Talk:Reuleaux tetrahedron

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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)