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I was reading lately that the quintisection of an angle is possible with paper folding (origami). Now, in contrast to the trisection of an angle, a problem which was discussed historically, and was actually successfully solved already by the Greeks (although not with a compass and a straghtedge, but with other means), the history of quintisection is less discussed. Arthur Baragar shows here that if one can construct a complex number $a$ with a compass and twice-notched straightedge then it belongs to a field $K$ that lies in a tower of fields having degrees of extensions 2,3 or 5. I assume hence that one can quintisect an angle with these apparatuses.

However, what was the history of the quintisection problem?

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  • $\begingroup$ Trisection is more famous exactly because it is more difficult, and thus its investigation led to important developments. $\endgroup$ – Alexandre Eremenko Mar 18 '17 at 20:39
  • $\begingroup$ Shouldn't it be "quinquisection"? $\endgroup$ – fdb Mar 18 '17 at 22:14
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    $\begingroup$ @fdb "Pentasection" is one more variant. $\endgroup$ – Conifold Mar 19 '17 at 1:03
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Not much history to it, I am afraid. It seems that methods of trisection rather obviously (to those who considered them) applied to quintisection as well, so the problem was of little theoretical interest in itself.

Without restriction on tools the problem of angle multisection is easily solvable. For instance, the quadratrix of Dinostratus, originally the trisectrix of Hippias (c. 420 BC), can be used to multisect any angle into any number of parts (and square the circle to boot, as discovered by Dinostratus), although it was originally meant to trisect. The Archimedean spiral (invented by his friend Conon) can also be used for the same purposes, which would have been obvious to Archimedes, if not Hippias, although he also only discusses trisection. Euclid in Elements of course describes a construction of regular pentagon with straightedge and compass, which is a case of quintisection for a special angle. In a discussion on the NCTM Math Forum another ancient case of quintisection is mentioned:

"It would appear that at Cydonia such quintisection is DONE for a very specific reason (which by now I cannot disclose or Dr. Lahoz would harm me): but the environment SUGGESTS a very SIMPLE construction (???) which so far escapes our analysis: I guess that it may be related to the Archimedean trisection."

Cydonia or Kydonia was an ancient city-state on the northwest coast of the island of Crete, that lasted from archaic to Byzantine periods, and is a site of modern archeological excavations. I was unable to confirm or find specifics on this mention. Angel Garcia suggested how the quintisection could be performed in the style of Archimedes with a marked straightedge (and avoiding towers of fields, etc.), see also Trisection and Pentasection of Line Segments and Angles, but we have no ancient sources describing such a construction.

In modern times quintisection was considered at least as early as Viète (1540 – 1603) and Bürgi (1552 - 1632), Briggs used it to construct his trigonometric tables. Wallis's Treatise of Angular Sections (1648, published 1685) also discusses the "quinquisection" explicitly. Here is from Roegel's Reconstruction of the Tables of Briggs and Gellibrand’s Trigonometria Britannica (1633):

"After having considered the triplication and quintuplication, Briggs considered again the equations, but now in order to divide an arc into $3$ (chapter 4 [11, pp. 5–10]), $5$ (chapter 6 [11, pp. 12–18]), and $7$ (chapter 7 [11, pp. 19–20]) parts. For instance, if p is the chord of an angle a, we have seen that the chord of 3a is $c(3a) = 3p-p^3$, and trisecting an angle amounts to solve a cubic equation [13, p. 461]. For a division by $5$, the equation is $c(5a) = 5p -5p^3 + p^5$. Then we have $c(7a) = 7p - 14p^3 + 7p^5 - p^7$. And so on. Even sections lead to the equations $c(2a) = \sqrt{4p^2-p^4}$, $c(4a) = \sqrt{16p^2-20p^4 + 8p^6-p^8}$, and so on. The general case is considered in chapter 8 [11, pp. 20–28] and the coefficients of all these equations can be obtained from a table given by Briggs [11, p. 23]. This table can easily be extended.

Briggs’ work is certainly partly inspired from François Viète’s Ad Angulares Sectiones which has such a table [83, p. 295]. Viète is in particular explicitly quoted on the cover of the Trigonometria Britannica. It is interesting to observe that Jost Bürgi also obtained another similar table for the same purpose, certainly independently, and described it in his “Coss,” probably around 1598 [50, pp. 33–35] [58, p. 77]... Like Bürgi before him [50], Briggs develops a method to find some roots of these equations by iteration... This happens to be exactly the so-called Newton-Raphson method, with the constraint that only one new digit is obtained at a time. The Newton-Raphson method actually goes back at least to Viète... The sixth chapter of the Trigonometria Britannica is devoted to the quintisection of arcs and expounds the same method. Briggs considers the equation $x^5-5x^3 + 5x = a$, a first approximation $b$ of a root, and he obtains a new approximation $L = b + c$ with $c=\frac{a-b^5+5b^3-5b}{ 5b^4+15b^2+5}$."

Even in the recent recreational literature the quintisection problem received little attention, especially compared to the endless recycling of the trisection. Application of Miller's Trisector to Quinquisection of any Angle appeared in Proceedings of the Royal Society of Edinburgh in 1902. Duncan and Barnier published one page note On Trisection, Quintisection,...,Etc. in the Monthly in 1982, their only reference is an algebra textbook. Balo-Roy's Trisection and Pentasection of any Angle by Dividers is not easily accessible, I am afraid. There is a bit more on the origami constructions.

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