# Talk:Reuleaux triangle

"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)

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)

"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 
  
    
      
        
          (
          
            
              
                π
                2
              
            
            −
            
              
                
                  
                    3
                  
                
                2
              
            
          
          )
        
        
          d
          
            2
          
        
      
    
    {\displaystyle \left({\pi  \over 2}-{{\sqrt {3}} \over 2}\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)

Narasimhachar from Bangalore:
pi*d is incorrectly, perhaps hastily, purported to be the AREA of a Reuleaux polygon, whereas it is the PERIMETER. The area, of course, is not uniquely determined by the width.  —Preceding unsigned comment added by 59.92.155.34 (talk) 19:08, 12 June 2009 (UTC) 

The first sentence, which defines the subject of the article, reads:
"A Reuleaux polygon is a curve of constant width - that is, a curve in which all diameters are the same length."
Exactly what does the word "diameter" mean here?  The diameter of a circle is a well-known concept, but this is not a circle.  (This usage may refer to a line segment or to its length.)  
Also, the diameter of a point-set is a well-known concept in geometry: It is the supremum of all distances between points of that set.  (This is usually a number, not a line having that number as its length.)
But what kind of diameter are we discussing here?  It can't be the circle defintion.  And it can't be the supremum definition, (or a line segment between two points in the set that are at a distance equal to that supremum) ... since then a straight line segment would fit the definition of a curve of constant width, as would an equilateral triangle.
So how about either 
a) defining this use of the word "diameter", 
or
b) rewriting the definition without using mystery concepts.
Choice b) is  strongly preferred.Daqu (talk) 02:20, 16 November 2008 (UTC)

The Reuleaux triangle can be generalized to regular polygons with an odd number of sides.
Even more generally, a curve of constant width can (I believe) be built on any set of lines provided that no two are parallel. —Tamfang (talk) 21:02, 31 January 2009 (UTC)

I have extended a different aspect of Reuleaux Triangles, now introducing the Blankenhorn Triangle where the triangle is circumvented by a continuous elliptical curve. Go see: http://ellipticaltriangle.blogspot.com/
No matter how relevant my math is and of the real economic figures that Wiki chooses to print even though they are lies put out by our government (including the real negative 10% GDP growth even though they claim it to be positive), I no longer edit front pages. It's up to the wiki editors to insert salient items I mention in the talk page. Thomas_Blankenhorn (talk) 14:52, 6 December 2013 (UTC)

A Reuleaux polygon is a curve of constant width - that is, a curve such that, if two parallel lines are drawn tangent to the curve in any orientation, the distance between them is fixed
...I'm pretty sure the distance between parallel lines is fixed anyway. I get the feeling that the author might be trying to convey something different from how I'm interpreting it, but I really don't know what it is. Anybody up to rephrasing that first sentence? --superioridad (discusión) 23:59, 19 April 2009 (UTC)

(Furthermore, the definition was duplicated - somebody copied and moved it sloppily, I suspect. I've eliminated the duplication.)
I wrote the definition in the lede (both the present version and the previous one complained of here), not noticing that it repeated what was there a couple of paragraphs down. —Tamfang (talk) 19:06, 21 April 2009 (UTC)
The problem with this article now is that it makes zero sense to the layman who stumbles across it; it's fine for somebody who already knows what a Reuleaux polygon is, but for those who are ignorant it's baffling. (Many technical definitions are like that.) Since this is an encyclopedia, not a technical manual or a textbook, it would be nice to have a simpler explanation to lead off, something that gives the "aha!" sense of how these are "sort of like" circles in a very unexpected way. I know the word "diameter" is mathematically incorrect but it sure conveys the sense of what makes these special. I'm not sure how to phrase it, however. - DavidWBrooks (talk) 10:42, 21 April 2009 (UTC)
How about width as a less-incorrect substitute for diameter? :P —Tamfang (talk) 19:06, 21 April 2009 (UTC)
Width is good. Maybe something like " ... is one of the few geometric figures that, like the circle, is always the same width when measured from any point to the opposite point." ... "opposite" is a good word for laymen, as it makes intitutive sense even if defining it mathematically might be tough. - DavidWBrooks (talk) 21:12, 21 April 2009 (UTC)
Hey, we could call it something like curve of constant width! Great! — I don't know about you but to me opposite implies that each point corresponds to one and only one other point, with respect to some center.  That's clearly false here: every point on each arc is 'opposite' to the same corner. (This is not true of every CCW; e.g. [1].)  —Tamfang (talk) 19:38, 23 April 2009 (UTC)
Yeah, you're right. I'm not sure there is a good way to say it in layman-accessible terms (often a very hard thing to do, which is why there are so many crappy textbooks). It's too bad, because the current wording doesn't really convey to casual readers the counter-intuitive charm of these shapes. - DavidWBrooks (talk) 01:45, 24 April 2009 (UTC)

