Hypercube Vertex First Cross Sections

To understand the images below, it may be helpful to visualize a series of cross sections produced by bisecting a cube with a plane perpendicular to the diagonal of the cube. Hold a cube so that one vertex is closest to you and so that a line drawn from that vertex to the opposite vertex is going directly away from you. Place a plane perpendicular to that line at a point where the nearest vertex just touches it. This cross section of the cube is a point. Push the plane into the cube a short distance. The cross section produced is an equilateral triangle. If you keep pushing the plane into the cube, the cross section becomes a larger and larger triangle, until you reach the next 3 vertices of the cube. Here the triangle stops growing. Continue pushing the plane into the cube, and the cross section becomes an irregular hexagon, with 3 short and 3 long edges. When your plane reaches the halfway point along the diagonal of the cube, the cross section is a regular hexagon. Continuing on, the crossection progresses through another series of irregular hexagons, regular triangles and finally a point when it reaches the far vertex.

The following images are a series of cross sections of a hypercube analogous to the 3-D series described above. The gray figures are orthographic projections of a hypercube. The rotation was chosen to reveal the relevent information about the crossections. The red figures are the cross sections.
The first cross section (not shown here, see animation) is a point or the first vertex of the hypercube.

The second cross section is a series of Tetrahedra that increase in size. 4 cubic cells surround each vertex of the hypercube. The 4 triangles of this tetrahedron are produced by pushing the plane into each of the cubic cells surrounding that vertex. This tetrahedron in regular in 4-space, but distorted when projected to 2-space.
This cross section is the largest Tetrahedron in the series. It is special in that it includes 4 vertices of the hypercube.
The next cross section is a Truncated Tetrahedron. This series of sections starts with just a small chunk of each of the tetrahedron's vertices cut off. As the series progresses larger and larger chunks are cut off, until the hexagonal faces become regular. The triangular faces are regular throughout. The hexagonal faces are produced in the same way as the 3-D mid-point cross section of the cube described above. They are the mid cross sections of each of the 4 cubic cells surrounding the first vertex.
With this Octahedral cross section we have moved half way through the hypercube. Progressing from the previous truncated tetrahedron cross section, the triangular faces get larger and the short edges of the hexagonal faces smaller and smaller, until the hexagonal faces become equilateral triangles. This cross section is special in that it contains 6 of the vertices of the hypercube.
From here, the cross sections repeat the first half of the series, except that they are reversed from there conterparts. The Truncated Tetrahedron is one of the semi-regular solids known to Archimedes in the 3rd Century B.C.
We return again to the Tetrahedron containing 4 of the hypercubes vertices. It is interesting that 2 of the cross sections found in this series, the Tetradedron and Octahedron, are part of the series of regular solids know as the Platonic solids.
The the Tetrahedron gets smaller until it becomes a point representing the virtex at the other side of the hypercube.
An animation (133K) of this series is also available. This series of sections is described by Thomas F. Banchoff in his book "Beyond the Third Dimension"

Last revised 10/20/2002