I took a look at Ed Pegg Jr's exact algebraic unit-length embedding of the Heawood graph at http://community.wolfram.com/groups/-/m/t/518461 Expressed in terms of the single real root z of a degree-21 polynomial with 9-digit coefficients, it surely is a considerable improvement over the one Dan reviewed earlier. There are minor difficulties with the webpage definition. Although his list "pts" of vertices specifies their coordinates components correctly, they appear in an order disobeying his bipartite classification via index parity: this may be corrected by reversing signs of the final 6 x-components. Furthermore his ordering ignores any 1-2-4 rule facilitating correspondence between points and circles. When modifying his Mathematica script for Maple I was therefore eventually obliged to radically unscramble them. Plotted in the form of a Fano plane, the embedding appears as https://www.dropbox.com/s/3qv94ot8t2jw58b/fano_pegg.gif?dl=0 Everything looks tidily spaced apart ... hang on while I polish my specs ... 'Ullo, 'ullo, 'ullo, wot's this lurking down near (-1.4, -0.7) then? Looks uncommonly like another three-circle intersection ?! Indeed there is an eighth triple point precisely at (-1-x, y) , where (x, y) denotes the Fano point near (0.4, -0.7) ; and dually, an eighth unit circle centred at (1+x, y) meeting all three Fano points with x > 0 & y < 0 ! Won't please DWW then, but might appeal to RWG instead. Fred Lunnon # Maple data: z := 'z': zpol := - 153027 + 1353114*z - 1986024*z^2 + 2763312*z^3 + 10497808*z^4 + 31063072*z^5 - 8457728*z^6 - 40590336*z^7 + 44468736*z^8 + 164975616*z^9 + 119705600*z^10 - 494067712*z^11 - 763912192*z^12 + 414924800*z^13 + 1154613248*z^14 + 48496640*z^15 - 864485376*z^16 - 376307712*z^17 + 253231104*z^18 + 271581184*z^19 + 89128960*z^20 + 10485760*z^21; z := fsolve(zpol, z, real); # 0.1304718735 --- single real root! # Centres and points are alternate Heawood vertices; # if (x,y) is a point then (-x,y) is a centre epjpnt := [ [z, 1/2], # 0 [-1/2, 1/2*(-1+sqrt(3-4*z-4*z^2))], # 1 [1/2, -(1/2)*(-1+sqrt(3-4*z-4*z^2))], # 2 [-sqrt(3-4*z^2)/(2*sqrt(1+4*z^2)), -((z*sqrt(3-4*z^2))/sqrt(1+4*z^2))], # 3 [1/2,-((z*sqrt(3-4*z^2))/sqrt(1+4*z^2))-sqrt((8*z^2+sqrt(3+8*z^2-16*z^4))/(2+8*z^2))], # 4 [-z, -1/2], # 5 [(1/2)*(-1+sqrt(3-4*z*(1+z)))*sqrt((7+8*sqrt(3-4*z*(1+z)) -8*z*(-1+z*(1+2*z*(2+z))))/(13+8*z*(1+z)*(-3+2*z*(1+z)))), -(1/2)*sqrt((7+8*sqrt(3-4*z*(1+z))-8*z*(-1+z*(1+2*z*(2+z))))/(13+8*z*(1+z)*(-3+2*z*(1+z))))], # ?? NULL]: epjpnt := evalf(epjpnt); # epjpnt := [ [.1304718735, .5000000000], [-.5000000000, .2762120950], [.5000000000, -.2762120950], [-.8284016255, -.2161662241], [.5000000000, -1.160704403], [-.1304718735, -.5000000000], [.3968937166, -.7184582495] ]: # eighth_point := [-1.396893718, -.7184582435]: