[math-fun] Waterman Polyhedra
The Waterman polyhedra are apparently the convex hulls of the centers of spheres in the fruitstand packing that lie within a fixed radius of a fixed point. For large distances, they closely approximate spheres. The original image is very large, and the lacework of polyhedron edges is clearly visible at full size. I have reduced it to a 1024x768 wallpaper, in which the polyhedral faces are not distinguishable, the overall impression is of a spherical fractal with with a network of great circles emphasized. http://astronomy.swin.edu.au/~pbourke/polyhedra/waterman/index.html - David W. Wilson "Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"
dwilson>http://astronomy.swin.edu.au/~pbourke/polyhedra/waterman/index.html Yow, DeathStars! Pentagons, heptagons, adjacent nonagons, 14- and 16-gons. Are all n-gons possible? And for what Radii^2/2 are all the vertices on the sphere? What an interesting approach to triangulation. --rwg
And think of the spherical packing puzzles a person might come up with by combining the individual Voronoi regions commanded by adjacent spheres, something along these lines... http://www.plambeck.org/oldhtml/mathematics/klarner/tombsupplement/index.htm Thane Plambeck http://www.plambeck.org/ehome.htm R. William Gosper wrote:
dwilson>http://astronomy.swin.edu.au/~pbourke/polyhedra/waterman/index.html Yow, DeathStars! Pentagons, heptagons, adjacent nonagons, 14- and 16-gons. Are all n-gons possible? And for what Radii^2/2 are all the vertices on the sphere? What an interesting approach to triangulation. --rwg
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dwilson>http://astronomy.swin.edu.au/~pbourke/polyhedra/waterman/index.html gosper>Yow, DeathStars! What an interesting approach to triangulation. It's striking as you look at the pictures of the more nearly spherical ones how there seem to be really strong "nodal" lines and points that are persistent across pictures--places that tend to be inside faces, hence don't get "colored" by the edge/vertex renderings. Is there any way to characterize these places? A lot seem to be located via bisections or similar simple constructions, but there's also all sorts of odd "twins" etc. What kind of function would we get if we were to assign values to each point on the limit sphere reflecting how "resistant to coloring" it was (kind of like how the Mandelbort set is often colored based on how "divergent" the points are)?
participants (4)
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David Wilson -
Marc LeBrun -
R. William Gosper -
Thane Plambeck