general method for Kruschke domes

[David Zhang]

After working on this for about a week, i finally figured out a method for generating Kruschke domes of any frequency, working off of a sketch up model and spreadsheet i found in this group from Gerry.

Here are the steps:

1.  Create an insphere of the base polyhedra  - i.e. a sphere that fits perfectly inside an icosahedron or octahedron.   it almost works with a tetrahedron but the corners get cut off.  

2.  subdivide the primary triangles on the polyhedra into the desired frequency.

3.  now we need to cut the insphere with parallel slices perpendicular to the 'up' direction to create the flat layers.  we can do this by using a different axis for each side of the primary triangle on the polyhedra.  on the polyhedra, we rotate a point at the center of the base of the primary triangle up the triangle's height plus half its side length to get the 'axis point'.  this axis point represent the 'north pole' if the triangle is on the equator, pointing up, and parallel with a vector that goes from the center of the polyhedra to the axis point.

4.   Now we apply this magic formula to get points on the insphere that corresponds with a parallel slice projected from the polyhedra.   Take the dot product of the axis point in step 3 and a point that corresponds with the subdivisions created in step 2.  Multiply this dot product by the ratio of the polyhedra's  insphere to its circumsphere side length, then take the arccosine of this number.     Now we have an angle that corresponds with the radius of a small circle on the surface of the insphere centered on the corresponding axis point .  this small circle is a parallel slice. 

5.  Now do this for all three sides of the triangle for each frequency subdivision  to get the intersecting small circle slices we see in the sketchup model.    Above 4v, these slices don't quite intersect - they form a 'window' like in the equal arc subdivision, so you need a rounding method to get the flat layers.  

Anyway, here's an animated gif demonstrating  my method creating 12v Kruschke sphere.