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Topic: Tesselated Helicoid
Replies: 1   Last Post: May 17, 2014 5:08 PM

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Posts: 359
Registered: 9/16/06
Tesselated Helicoid
Posted: May 16, 2014 9:03 PM
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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.

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

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