Talk:Deltahedron

Image?

If somebody could find/generate an image with all of the deltahedrons displayed, that would make this page far more useful and comprehensible.Matt gies 23:46, 28 Feb 2004 (UTC)

18-faced deltahedron

Any proof that we can't have an 18-face deltahedron where some vertexes have 5 and some have 4?? Georgia guy (talk) 21:59, 1 May 2014 (UTC)

It's easy to list all the topologies with valences <6; none of them have 11 vertices. —Tamfang (talk) 05:46, 4 May 2014 (UTC)

I wonder where I buried that list! —Tamfang (talk) 07:57, 9 October 2024 (UTC)

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circumspheres

I am uneasy about saying the deltahedra appear in solutions of the Thomson problem and Tammes problem. Only the three regulars have vertices all lying on a sphere! Some other solutions have convex hulls topologically equivalent to the convex deltahedra, but these are not deltahedra. —Tamfang (talk) 22:45, 4 September 2024 (UTC)

Okay. I will remove it. Dedhert.Jr (talk) 01:48, 5 September 2024 (UTC)

convexity

A non-convex deltahedron is a deltahedron that does not possess convexity, meaning it has coplanar faces or collinear edges.

Consider an octahedron and an icosahedron sharing a face. Are they not a non-convex deltahedron? Do they have coplanar faces or collinear edges?

A figure with coplanar faces may still be convex, but not strictly convex like the eight. —Tamfang (talk) 19:19, 14 September 2024 (UTC)

Hmm! Some sources use "strictly convex", whereas Cundy and Litchenberg sources use "convex" instead to classify those eights. Google Scholars shows the "strictly convex deltahedron" but they do not explain what are those specifically, and the worst part is, Google Books does not shows anything about "strictly convex", but "convex" instead . Dedhert.Jr (talk) 00:45, 15 September 2024 (UTC)

Eulerizing

The number of possible convex deltahedrons was given by Rausenberger (1915), using the fact that multiplying the number of faces by three results in each edge is shared by two faces, by which substituting this to Euler's polyhedron formula. In addition, it may show that a polyhedron with eighteen equilateral triangles is mathematically possible, although it is impossible to construct it geometrically.

The second sentence is false: it is topologically impossible, not only geometrically.

The first sentence is ungrammatical; perhaps something like

Using Euler's polyhedron formula, Rausenberger (1915) showed that the number of a triangle-faced solid's vertices is determined by the number of faces, and this allowed an enumeration ...

—Tamfang (talk) 22:12, 1 July 2025 (UTC)

@Tamfang

Re first sentence: That was paraphrased for showing that Litchenberg stated there are eight possible convex deltahedron, starting from the Euler's polyhedral formula. Each face of deltahedron is bounded by three edges, and multiplying the number of faces by three the each edge is twice "because each edge is shared by two faces". See the Litchenberg source, particularly at p. 263.

Re second sentence: See the Litchenberg source, particularly at p. 265: "Going back to equations $3F=2E$ (2) and $3V_{1}+4V_{2}+5V_{2}=2E$ (5), we see that if we substitute $3F$ for $2E$ in (5) and solve for $F$, we have ${\textstyle F={\frac {1}{3}}(3V_{1}+4V_{2}+5V_{3})}$. Using this result we see the solution 2 [$(0,1,10)$] of equation $3V_{1}+2V_{2}+V_{3}=12$ (7) seems to produce a deltahedron for which $F=18$. But as we suggested earlier, it can be shown that this configuration is geometrically impossible." And by the way, how is that impossible topologically as well? Dedhert.Jr (talk) 00:29, 2 July 2025 (UTC)

First: I question not the substance but the wording. "results in [sentence]" is ungrammatical, "by which substituting" invites us to expect a clause that never comes.

Second: Years ago I wrote a program to search for abstract triangulations of closed surfaces with degree <6; it found none with 11 vertices / 18 faces. —Tamfang (talk) 00:36, 2 July 2025 (UTC)

@Tamfang Oh sorry for miscommunication. For the first reply, I won't mind if you change the writing. Dedhert.Jr (talk) 02:58, 2 July 2025 (UTC)

David Eppstein is bolder than I. —Tamfang (talk) 02:51, 9 July 2025 (UTC)