Rayne's n-6-3 acrohedra

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Rayne's n -6-3 acrohedra are an infinite family of self-intersecting acrohedra discovered in early 2023.[1] Nondegenerate acrohedra exist for n  = 5 and n  ≥ 7.

For even n , the process for building an acrohedron is as follows:

  • Start with a regular n -gon as the base.
  • Alternately attach regular hexagons and equilateral triangles to the sides of the base, closing up all open edges of triangles and resulting in n  copies of n -6-3 acrons.
  • Each hexagon now has three open edges, one of which is parallel to the base. Attach an equilateral triangle to the other two, coplanar to the hexagon.[note 1]
  • Add n  equilateral triangles to seal up the 2n  open edges that are not parallel to the base.
  • All open edges are now in the same plane. Mirror the figure about that plane to close up all edges.
  • If coplanar faces are not allowed, resolve them by excavating tetrahedra.

The resulting figure has the same symmetry as the uniform di--gonal prism.

For odd n, the process is:

  • Start with a regular n -gon as the base.
  • Alternately attach regular hexagons and triangles to the sides of the base such that there are 2 adjacent edges of the n -gon that are connected to triangles, creating n -1 copies of n -6-3 acrons.
  • Each hexagon now has three open edges, one of which is parallel to the base. Attach an equilateral triangle to the other two, coplanar to the hexagon.
  • Add n -2 triangles to seal up the 2n -2 open edges that are not parallel to the base or colinear with the open edges of the triangles connected to the base.
  • The edges of the hexagons opposite the base and the open edges of the triangles connected to the base form 2 perpendicular planes. Reflect around these planes to get 4 copies of the previous section.
  • Add 4 triangles to close the remaining open edges.
  • If coplanar faces are not allowed, resolve them by excavating tetrahedra.

The resulting figure has brick symmetry.

See also[edit | edit source]

Notes[edit | edit source]

  1. This is the step where n = 6 fails, resulting in a figure with three open edges that can be closed to form a truncated tetrahedron.

References[edit | edit source]