I am looking for the origin of the symbol $S_n$ used by chemists to denote the symmetry operation consisting of a $\smash{\frac{2\pi}n}$ rotation ($C_n$) about an axis and a reflection in a plane perpendicular to that axis.

Google Scholar traces earliest use of the operation’s name “Drehspiegelung” to A. Schoenflies’, Krystallsysteme und Krystallstructur (1891). However, that book does not seem to denote it $S_n$.

Physics books on molecular structure had $S_n$ by the 1930s: an early use can be seen in L. Tisza, Zur Deutung der Spektren mehratomiger Moleküle (1933, p. 59), so it must have started earlier.

(N.B.: Symbols like $\mathrm C_n$, $\mathrm S_n$ can also denote symmetry groups. This should not be confused with the notation for symmetry operations or “elements” — see e.g. this table.)

  • 1
    $\begingroup$ Burckhardt (1971) quotes Schönflies as using gothic $\mathfrak S(\omega)$ in correspondence with Fedorov, you should probably look at the many references of that paper. $\endgroup$ Commented Nov 3, 2018 at 23:59
  • $\begingroup$ I had downloaded Schönflies' book from Internet Archives. What I am not sure is that he uses Gothic S for pure reflection. How is he distinguishing pure reflection and rotary-reflection? The S(ω) is not associated with Drehspiegelung. $\endgroup$
    – ACR
    Commented Nov 4, 2018 at 0:06
  • 1
    $\begingroup$ Book and Encyklopädie (1906, p. 444) use $𝔖$, $𝔄(α)$, $\overline{𝔄}(α)$ for Spiegelung, Drehung, Drehspiegelung. The correspondence points to (1889, p. 175) where Drehspiegelung was $𝔖(α)$. Earlier (1888, p. 489) had used $𝔖$ for symmetrische Operation. $\endgroup$ Commented Nov 4, 2018 at 5:06
  • $\begingroup$ Thank you (will acknowledge your input). This confirms my observations. Drehung symbol is equivalent to A. In the 1906 encyclopedia, he gives reference to his book Krystallsysteme und Krystallstructur (1891) in the footnote, which uses the same symbols. My German is rudimentary so I take help from machine translation after finding the paragraph of interest. $\endgroup$
    – ACR
    Commented Nov 4, 2018 at 5:24
  • $\begingroup$ Can you summarize in an answer to take it off the unanswered list? (Also, did you mean to say that people later switched back and standardized on $\mathfrak S(\alpha)$ for rotation-reflection?) $\endgroup$ Commented Nov 4, 2018 at 5:30

2 Answers 2


Some data points from the convoluted history of rotatory-reflection (RR) and its notation:

  • Bravais (a classmate of Galois) was among the first to consider orthogonal symmetry of crystals. His papers (1849, p. 144; 1850, p. 93) introduce notation for the rotations ($\Lambda^n$ or $\mathrm L^n$ according as the axis is “principal” or not) and also for plane and central reflections ($\Pi$, $\mathrm P$, $\mathrm C$). But none for RR, which he deliberately disregards on grounds that the extra crystal class it allows “doesn’t seem to occur in Nature” (1851, pp. 229–230, 275), though he does mention (p. 237) a shape with such symmetry, viz. Naumann’s sphenoid (a.k.a. Breithaupt’s tetragonal sphenohedron, from σφήν $=$ wedge):


  • Gadolin (often credited with first deducing the 32 crystal classes) does consider RR under the name sphenoidal symmetry (and thus gets the “tetragonal-(di)sphenoidal class” Bravais had missed) (1871, pp. 18, 28). He apparently has no symbol for the operation.

  • Curie (also credited with deducing the 32 classes) completes Bravais’ notation by writing $\Pi^n$ for RR, which he calls plane of alternating symmetry of order $n$ (1884, pp. 430, 442, 453; 1885):


  • Minnigerode (also credited) calls RR a symmetry of the second kind. He has no symbol for it, as he prefers to use rotatory-inversion (1884, pp. 203, 213; 1887, pp. 151, 162).

  • Schoenflies briefly calls RR an operation $\smash{\mathfrak S(\frac{2\pi}n)}$ in (1889, p. 175), then switches to $\smash{\overline{\mathfrak A}(\frac{2\pi}n)}$ and the name Drehspiegelung in his influential book (1891, pp. 29, 49) and Encyklopädie article (1906, p. 444). His later papers keep that notation, e.g. (1926, p. 219).

  • Fedorov calls RR combined symmetry, without symbol (1892, pp. 27–28). This terminology is adopted by e.g. Swartz (1909, pp. 374–377), Jellinek (1915, pp. 355–361).

  • Viola calls an RR axis Spiegeldrehachse, without symbol (1904, p. 246).

