2,3)1, D14(2,2,2,3)2, D14(2,2,2,3)3, D14(2,3,0,4), D14(3,0,3,3)1, D14(3,0,3,3)2. D14(1,4,2,1,1)1, D14(1,4,2,1,1)2, D14(1,5,1,0,2), D14(2,1,5,0,1), D14(2,2,3,1,1)1, D14(2,2,3,1,1)2, D14(2,2,3,1,1)3, D14(2,3,1,2,1)1, D14(2,3,1,2,1)2, D14(2,3,1,2,1)3, D14(2,3,2,0,2)1, D14(2,3,2,0,2)2, D14(3,1,2,2,1)1, D14(3,1,2,2,1)2, D14(3,1,3,0,2), D14(3,2,0,3,1)1, D14(3,2,0,3,1)2, D14(3,2,1,1,2)1, D14(3,2,1,1,2)2, D14(4,0,2,1,2), D14(4,1,0,2,2), D14(2,2,4,0,0,1), D14(2,3,2,1,0,1)1, D14(2,3,2,1,0,1)2, D14(2,4,0,2,0,1), D14(2,4,1,0,1,1), D14(3,1,3,1,0,1), D14(3,2,1,2,0,1)1, D14(3,2,1,2,0,1)2, D14(3,2,2,0,1,1)1, D14(3,2,2,0,1,1)2, D14(4,0,2,2,0,1). Please note that D14(2,1,4,2)1 and D14(2,1,4,2)2 share the same numbers of vertices and edges, that is isomorphism graph.

Currently, I am working on counting deltahedra with sixteen (16) faces. Since having known the count of forty-seven (47) graphs of deltahedra with fourteen (14) faces, by single-face augmentation and multiple-face augmentation methods we discovered 226 graphs of deltahedra with sixteen faces including twelve isomorphism graphs of D16(3,2,1,1,1). As per Prof. Naoya Tsuruta’s paper (Random Realization of Polyhedral Graphs as Deltahedra, J. for Geometry Graphics 227-236, 2015) there exists 233 graphs including 154 Deltahedral Graphs and 79 Non-Deltahedral Graphs. I will continue construct and confirm with Prof. Tsuruta for all graphs of deltahedra with sixteen (16) faces.