Cost Efficiency Optimization (more math version)

Inspired by Almaravarion's design, but generated numerically. I hate how the grid leaves the hexagons irregular in that design. Making them regular will slightly reduce structure points and increase cell points at large radii. The cost increases very slightly but the power increases more. More importantly, it has better symmetry!

The reason this design is cost optimal is a combination of 3 things:
  1. Cell points are significantly less expensive than structure points. This means that minimizing structure to area ratio is more cost effective.
  2. Goldberg icosahedral polyhedra have sides with at least 5 vertices. The more vertices per face, the more area (cells) to perimeter (structure) you have.
  3. A spherical projection of a class I, frequency 2, icosahedral Goldberg polyhedral has the largest faces (with the most vertices) possible before shells become too large to fill with cells.
Do remember that per reason 1, this design is more cost efficient the larger your radius. Watch out though, with only 80 nodes, it will take hours to fill at ~40 sails/s.

Creation methodology:
  1. Generate vertices of an icosahedron centered on the origin.
  2. Generate triangular faces from the icosahedron
  3. For each face:
    • Calculate the centroid of the face. Save as point.
    • Calculate the midpoint between the centroid and each vertex. Save as point.
  4. For each point:
    • Normalize to a unit vector. This projects it onto the sphere.
    • Translate coordinates to longitude (atan of y and x) and latitude (asin of z).
  5. Enter all 80 node coordinates into the game.
Possible follow up: calculate version with additional nodes along the structure to increase cell consumption rate.



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