On 2015-04-26 06:42, rwg wrote:
On 2015-04-25 22:33, Dan Asimov wrote:
On Apr 25, 2015, at 4:37 PM, rwg <rwg@sdf.org> wrote:
On 2015-04-25 14:42, Mike Stay wrote:
I guess they're not-quite-but-approaching regular? On Sat, Apr 25, 2015 at 11:43 AM, Bill Gosper <billgosper@gmail.com> wrote:
gosper.org/gridheptagons.png
Yes. They're actually Minsky "circles". --rwg
Which is the triangular lattice version of what the Bresenham algorithm is for the square lattice?
--Dan
I seriously doubt it. The "triangular lattice" is just a square one skewed by {x,y} → {(x Csc[π/7] - 2 y Sin[π/7])/(Sqrt[2] (-1 + Csc[π/7]^2)^(1/4)), Sqrt[2] y (-1 + Csc[π/7]^2)^(1/4) Sin[π/7]} , (http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition , p14) which circularizes Minsky ellipses computed by the iteration x[n+1] = x[n] - Floor[4 Sin[π/7]^2 y[n]] ; y[n+1] = y[n] + x[n+1]
This recurrence has period 7 for {x0,y0} among ..., {-7, -1}, {-9, -6}, {-11, -5}, {-12, -5}, {-6, -2}, {-6, -1}, {-9, -4}, {-5, -5}, {-5, -3}, {-5, -1}, {-4, -2}, {-9, -2}, {-2, -2}, {-2, -1}, {-8, -7}, {-12, -10}, {-11, -10}, {-10, -9}, ...
But, Oh, SHOOT!, had I instead used
x[n+1] = x[n] - y[n] ; y[n+1] = y[n] + Floor[4 Sin[π/7]^2 x[n+1]] (not x[n])
the grid becomes nearly equilateral (~ 7:8:8) and the heptagons http://gosper.org/heptagrid.png look regular enough for a 1 April hoax! --rwg
In case anyone worries that they really might be regular, here are two pairs of small ones, http://gosper.org/hexoct.png , each pair with a permutation of the same four side lengths. Note the illusion of an octagon in a hexagon, only possible because the small ones each have two parallel sides. (tsk.) --rwg