  • Auerbach in Handbuch der Physik calls an RR axis Spiegeldrehachse, symbol $\color{red}{S_n}$ (1908, p. 58).

  • Voigt calls an RR axis in direction $r$ Spiegelachse, symbol $S_r$; or Drehspiegelachse, symbol $\mathfrak S$ (1910, pp. 66–77; 1918, pp. 245, 333–336).

  • Rinne calls rotations Gyralen and RR Gyroiden, without symbol; in a later edition he denotes them $\mathrm G_n$ and $\underline G{}_n$ (1912, p. 1; 1922, p. 1).

  • Rogers calls RR rotatory-reflection, symbol $ap_{2\pi/n}$ (1912, pp. 5, 7; 1926, pp. 167–171; 1935, p. 478).

  • Ewald in Handbuch der Physik calls an RR axis Drehspiegelachse, symbol $\color{red}{S_n}$. In a footnote about the nomenclature (of both operations and groups), he adds: “As experience showed that misunderstandings can hardly result, a simplified spelling of Latin letters and subscripted symmetry indices established itself, which has also been applied here” (1927, pp. 199, 203, 206). He edited Zeitschrift für Kristallographie.

  • Wigner adopts Ewald’s Handbuch notation (1930, p. 139).

The linked papers of Swartz (1909) and Rogers (1926) have nice historical accounts in English. In decreasing order of likelihood, I’d say that the $S$ in $S_n$ seems to come from 1) Spiegeldrehachse, 2) sphenoidal, or 3) Schoenflies’ ephemeral $\mathfrak S$.

  • $\begingroup$ Thanks. I also think that Ewald is behind modernization of these symbols. One cannot find any previous notion of $S_n$ in any publication. $\endgroup$
    – ACR
    Commented Nov 12, 2018 at 20:42
  • $\begingroup$ @M.Farooq Why do you say that? Auerbach 1908 is right above, and I would expect that googling will reveal more like it. $\endgroup$ Commented Nov 12, 2018 at 22:41
  • $\begingroup$ Sorry, I missed Auerbach. I wanted to say the Ewald 's set symbolism is still used today. Auerbach's $S_n$ is there but then his rotational axes are same as Schoenflies and other symbols don't match. $\endgroup$
    – ACR
    Commented Nov 12, 2018 at 23:07

In the 1906 Encyklopädie (see comments above), Schonflies gives reference to his book Krystallsysteme und Krystallstructur (1891) in the footnote. His book uses the same symbols, namely $\mathfrak S$, $\mathfrak A$(α), $\overline{\mathfrak A}(\alpha)$ with a bar for Spiegelung, Drehung, Drehspiegelung respectively. However, in the modern literature, the $S_n$, where n indicated the 360/n rotations is used. It seems Schonflies was switching back and forth in his symbolism. An early use of modern symbol can be seen in Tisza's 1930s article: Zur Deutung der Spektren mehratomiger Moleküle https://link.springer.com/article/10.1007/BF01342114

  • $\begingroup$ I changed the symbol with some trials. I feel (personal opinion) it was just a matter of typesetting where the symbol $\mathfrak S$(\alpha) became $S_n$. Is there an email option for you? $\endgroup$
    – ACR
    Commented Nov 6, 2018 at 2:02
  • $\begingroup$ After doing an extensive search for pre-1927, I strongly feel that the Ewald in Handbuch der Physik is perhaps the first one to modernize it $\mathfrak S\ to S_n. $\endgroup$
    – ACR
    Commented Nov 7, 2018 at 7:35
  • $\begingroup$ I will check Zeitschrift fur Kristallographie. I think we have to distinguish several things. Symmetry element, Symmetry operation, and the point group. There is too much overlap in symbolism. Schoenflies does $C_n$ but for group of crystals in the table. The $S_n$, I am searching is the symmetry element. Have a look at the table here sites.cns.utexas.edu/jones_ch431/…. Our decades are correct. It is in between 20-25, before Wigner. I will check Lehrbuch der Algebra by Weber. He wrote that he was inspired by it. $\endgroup$
    – ACR
    Commented Nov 8, 2018 at 0:37
  • $\begingroup$ I think only Weyl's work is remaining. I checked the terms symmetrielemente in the Zeitschrift (vol 60-61). They are still using old symbols. So it is now Weyl or Wigner for crosschecking. Weyl wrote his book in 1925. Wigner wrote his in 1930s both are extremely difficult books for chemists, but I can still search the symbols. Fortunately, translations are also available. $\endgroup$
    – ACR
    Commented Nov 8, 2018 at 6:33
  • $\begingroup$ With apologies, I am no longer sure the answer should even involve Schoenflies, if it’s a matter of finding who started “$S_n$”. I wrote an answer as this is too long for comments. $\endgroup$ Commented Nov 12, 2018 at 20:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.