Talk:Reuleaux tetrahedron

Is there really enough mathematical content here, beyond the pretty pictures (and they are quite pretty) to justify an article? Doing a search for some external links to justify including it, I found:

• MathWorld: Reuleaux Tetrahedron. Similar material to here, but more thorough, including detailed calculations of the arc length and volume. The only motivation given is analogy to the triangle, despite the observation that the 3d construction doesn't lead to a constant width shape the way the 2d construction does Claims credit for coining the name "Reuleaux Tetrahedron", which I'm not convinced of — it's an obvious enough name and obvious enough construction that probably many people have come up with both, and I don't see much reason to believe that Weisstein was first of them. But in the absense of evidence to the contrary we should probably at least mention his claim, if we continue to keep this article.
• Bazylevych and Zarichnyi: On Convex Bodies of Constant Width. Claims (bottom p.4) that it is well known that this shape has constant width, apparently erroneously forgetting to consider pairs of support planes through opposite edge pairs. So not a good reference to include.
• A promotion for the Colorado School of Mines (p.29 of the pdf file) in which marbles in this shape are inscribed with the school logo.
• Creating a Social Robot for Playrooms in which the rounded shape is used to make a puppet for some social interactivity experiments. No math.

Google Scholar turned up a few more:

• Rote, Curves with increasing chords. Dated April 1993, so a strong contender as a counterexample to Weisstein's coinage claim. Uses a Reuleaux tetrahedron as a lower bound example for a geometric construction, but shows that a modified shape provides a stronger lower bound.
• Tokieda, A Mean Value Theorem. The American Mathematical Monthly, 1999. Repeats the mistake that this shape has constant width.
• Glicksman, Analysis of 3-D network structures and Glicksman, Energetics of Polycrystals. Something about modeling foams and biological cell structures. Seems to require 120 degree dihedrals but includes (fig.6) a Reuleaux tetrahedron as an example of one possible foam cell shape even though its dihedrals are wrong.

And, excluding other pages that seemed to be copies of links to here or MathWorld, that was pretty much it. Not promising for a page that seems to intend to be about mathematics. The mathematical content seems to be limited to: some people thought the Reuleaux triangle could be generalized in this way, but it turned out to be a mistake. Is that enough?

—David Eppstein 07:02, 12 September 2006 (UTC)

Update: books.google.com found

Partial Differential Equations and Applications: Collected Papers in Honor of Carlo Pucci - Page 44

by Paolo Marcellini, Talenti Talenti, Giorgio Talenti - Mathematics - 1996 - 392 pages

In particular, as a solution of Problem A, Bonnesen and Fenchel (1934) conjectured

a sort of Reuleaux tetrahedron constructed by Meissner (1912) (see also ...

I will have to go to my library to view a physical copy of this book but it's looking more likely that the history of discovery of its shape and the repeated mistake about its width will make for sufficient content for an article despite the limited mathematical content. —David Eppstein 18:31, 12 September 2006 (UTC)

Equivalents for other solids?

File:UAE 50 Fil Coin (Front).jpg

Reuleaux heptagon coin

Would I be correct in surmising that there woukd be an equivalent Reuleaux and Meissner shape for each the other four platonic solids? — Preceding unsigned comment added by 64.121.1.13 (talk) 01:10, 31 August 2020 (UTC)

Uh, no? There is also no Reuleaux square in the plane. You can make constant-width curves in the plane, using the same idea as the Reuleaux triangle, for polygons with an odd number of sides, and some of these are used as coins. But this works because the odd polygons always have a vertex opposite each side that can be used as the center of each circular arc. In 3d, the tetrahedron has a vertex opposite each face, and the Reuleaux tetrahedron uses a sphere centered at that vertex. But none of the other regular polyhedra have vertices opposite their faces; instead they have a face opposite each face. You can still make shapes that look like cubes (say) with spherical patches as its faces instead of flat faces, but there's no particular reason to call them "Reuleaux", just like rounded triangles whose edges are arcs centered at other places than the opposite vertex are rounded triangles but are not Reuleaux traingles. —David Eppstein (talk) 01:49, 31 August 2020 (UTC)

Constant-width with tetrahedral symmetry

The article explains that the Meissner tetrahedron is three-dimensional object of constant width. However, because it has three of its six edges rounded a bit, it has no tetrahedral symmetry.

I recently came across what appears to be a self-published paper from 2012 by a fellow named Patrick Roberts, who proved that it is possible to create a constant-width tetrahedron that is perfectly symmetric. There is also a web page with a link to the paper here: http://www.xtalgrafix.com/Spheroform2.htm

I found this fascinating, and I would like to mention this object in the article, but due to the original-research nature of the paper, I thought I'd ask here first.

@David Eppstein: you seem to be the only regular here on this talk page. What say you? ~Anachronist (talk) 06:22, 7 May 2022 (UTC)

That does sound like a very interesting construction to include. Unfortunately, I don't think the paper you link can count as a reliable source. I didn't find anything relevant by Roberts in MathSciNet, nor by the other mathematician he mentions, Bergerud. So I think we need to wait unless/until better sources turn up. —David Eppstein (talk) 06:47, 7 May 2022 (UTC)

It's a good paper, and it's unfortunate we can't use it. I couldn't find anything else either. It seems worthy of publication to me. I wish he had submitted it somewhere.

However, perhaps a brief mention in the 'external links' section would be OK. I'll add it. ~Anachronist (talk) 06:51, 7 May 2022 (UTC)