Cross Sections of a Regular
4-dimensional 8-cells Polytope.
Cross Sections of a Regular
4-dimensional 8-cells Polytope (Hypercube).
The definition of polytope of 4-dimensional space is similar to the
definition of polyhedron of 3-dimensional space. Faces of polytope are called
‘cells’.
Hypercube of 4-dimensional space is a regular polytope of 8 cells. Each cell of
hypercube is a regular hexahedron (see Plato’s polyhedrons). Cross sections by a
3-dimensional hyperplane containing the center of a hypercube will be called
central cross sections
.
Central cross sections of a hypercube by a plane parallel to any cell of a
hypercube is a regular hexahedron.
Let G and H be parallel cells of a hypercube. Assume that a plane of a central
cross section does not cross G and H. Then cross section is a polyhedron of 6
faces and each face is a parallelogram. We’ll not present pictures of such cross
section, because they are easy to imagine. However the following polyhedron
differs very little from such types:
This is also central cross section of a hypercube.
If we proceed further we’ll receive another polyhedron (of course – central cross
section of a hypercube):
This is a prism. The basis and the top – these are regular hexagons. Each regular
hexagon is a central 2-dimensional cross section of 3-dimensional cell of a
hypercube (as it was mentioned before – this cell is a regular hexahedron).
Notice that the hyperplane of this cross section crosses each cell of a hypercube.
The last cross section of a hypercube we want to present is also a prism:
There are central cross sections of a hypercube, which are not prisms. One of them
is a regular octahedron (see Plato's polyhedrons.)
It is well seen on the applet of 'Literka' Cross sections of two regular polytopes.
The hyperplane of this cross section (it is normal to the vector <1,1,1,1>) crosses
all cells of a hypercube. It is left to the reader to decide how it crosses these
cells, which are regular 3-dimensional cubes.
