Benzels are regions in the hexagonal grid that are analogous to Aztec diamonds in the square grid vis-a-vis the theory of tilings. Here is an example of a benzel that has been tiled by trihexes (unions of three adjacent hexagonal cells):
(The picture has been superimposed with its graph theoretic dual
in which cells become vertices and tiles become
collections of edges connecting nearby vertices.
"Stones" are the tiles Conway and Lagarias called T2 tiles
and "bones" are the tiles they called L3 tiles;
see the article
Tiling with polyominoes and combinatorial group theory
by Conway and Lagarias and the follow-up article
by Thurston.)
I gave my first talk on benzels online on November 29, 2021
as part of the
CAP21
(Combinatorics and Arithmetic for Physics) workshop
sponsored by IHES. (I had only two hours to throw the talk together,
so please forgive the lack of slides!) The talk was called
"Conjectural Enumerations of Trimer Covers of Finite Subgraphs
of the Triangular Lattice" and
the video
is available on YouTube.
In 2022 I gave several talks about trihex tilings of benzels:
In addition, the following writeups are available:
In 2024 I gave two talks that touched on stones-and-bones and/or benzels:
Spaces of tilings
(presented on August 8 at a birthday conference for Nicolau Saldanha) and
Tilings? Again?
(presented on October 23 at the UMass Lowell "Working On What" seminar).
In addition to reading the slides, you can view
a video
of the latter talk or listen to
the audio track.
Both talks focused more on stones-and-bones tilings of triangles
(TN regions in Conway and Lagarias' notation)
rather than benzels.
For those who are interested in tilings of triangles, see the article
A polyomino tiling problem of Thurston and its configurational entropy
by Lagarias and Romano as well as the OEIS entries
A334875 and
A377309.
Regarding the twenty open problems mentioned in
"Trimer covers in the triangular grid",
there has been progress on about half of them as of October 2024.
In particular:
I coined the term "benzel" in 2021 in honor of
the chemical element benzene (whose hexagonal structure
reflects the hexagonal cells of which benzels are composed),
the Mercedes-Benz car company
(whose logo is reminiscent of the way three hexagonal cells meet),
the inventor Gustav Benzel
(whose 1870 innovation, the merry-go-round, undergoes rotation
in a manner vaguely reminiscent of
the three-fold rotational symmetry of benzels),
and author Carl Sagan (whose novel Contact introduced the word "benzel"
as the name of a spinning gadget of extraterrestrial design).
Here "DLPY" refers to the published article by
Colin Defant, Rupert Li, James Propp, and Benjamin Young,
"DFLPY" refers to the preprint by
Colin Defant, Leigh Foster, Rupert Li, James Propp, and Benjamin Young,
and "BCL" refers to the preprint by Seok Hyun Byun, Mihai Ciucu, and Yi-Lin Lee.