Okay, idea: how about a new lead image one that compares two Reuleaux triangles, each with a pair of parallel tangent lines drawn on it, showing that each pair is the same distance apart? --superioridad (discusión) 14:45, 1 May 2009 (UTC)

Looking at the page again, that rotating image has a lot of visual impact that conveys something - maybe we should just move it up - DavidWBrooks (talk) 16:22, 1 May 2009 (UTC)

Why does the method of construction say "start with an equilateral triangle"?
Try this:

Choose a constant radius.
Draw a circle (or such arc as sense indicates sufficient), say the blue one in the diagram,
With centre on that circle draw an arc, say of the green circle, to intersect the first circle,
Draw an arc of a third circle, say the red one, centred on the point of intersection of the first two circles. to intersect them.
Voilà, surely, the bulgy bit is a Reuleaux triangle with no equilateral triangle drawn.--SilasW (talk) 18:17, 23 September 2009 (UTC)
reply: those three places you just stuck your compass point, that's an equilateral triangle with sides length r. You didn't measure the length of the arcs, you measured the sides of a triangle. 96.224.41.4 (talk) 18:58, 21 January 2012 (UTC)

Under "Other Uses", the article mentions the following:

A Reuleaux triangle rolls smoothly and easily, but makes a poor wheel because (...)
Several pencils are manufactured in this shape, (...) having the advantage of not rolling off tables.

So which is it? Do they roll smoothly and easily, or do they not roll off tables? McGravin (talk) 03:41, 2 February 2010 (UTC)

HA! Good catch. I've watered down both statements; does that seem enough? - DavidWBrooks (talk) 14:12, 2 February 2010 (UTC)
There's no contradiction -- when there are two or more "rollers" of constant width (including Reuleaux triangles) resting on a flat surface with a flat sheet of material on top of them (File:Reuleaux_triangle_54.JPG), then if you pull the top material forward, it will move purely horizontally (without any up-and-down bumpiness).  On the other hand, a Reuleaux triangle under gravity has a preferred (i.e. minimum potential energy) orientation with one point up; to get a Reuleaux triangle pencil rotating, you have to apply additional force beyond that necessary to overcome static friction, in order to raise its center of gravity.  However, in the case of a circular pencil, once you overcome static friction, it will start merrily rolling away... AnonMoos (talk) 12:57, 13 December 2010 (UTC)

It would be nice if the two articles could be better coordinated. AnonMoos (talk) 12:59, 13 December 2010 (UTC)

yes, especially because this article says that these things "can be rolled" as if square wheels can't also be rolled. I don't mind if there is a semantic meaning behind this idea that says that square wheels cannot roll, but it's not obvious to me what it is. 96.224.41.4 (talk) 19:01, 21 January 2012 (UTC)
Squares are not useful as rollers -- that is, when you rotate them, something placed on the top of them is not kept a constant distance off the ground.  Curves of constant width are useful rollers.  Consult File:Reuleaux_triangle_54.JPG, and imagine what the difference would be if the upper sheet of material were resting on pegs of square cross-section... AnonMoos (talk) 05:36, 24 April 2012 (UTC)
A counterexample. —David Eppstein (talk) 06:40, 24 April 2012 (UTC)
"Rollers" does not equal "Wheels". A roller has something on top of it as well as a surface contacting the bottom, and has no axle. --Guy Macon (talk) 21:02, 2 February 2018 (UTC)

