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Tesselated Helicoid
Posted:
May 16, 2014 9:03 PM


Hope it is an interesting exercise in Hyperbolic geometry.
At any node/vertex the sum of angles is 420 degrees,an extra equilateral triangle ( any side is of unit length) is pushed in. This forms a warped polyhedral surface. (Six triangles can be assembled to make a flat hexagon). Around a vertex 3 pairs of normals meet above and 4 pairs meet below the polyhedral surface.
Find angle made by normals of adjacent faces.
Find coordinates of all vertices.
To better imagine spiraling of edges of helicoids or bending of a spine I worked with cardboard or plastic equilateral triangles and joined edge pairs with duct or cello tape.
http://i62.tinypic.com/34958g1.jpg
I have started with three points of equilateral triangle in first quadrant:
Rectangular coords (0,0,0), (1,0,0),(1/2, 1/sqrt(12), sqrt(2/3) or Spherical coords (0,0,0),(1,0,0),(1, pi/6, arctan(sqrt(2)).
Not able to further find coordinates for other points in the triangulation/tessellation.
My main motivation in starting this is that.. so many symmetric tesselations around origin for elliptic geometry exist but ( as much as I know ) none in hyperbolic geometry except those forming on surfaces of revolution...
Best Regards Narasimham



