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A cube has some hexagonal cross-sections.

A cross-section is the intersection of a polytope with a hyperplane.

Taking the cross-section of a circumscribable polytope near its vertex, in an appropriate direction, results in the polytope's vertex figure.

For viewing higher-dimensional polytopes[edit | edit source]

The structures of higher-dimensional polytopes can be difficult to understand. Viewing cross-sections of these polytopes can provide insights into how they're laid out. Projections are another way to do this.

For example, this is a series of cross-sections of the icositetrachoron. The cross-sections span (from left to right, then to the start of the next row down) from an octahedral cell up to the widest point on the icositetrachoron (halfway through it, where its cross-section is a cuboctahedron). All are taken in parallel hyperplanes to the starting octahedral cell.

The cells we can see are:

  • An octahedral cell at the start
  • Six halves of octahedral cells coming off of the first cell's vertices. These are visible as growing squares.
    • In this direction, squares are the cross-sections of the octahedron.
  • Eight octahedral cells coming off of the first cell's faces. These are visible as triangles "morphing" into dual triangles.
    • In this direction, triangles and (isogonal) hexagons are the cross-sections of the octahedron.

The reason that the cube, the vertex figure of the icositetrachoron, isn't visible among these cross-sections, is because the hyperplane that the cross-sections are taken in is parallel to a facet of the icositetrachoron. A different selection of hyperplane could result in some cube-shaped cross-sections.

Cross-sections can also be animated.

See also[edit | edit source]