The signs are said to represent a prospector's shovel.  They do coincidentally resemble a Reuleaux triangle, but I don't think they're equilateral. —Tamfang (talk) 01:13, 7 February 2011 (UTC)

Could you provide a pronunciation key for Reuleaux? Thanks, Yoninah (talk) 00:20, 25 June 2015 (UTC)

I don't know a good source for the pronounciation. It looks like it should be pronounced in the French way (roo-LOW) but Reuleaux was actually German so it wouldn't surprise me if he used a different pronounciation. —David Eppstein (talk) 05:52, 25 June 2015 (UTC)

I am somewhat surprised to see no reference to Mazda's iconic rotary engine. Does it not harness the shape of Reuleaux's triangle? I wish to add reference to it, but hesitate as I've never butchered a "Good Article" before. - ArchitectOfIdeas (talk) 20:45, 2 February 2018 (UTC)

Not a Reuleaux triangle. "The rotor is similar in shape to a Reuleaux triangle with sides that are somewhat flatter." -- Wankel engine
--Guy Macon (talk) 20:53, 2 February 2018 (UTC)
It is not true that there is no reference to it in this article. In the "mechanism design" section, we say "The rotor of the Wankel engine is shaped as a curvilinear triangle that is often cited as an example of a Reuleaux triangle.[9][44] However, its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width." —David Eppstein (talk) 20:58, 2 February 2018 (UTC)

The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:

UAE 50 Fil Coin (Front).jpg
Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 17:03, 16 April 2021 (UTC)

Swapped out for a different coin. —David Eppstein (talk) 18:41, 16 April 2021 (UTC)

I'm in no way an expert but given that the defining factor in manhole cover shape is "bigger than the hole", and the only reason the conversation came up (at least that I can find) is that Microsoft used "Why are manhole covers circuluar?" as an interview question, should we really keep the line about it in the article? I'm not going to just remove it as irrelevant partially because I'm happy to hear it defended and partially because if I did so with my non-account it'd likely just get bot reverted as vandalism, but to me it seems out of place at best and confusing at worst.

It's not that it's "bigger" it's that it has a constant diameter and therefore there's no way to maneuver it so it will fit through the hole - unlike, say, a square, which can be dropped through its own hole if you orient it properly. - DavidWBrooks (talk) 11:41, 29 April 2021 (UTC)
This really makes no sense. Any shape can fall down through its own hole if the shape is not bigger than the hole below it. 77.241.129.128 (talk) 10:03, 24 May 2025 (UTC)
The cover is always slightly larger than the hole. But with a square hole and cover, you can lift the cover up, turn it on edge, and then drop it down through the diagonal of the hole. With a circular (or Reuleaux triangle) hole and cover, this is not possible. –jacobolus (t) 15:51, 24 May 2025 (UTC)

I wonder if this symmetric solid is worth mentioning. The source is a 2012 self-published website by Patrick Roberts, but this is in my opinion more beautiful than Meissner's quasi-tetrahedral solid:

Roberts, Patrick (2012). "Spheroform with Tetrahedral Symmetry". Xtal Grafix. Corvalis, Oregon. (also see "Proof of Constant Width of Spheroform with Tetrahedral Symmetry")
A literature search turned up a mention in this nice paper:

Arelio, Isaac; Montejano, Luis; Oliveros, Déborah (2023). "Peabodies of constant width". Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry. 64 (2): 367–385. arXiv:2107.05769. doi:10.1007/s13366-022-00637-z..
(It was also added a couple years ago to Reuleaux tetrahedron § External links.)
–jacobolus (t) 16:18, 16 July 2024 (UTC)