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Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves distance and $F(X) = X$).

Thanks!

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    $\begingroup$ Doesn't the essential idea---that symmetries have to do with invariance under certain reflections or rotations---go back to the ancients? Although they didn't have our function concept, they did understand all the isometries of the plane. $\endgroup$ – Joel David Hamkins Dec 25 '14 at 20:17
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    $\begingroup$ Well, to be more precise, I would like to know the first published book or article where symmetry in Geometry is defined as a function ... Thanks. $\endgroup$ – Humberto José Bortolossi Dec 26 '14 at 0:25
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    $\begingroup$ This would be a good question for the new hsm.stackexchange.com. $\endgroup$ – Paul Siegel Dec 30 '14 at 4:13
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    $\begingroup$ The reference I posted in my answer seems to refute my comment above: evidently, the ancients did not use the concept of symmetry that way. $\endgroup$ – Joel David Hamkins Jan 2 '15 at 16:09
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If I understand correctly, you mean the definition of an object’s {symmetries} as its stabilizer under some group action on an ambient space. While a clearcut “first” might not exist, I’d say Jordan has early approximations to that, provided you are willing to replace “function” by “transformation”, “substitution”, or “motion”. E.g. in his Traité des substitutions (1870, p. 50):

§V. — Symmetry of rational functions.

The link between groups and functions.

    60. Let $\mathrm F_1$ be an arbitrary rational function of $k$ letters $a, b, c,\dots$; $\mathrm F_1, \mathrm F_\alpha, \mathrm F_\beta$ the $1.2.\dots k$ functions obtained by letting the $1.2.\dots k$ substitutions $1, \alpha, \beta,\dots$ operate on these letters (...) The $M$ substitutions $1,\alpha,\dots$ which don’t alter the function $\mathrm F_1$ obviously form a group, which one can call the function’s group.

(Earlier Serret’s Cours d’algèbre supérieure (1866, p. 387) wrote that the “substitutions admises” by a function form a “système conjugué” ($=$ Cauchy’s term for group, p. 251). I’d have thought crystallographic groups were defined as stabilizers before this, but it seems not.)


EDIT: To answer the question as recast in your comment (“first published book or article where symmetries in geometry are defined as transformations”), I think a problem is that early sources won’t call any maps symmetries: instead you’ll find expressions like mouvements qui superposent à lui-même (Jordan 1867, p. 230), Transformationen in sich (Klein & Lie 1871; Klein 1893, p. 326), Deckbewegungen (Sohncke 1875, p. 115), Deckoperationen (Schoenflies 1891, p. 13; Curie 1894, p. 395; Love 1906, p. 147), or at best symmetry-operations (Hilton 1903, p. 32).

While these references all contain the main idea, the exact terminology you want does not seem to appear until A. Speiser, Die Theorie der Gruppen von endlicher Ordnung (2nd ed., 1927, p. 78):

Eine kongruente Abbildung des Gitters auf sich selber nennen wir eine Symmetrie des Gitters.

Interestingly, this sentence is absent in the first edition (1923, p. 53). In English, the earliest such definition I can find is by H. S. M. Coxeter in Regular skew polyhedra in three and four dimensions and their topological analogues (1937, p. 35):

We define a symmetry (or "symmetry operation") of any figure as a congruent transformation of the figure into itself (i.e., a combination of translations, rotations, and reflections).

Coxeter later repeats it in e.g. Mathematical Recreations and Essays (1939, p. 130) or Regular complex polytopes (1974, p. 2).

