[math-fun] OK, so where did Betsy Ross get cos(2π/13)?
She probably neither knew nor cared that Out[87]= 1/24 (-2 + ( 2 ((-13)^(2/3) 2^(1/3) + (-13 - 195 I Sqrt[3])^(1/3)))/(-5 + 3 I Sqrt[3])^(1/3) + Sqrt[ 6 (26 + 2^(2/3) (1703 - 195 I Sqrt[3])^(1/3) + 2^(2/3) (1703 + 195 I Sqrt[3])^(1/3))]) In[89]:= ArcCos@%% // FullSimplify Out[89]= (2 \[Pi])/13 In[53]:= MinimalPolynomial[Cos[2 \[Pi]/13]] Out[53]= -1 + 6 #1 + 24 #1^2 - 32 #1^3 - 80 #1^4 + 32 #1^5 + 64 #1^6 & These are guaranteed to solve in radicals. (Why? Well trivially, In[27]:= ToRadicals[Cos[2 \[Pi]/13]] Out[27]= -(1/2) (-1)^(11/13) (1 + (-1)^(4/13))) A great source of tests for Julian's and my sextic solver. —Bill
Possibly she used origami, or had some other kind of angle trisector? Wikipedia's "Tridecagon" entry gives this link to Gleason's Monthly article on the topic; see pages 192-193. https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.t... NDE
My guess is that she used a marked thread. Make 12 marks on this thread using a ruler, put it down on the field to make a circle with the help of this plate here - done! --Ed Pegg On Thursday, July 4, 2019, 09:28:24 AM CDT, Elkies, Noam <elkies@math.harvard.edu> wrote: Possibly she used origami, or had some other kind of angle trisector? Wikipedia's "Tridecagon" entry gives this link to Gleason's Monthly article on the topic; see pages 192-193. https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.t... NDE _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Plus maybe a paper cone! Nice. Another question: Were Betsy's stars "points out"? https://en.wikipedia.org/wiki/Betsy_Ross_flag#/media/File:Flag_of_the_United... or "points up."? https://en.wikipedia.org/wiki/Betsy_Ross_flag#/media/File:Betsy-Ross-Flag.jp... Wikipedia says Ross's most verifiable contribution to the design was pentagrams replacing hexagrams. —rwg On Thu, Jul 4, 2019 at 7:35 AM ed pegg <ed@mathpuzzle.com> wrote:
My guess is that she used a marked thread. Make 12 marks on this thread using a ruler, put it down on the field to make a circle with the help of this plate here - done!
--Ed Pegg
On Thursday, July 4, 2019, 09:28:24 AM CDT, Elkies, Noam < elkies@math.harvard.edu> wrote:
Possibly she used origami, or had some other kind of angle trisector?
Another chance to mention Julian's lovely quadricuspid cycloids pentasectrix and trisectrix <http://gosper.org/tripenta.gif>:
Wikipedia's "Tridecagon" entry gives this link to Gleason's Monthly article on the topic; see pages 192-193.
https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.t...
NDE
participants (3)
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Bill Gosper -
ed pegg -
Elkies, Noam