teaching machines

Icosahedron, Part 2

December 31, 2017 by Chris Johnson. Filed under madeup, public.

Several hours later, I have now found the difference between an octahedron and an icosahedron. I had been stuck on generating the coordinates of the octahedron. A little reading and experimentation directed my attention to the cube circumscribing the icosahedron. The way I’ve set things up, its vertices are all [±u, ±u, ±u], where u is t + 1. (And curiously, since t is the golden ratio, t + 1 is the same as t * t.)

An octahedron is a dual of a cube, meaning its six vertices are just the six centroids of the circumscribing cube. With those 6 vertices stitched together to form eight faces, I had an octahedron, from which I subtracted the icosahedron.

As Dr. Ward used to say, “Viola!” (sic):

  

    to wireframe vertices, faces
  for fi to size faces
    face = faces[fi]
    for vi to 3
      p = vertices[face[vi]]
      moveto p[0], p[1], p[2]
    end
    home
    forget
  end
end

t = (1 + 5 ^ 0.5) / 2.0
u = t + 1

isoVertices = {
  {-1, t, 0},
  {1, t, 0},
  {-1, -t, 0},
  {1, -t, 0},
  {0, -1, t},
  {0, 1, t},
  {0, -1, -t},
  {0, 1, -t},
  {t, 0, -1},
  {t, 0, 1},
  {-t, 0, -1},
  {-t, 0, 1}
}

isoFaces = {
  {0, 11, 5},
  {0, 5, 1},
  {0, 1, 7},
  {0, 7, 10},
  {0, 10, 11},
  {1, 5, 9},
  {5, 11, 4},
  {11, 10, 2},
  {10, 7, 6},
  {7, 1, 8},
  {3, 9, 4},
  {3, 4, 2},
  {3, 2, 6},
  {3, 6, 8},
  {3, 8, 9},
  {4, 9, 5},
  {2, 4, 11},
  {6, 2, 10},
  {8, 6, 7},
  {9, 8, 1}
}

octaVertices = {
  {0, 0, u},
  {u, 0, 0},
  {0, 0, -u},
  {-u, 0, 0},
  {0, u, 0},
  {0, -u, 0}
}

octaFaces = {
  {0, 1, 4},
  {1, 2, 4},
  {2, 3, 4},
  {3, 0, 4},
  {0, 3, 5},
  {3, 2, 5},
  {2, 1, 5},
  {1, 0, 5}
}

wireframe isoVertices, isoFaces
wireframe octaVertices, octaFaces

.rgb = {1, 1, 0}
iso = trimesh isoVertices, isoFaces
.rgb = {1, 0, 1}
oct = trimesh octaVertices, octaFaces

-- iso -- uncomment to yield the icosahedron
oct - iso

    
    
    
  
  
    

    to wireframe vertices, faces
  for fi to size faces
    face = faces[fi]
    for vi to 3
      p = vertices[face[vi]]
      moveto p[0], p[1], p[2]
    end
    home
    forget
  end
end

t = (1 + 5 ^ 0.5) / 2.0
u = t + 1

isoVertices = {
  {-1, t, 0},
  {1, t, 0},
  {-1, -t, 0},
  {1, -t, 0},
  {0, -1, t},
  {0, 1, t},
  {0, -1, -t},
  {0, 1, -t},
  {t, 0, -1},
  {t, 0, 1},
  {-t, 0, -1},
  {-t, 0, 1}
}

isoFaces = {
  {0, 11, 5},
  {0, 5, 1},
  {0, 1, 7},
  {0, 7, 10},
  {0, 10, 11},
  {1, 5, 9},
  {5, 11, 4},
  {11, 10, 2},
  {10, 7, 6},
  {7, 1, 8},
  {3, 9, 4},
  {3, 4, 2},
  {3, 2, 6},
  {3, 6, 8},
  {3, 8, 9},
  {4, 9, 5},
  {2, 4, 11},
  {6, 2, 10},
  {8, 6, 7},
  {9, 8, 1}
}

octaVertices = {
  {0, 0, u},
  {u, 0, 0},
  {0, 0, -u},
  {-u, 0, 0},
  {0, u, 0},
  {0, -u, 0}
}

octaFaces = {
  {0, 1, 4},
  {1, 2, 4},
  {2, 3, 4},
  {3, 0, 4},
  {0, 3, 5},
  {3, 2, 5},
  {2, 1, 5},
  {1, 0, 5}
}

wireframe isoVertices, isoFaces
wireframe octaVertices, octaFaces

.rgb = {1, 1, 0}
iso = trimesh isoVertices, isoFaces
.rgb = {1, 0, 1}
oct = trimesh octaVertices, octaFaces

-- iso -- uncomment to yield the icosahedron
oct - iso

  
  expand