My newest design on Shapeways is a model of the Lawson-Klein surface : a stereographic projection of
( cos(u)cos(2v), cos(u)sin(2v), sin(u)cos(v), sin(u)sin(v) )
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My newest design on Shapeways is a model of the Lawson-Klein surface : a stereographic projection of
( cos(u)cos(2v), cos(u)sin(2v), sin(u)cos(v), sin(u)sin(v) )
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The formula is symmetrical enough that I thought the ‘best’ projection point — the point on the 3sphere farthest from the LK surface — would probably be one of the 80 points whose coordinates are permutations of
i.e. the vertices of a hypercube, the midpoints of its edges and so on.
I now think that ain’t necessarily so, but don’t know how to search efficiently for a better one.
In stereographic projection from (0,0,0,1), the LK surface is asymptotic to
which in turn is tangent to the planes x′=x/(1-z)=±2, which is the projection of a sphere centred at (0,±1,0,2)/√5; so I tried projecting from there, and am pleased with the result.