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  • $\begingroup$ Thanks Ziegler. And what about euclidean geometry? Indeed, being more precise, I would like to know the first published book or article where symmetries (in Geometry) are defined as transformations. $\endgroup$ – Humberto José Bortolossi Dec 26 '14 at 0:27
  • $\begingroup$ You might check: Rogers, A. F. (1926). A mathematical study of crystal symmetry. Proceedings of the American Academy of Arts and Sciences (pp. 161-203). American Academy of Arts and Sciences. (In particular, "Symmetry Operations" beginning on p. 162 and the references contained therein.) $\endgroup$ – Benjamin Dickman Dec 31 '14 at 23:32
  • $\begingroup$ @BenjaminDickman Why? I don't notice anything in Rogers that's not already in the above-quoted Hilton (1903)... $\endgroup$ – Francois Ziegler Jan 1 '15 at 17:11
  • $\begingroup$ I had wondered specifically if there was anything of value in foot-note 3 of p. 163: The source is in German (Wulff, 1897) but I could not track down that particular reference... You are right that it (and other sources) are mentioned in Hilton (1903); I had not searched through Hilton for that reference, though it makes sense reading Rogers' acknowledgement at the end of the paper. (In any event: your answer here is very nice!) $\endgroup$ – Benjamin Dickman Jan 1 '15 at 19:24
  • $\begingroup$ @BenjaminDickman Thanks. I found the Wulff paper here -- I may be wrong but I'm not finding more about the OP's question in it. $\endgroup$ – Francois Ziegler Jan 1 '15 at 20:06
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From the introduction of Legendre's Revolution (1794): The Definition of Symmetry in Solid Geometry, Giora Hon and Bernard R. Goldstein, Archive for History of Exact Sciences, Vol. 59, No. 2 (January 2005), pp. 107-155.

The modern scientific concept of symmetry is, in the first place, a mathematical idea. It refers to an intrinsic property of a mathematical object which makes that object invariant under certain classes of transformations, such as rotation, reflection, inversion, or other operations. These invariant properties are the subject of group theory - a mathematical theory which explores, systématises and formalises these permanent features that are preserved by the object despite the transformation that it undergoes. The pervasiveness of symmetry in modern mathematics and science is well known, and it will be taken for granted here.

In antiquity the fundamental meaning of summetrian Greek was proportion which in mathematics also meant commensurability. Clearly, the meaning of symmetry (Latin: symmetrid) changed over time, and it will be the goal of this paper to describe the introduction of this term into solid geometry at the end of the 18th century with an entirely new meaning. We put aside for another occasion the aesthetic sense of summetria/symmetria in antiquity as well-proportioned (notably in Vitruvius's De architecture^ and its subsequent applications in architecture and art (and, to some extent, in scientific contexts).

In fact, in the period from 1794 to 1815 three scientists claimed to use symmetry in a new way: Adrien-Marie Legendre (1752-1833) in solid geometry (1794), Sylvestre François Lacroix (1765-1843) in algebra (1797), and René- Just Haüy (1743-1822) in crystallography (1815). In a series of papers now in preparation we will discuss the introduction of symmetry in algebra and in crystallography, and then argue that the modern usages of symmetry in all scientific domains flow from these three seminal figures.

...

In this paper we investigate Legendre's work on solid geometry where he introduced a new definition of symmetry that, we claim, has served as the basis for the modern scientific concept of symmetry.

This text seems to be an extremely thorough account of the history of the modern concept of symmetry.

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    $\begingroup$ Hon and Goldstein have expanded their thesis into a book which has interesting reviews (one, two): there «their argument that Legendre, the “hero” of their saga, (1) gave a “definition” of “symmetry” which was at once (2) a drastic break with the past and (3) a seminal influence on the future» is hotly contested on all 3 counts. $\endgroup$ – Francois Ziegler Jan 2 '15 at 20:08
  • $\begingroup$ @FrancoisZiegler Thanks for those reviews, which are indeed interesting. It seems the history of this concept is slippery; I'm inclined to agree with a remark mentioned in one of the reviews, that symmetry is a concept without a history, because it is innate. $\endgroup$ – Joel David Hamkins Jan 2 '15 at 20:33
  • $\begingroup$ Other reviews, to which I don't seem to have access, may be more positive: (three, four, five, six)... $\endgroup$ – Francois Ziegler Jan 2 '15 at 20:43

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