Cross Sections of a Regular
4-dimensional 16-cells Polytope.
See a page of ‘Literka’ Cross Sections of Hypercube for basic
definitions of polytopes. One of the types of regular 4-dimensional polytopes is
16-cells regular polytope. All cells of this polytope are regular tetrahedrons.
This polytope is derived from a regular octahedron
(see Plato’s polyhedrons). Let
us explain what we mean by this. We need few remarks:
Let W be a regular polygon such that the maximal distance between vertices is
smaller than doubled length of its sides. Than we can construct a regular
pyramid P such that basis of P is W and all edges of P have equal length.
The same is true for polytopes. Take a regular polyhedron W such that the maximal
distance between vertices is smaller than doubled length of its edges. Than we
can construct a regular 4-dimensional pyramid P such that basis of P is W and all
edges of P have equal length. This pyramid P will consist of W and n 3-dimensional
regular pyramids, where n is number of faces of W. In the case W consists of
equilateral triangles, all cells of P are regular tetrahedrons.
To receive a regular 16-cells polytope take a regular octahedron, build a regular
4-dimensional pyramid P as it was described before. Take another exact copy of P.
You will have two the same 4-dimensional pyramids. Glue them basis-to-basis in
4-dimensional space. Obtained polytope is a regular polytope of 16 cell.
From the construction it follows that all cells of a regular 16-cells polytope
are regular tetrahedrons (see Plato’s polyhedrons).
Let us also mention that a polytope such that each vertex is a center of a cell
of regular hypercube (see Cross Sections of Hypercube) is a regular polytope of
16 cells. Polytope such that each vertex is a center of a cell of regular
16-cells polytope is a regular hypercube.
Take a 3-dimensional hyperplane defined by a regular octahedron, which we used to build 16-cells regular polytope R. It follows that this regular octahedron is a cross section of R defined by this hyperplane. It is a central cross section.
Most of central cross sections of regular 16-cells polytope remind deformed regular octahedrons. However we can find also 12-faces cross sections. Let us present a picture of most regular among them:
This polyhedron consists of two regular pyramids glued basis-to-basis. Bases of
these pyramids are regular hexagons.
At the end let us present a picture of a cross section showing how 12-faces cross
sections are becoming 8-faces cross sections:
It is also a central cross section. It is a 12-faces polyhedron reminding a
regular octahedron.
A page of ‘Literka’ Cross Section of Regular 120-cells Polytope presents a
picture of a cross section of more sophisticated polytope.
