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Peter Taylor pointed out at MathEduc that some BD$1 coins from 1997 are Reuleaux triangles:

          Coin
          (Image from de.ucoin.net.)
Does anyone know why they were shaped this way? Was there some pragmatic reason connected to its constant-width property? Or was it just a design/aesthetic decision?

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  • $\begingroup$ I've noticed the same thing on a British £0.20, it is a constant width pentagon $\endgroup$
    – Cato
    Jun 20, 2018 at 11:08
  • $\begingroup$ heptagon, sorry $\endgroup$
    – Cato
    Jun 20, 2018 at 11:11
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    $\begingroup$ FWIW the coin in the image is not British. $\endgroup$
    – Peter Taylor
    Jun 20, 2018 at 14:31
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    $\begingroup$ @Cato not just the UK 20p coin, but also the 50p. $\endgroup$
    – alephzero
    Jun 20, 2018 at 15:21
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    $\begingroup$ This question might have been better asked on the History SE. It seems likely that it was to save on metal, but then again it could have simply be because someone of high authority thought that having cooler-looking, math-smart coins was important for some political reason, like improving the nation's image as a one of intellectuals, or maybe good to raise interest of mathematics in the public. Why things happen can be very arbitrary. I think math-history is meant for the history of math concepts, not their applications. $\endgroup$
    – JoL
    Jun 21, 2018 at 15:58

4 Answers 4

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The Blaschke-Lebesgue Theorem states that among all planar convex domains of given constant width $B$ the Reuleaux triangle has minimal area.$^\dagger$

The area of the Reuleaux triangle of unit width is $\frac{\pi - \sqrt{3}}{2} \approx 0.705$, which is approximately $90\%$ of the area of the disk of unit diameter. Therefore, if one needs to mint (convex) coins of a given constant width and thickness, using Reuleaux triangles allows one to use approximately $10\%$ less metal.

$\dagger$ Evans M. Harrell, A direct proof of a theorem of Blaschke and Lebesgue, September 2000.

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    $\begingroup$ This is an astute observation. I did not know the B-L theorem. Thanks. $\endgroup$
    – Joseph O'Rourke
    Jun 20, 2018 at 18:51
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    $\begingroup$ while this is pretty cool (and the best answer from a mathematical point of view) is worth noting that we don't know if this was actually the reason for this choice $\endgroup$
    – Ant
    Jun 21, 2018 at 10:19
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    $\begingroup$ It's far from clear from this answer why a coin would require a) a shape of constant width or b) a specific value of this width, nor why the need to minimize the amount of metal used would apply to some coins but not the vast majority. $\endgroup$
    – jwg
    Jun 21, 2018 at 10:54
  • $\begingroup$ @jwg I agree. A reason for a) might be the use in coin operated machines which do not have to reorient the coin to measure the width and therewith its validity. $\endgroup$
    – M. Winter
    Jun 28, 2018 at 13:51
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The reason to not have a circle is likely purely aesthetic.

Once you've decided to mint a coin which isn't a circle, it's pretty important to still make one of constant width so it doesn't get stuck in machines. Also, many machines use the width of the coins to sort them (see for instance this youtube video showing one in action). That's a lot easier to do if you only have a single width for each coin instead of a range.

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    $\begingroup$ Good point and nice video! $\endgroup$
    – Joseph O'Rourke
    Jun 20, 2018 at 18:52
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    $\begingroup$ Also, differently shaped coins are a tremendous help to blind people fetching change from their pockets. $\endgroup$
    – Lee Daniel Crocker
    Jun 20, 2018 at 19:19
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    $\begingroup$ @LeeDanielCrocker Not only to blind people. Having coins that feel and look different one from the other is very useful to tell them apart. $\endgroup$
    – Federico Poloni
    Jun 21, 2018 at 11:38
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    $\begingroup$ The linked Wikipedia article mentions that some of the coins were called "Bermuda Triangles", in which case the triangular shape may be because of the name (Bermuda Triangle being a famous phrase). The Reuleaux shape is probably better than a pure triangle for reasons already mentioned (constant width, save metal, etc.) $\endgroup$
    – Brian Minton
    Jun 21, 2018 at 14:44
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    $\begingroup$ Since they are apparently struck as commemoratives instead of circulating coinage, I doubt that the ability for machines to sort them was a factor in their design. $\endgroup$
    – Dennis Williamson
    Jun 22, 2018 at 0:32
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Could it be to help the blind. I know it's not a uk coin, but the 50p and 20p are septagons to aid the blind. The old 10p was a similar size to the 50p; and the of 5p was a similar size to the 20p, so they made them a different shape. Also the 2p was a similar size to the 10p as well, so the 10p have grooves around the edge and the 2p is smooth. Same for the 1p (smooth)and it's similar size to the 5p (groved).

To summarize two groups of coin are a similar size:

1p (smooth edges), 5p (grooved edges), 20p (septagon shape). And the same for 2p, 10p and 50p, respectively.

The designs were to aid the blind who could feel the coin to know what is was as the sizes were too similar to distinguish.

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I remember when the 50p coin was introduced in the UK. I don’t know why they didn’t make it circular. It was quite a bit bigger than other coins, so helping the blind doesn’t seem like a relevant reason.

But, then, once you’ve decided to abandon the usual circular shape, why choose a shape of constant width? The reason that I recall is that this would make it easier for machines to handle the coins.

All purely from memory, and it was a long time ago, so no guarantees of correctness.

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