Community RSS Feed https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions in tag Visual Arts sorted by active [GIF] Off the End (Stereographic projections of rotating regular polygons) https://community.wolfram.com/groups/-/m/t/1710318 ![Stereographic projections of rotating regular polygons] **Off the End** Each horizontal row shows the stereographic projection to the line of (the vertices of) a rotating regular polygon. The middle row shows the image of the vertices of an equilateral triangle, the rows above and below show a square, the rows above and below that a regular pentagon, etc. The speeds are all chosen so that one vertex passes through the south pole of the unit circle every $\frac{4}{5}$ of a second, which obviously means the triangle is spinning much faster than the regular 60-gon that shows up in the first and last rows. Here&#039;s an unoptimized animation showing all the rotating polygons to get a sense of their relative speeds: ![Rotating regular polygons] Of course, this uses stereographic projection: Stereo[p_] := p[[;; -2]]/(1 + p[[-1]]); The actual image is set up as a GraphicsGrid; note that offset just a small irrational number to ensure that none of the polygon vertices pass through the north pole (where stereographic projection is undefined) at any of the time steps. With[{bg = GrayLevel[.2], max = 60, offset = Sqrt[2.]/1000000}, Manipulate[ GraphicsGrid[ Table[ {Graphics[ {White, PointSize[.005], Table[ Point[Append[Stereo[ReIm[Exp[I (-2 π t/n + π/2 + 2 π i/n)]]], 0]], {i, 1, n}]}, Background -&gt; bg, PlotRange -&gt; {{-2, 2}, {-.0125, .0125}}]}, {n, Join[Reverse[Range[4, max]], Range[3, max]]}], ImageSize -&gt; 540, Background -&gt; bg, Spacings -&gt; 1, PlotRangePadding -&gt; None], {t, offset, 1 + offset - #, #}] &amp;[1/40] ] : https://community.wolfram.com//c/portal/getImageAttachment?filename=proj11r.gif&amp;userId=610054 : https://community.wolfram.com//c/portal/getImageAttachment?filename=rot1.gif&amp;userId=610054 Clayton Shonkwiler 2019-06-24T16:14:52Z