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In the memoirs of the Polish mathematician Hugo Steinhaus ("Mathematician for all seasons", vol. I, English translation, Springer, 2015) he recollects a conversation with Henri Lebesgue in 1938:

"He told us that it was a mathematician at the Sorbonne in the thirteenth century who had first observed that the two feet of a normal human being are not congruent by means of an orientation-preserving Euclidean transformation, and only then did shoemakers begin making different left and right shoes.

Can this be true?? I remember seeing a few surviving samples of ancient shoes in museums, but did not pay attention on this detail.

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    $\begingroup$ Interesting, specially if it is indeed related to some mathematical publication, but I guess this could receive better answers in History Stack Exchange $\endgroup$
    – Mauricio
    Sep 27 at 21:47
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    $\begingroup$ A thirteenth century mathematician referring to "orientation-preserving Euclidean transformation" is priceless. The only mathematician from that time associated with the University of Paris (they did not call it Sorbonne until later) I can think of is Alexander of Villedieu. But he is only known for Latin verses on doing arithmetic featuring lines like "always extract the square root by starting from the left". $\endgroup$
    – Conifold
    Sep 28 at 8:53
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    $\begingroup$ As a matter of fact, the mathematician had no work to do since footwear was made left/right until "at the end of the 16th century the heel made its appearance and became quite high for both men and women... The shape of shoes changed with the introduction of the heel, they became straights, i.e. they could be worn on either feet, no longer were they rights and lefts. This lasted for about 200 years and rights and lefts did not return until around 1800", Swallow, The history of shoes. $\endgroup$
    – Conifold
    Sep 28 at 9:16
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While some sources claim it is a modern invention, Encyclopedia Brittanica claims that Romans already differentiated right from left footwear, see shoe. Look also at these sandals from Ancient Egypt.

Looking on the web, it is a mess as is full of blog/news articles without sources pointing to somebody in the 1800s. For English sources it is sometimes claimed that left/right shoes were invented somewhere in Philadelphia, Pennsylvania. Some French sources claim that it was Alexis Godillot, an industrial man that introduced some famous godillot boots that differentiated left from right.

The closest source I got so far was from German Wikipedia (which also confusingly) claims that it was a German academic called Georg Hermann von Meyer who apparently advocated for chiral shoes under a work (yet to be found) called Über die richtige Gestalt des Schuhes (1858). Some of von Meyer's work is freely available (translated) in Wikisource, here. This observation implies that about 1858, chiral shoes were at least as popular as symmetrical shoes.

Anyway most of the stories seem like oversimplified anecdotes (including Lebesgue's story) and I don't think there is a clear source but I will update this answer if something more concrete comes up.

Edit: Conifold provided this academic paper by A. Swallow History of Shoes (1987), which does not cite any of the men above but clearly states that chirality in shoes has been reintroduced many times in history depending on the kind of footwear.

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    $\begingroup$ Chapter 6, "Die richtige Gestalt des Schuhes", in Georg Hermann von Meyer, "Die richtige Gestalt des menschlichen Körpers: in ihrer Erhaltung und Ausbildung für das allgemeine Verständniß dargestellt." Stuttgart: Meyer & Zeller 1874, pp. 115-139. Von Meyer was a professor of anatomy. As described in the book, symmetrical construction was common for children's shoes, slippers, and fancy lady's footwear, while men's shoes typically used chiral left/right construction. $\endgroup$
    – njuffa
    Sep 28 at 0:17
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    $\begingroup$ @user253751: To see the advantage, try to switch your left and right shoes. $\endgroup$ Sep 28 at 12:41
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    $\begingroup$ @AlexandreEremenko but my shoes are asymmetrical. $\endgroup$
    – user253751
    Sep 28 at 13:01
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    $\begingroup$ I have one set of athletic socks with L and R labels on them, and I've been wondering why this isn't more common. $\endgroup$
    – Barmar
    Sep 28 at 14:46
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    $\begingroup$ @DarrelHoffman fairly common for athletic socks: sockgeek.com/collections/left-right-fit-socks $\endgroup$
    – eps
    Sep 28 at 17:33
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I would speculate that the first shoes were chiral and that symmetrical shoes are a relatively recent invention.

  1. I believe that ancient cobblers would have made the shoe custom for the patron's foot. They would not have even had to think about it, the pair of shoe would be chiral.

  2. Symmetrical shoes would have been introduced with mass production. It must be easier to tool a production line to produce one type of shoe than two types of shoes.

That is my speculation. In support of this consider https://bellatory.com/fashion-industry/The-History-of-Shoes-Early-Footwear, which says that the earliest shoes were sandals. Just try and imagine symmetrical sandals. It does not make any sense.

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