Talk:Reuleaux triangle

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"By the Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width." - Doesn't a circle have less area? Or is this assuming an unspoken "nontrivial"? Or am I not understanding something (quite possible - I have only a sketchy layman's understanding of the topic!) - DavidWBrooks 19:01, 6 Apr 2004 (UTC)

Actually, no. See mathworld's article for a calculation of the area of the Reuleaux triangle (we should probably add that here too).

Barbier's theorem says that for curves of constant width, perimeter is uniquely determined by width. Consequently, the isoperimetric inequality (the circle encloses the most area for a curve of given perimeter) says that the circle encloses the most area of any curve of given constant width. Dan Gardner 23:15, 6 Apr 2004 (UTC)

  • Most area - of course! Knew it was one end of the spectrum.
  • As to this introduction: '... simplest nontrivial example ..." would the circle be the simplest overall example? If so, the article should probably say it. DavidWBrooks 01:12, 7 Apr 2004 (UTC)

The intersection of the balls of radius s centered at the vertexes of a regular tetrahedron with side length s is called the Reuleaux tetrahedron, but is not a surface of constant width. It can, however, be made into a surface of constant width in two ways.

So, what would those ways be? Am I missing an explanation? There's certainly no explanation at the (nonexistent) Rouleaux tetrahedron page... Azure Haights 03:49, Nov 6, 2004 (UTC)

This page's explanation could be made much clearer by the addition of a few pictures, in my opinion; compare this page to the bicycle article. Does anyone have public domain graphics at hand? Wyvern 19:25, 7 May 2005 (UTC)

[edit] Diameter

What does "diameter" mean, for a curve? I think the diameter page doesn't help. It only tells about the diameter of various geometric objects, making the statement "a curve in which all diameters are the same length" meaningless. Maybe the authors of this page could add something to Diameter. --fudo 12:50, 7 January 2006 (UTC)

Interestingly enough, my "Dictionary of Mathematics" (Millington and Millington) doesn't define "diameter" by itself, only in specific uses ("Diameter of a conic", "diameter of a sphere"). Is there no general meaning of the term? I can't believe that.
Further interestingly enough: The dictionary does have a definition of "radius," although it seems to apply only to circles. (Can you have a radius of anything else?) Anyway, I've replaced the second reference of "diameter" in this article with a more exact definition, taken from curve of constant width- DavidWBrooks 13:41, 7 January 2006 (UTC)

[edit] Area

"By the Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. In particular, the area is given by pi*d, where d is the diameter."

The dimensions of "the area" and "pi*d" are not the same. Can this be possible?

--Tonyho 06:21, 30 June 2006 (UTC)

That's bogus. By my calculations, the area is \left({\pi\over2} - {\sqrt3\over2}\right)d^2. Anyone disagree? —Keenan Pepper 07:09, 30 June 2006 (UTC)
Well, MathWorld agrees with me, so it's going in. —Keenan Pepper 07:27, 30 June 2006 (UTC)
Here, here. --Tonyho 08:21, 30 June 2006 (UTC)
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