4-dimensional Pyramids.

Above is a picture of a net of 4-dimensional pyramid.

The simplest example of 4-dimensional pyramid is a regular 5-cells polytope (see net of 5-cells regular polytope and cross section of 5-cells regular polytope). The base of this polytope is a regular (3-dimensional) pyramid.
Let us notice that 16-cells regular polytope (see cross sections of regular 16 cells polytope ) is a 4-dimensional bipyramid. This means that this polytope consists of two pyramids glued base-to-base. These bases are regular octahedrons. A page of ‘Literka’ net of 16-cells regular polytope presents a picture of a net of such pyramid (with regular octahedron as its base). There are some remarks and description of it. Let us present a little different picture:

It is called ‘stellated octahedron’.

Let us take now a regular icosahedron (see ICOSAHEDRON) ) as a base of 4-dimensional pyramid. Picture of a net of this polytope is presented at the top of this page. A non-convex polyhedron seen on this picture is called a stellated icosahedron.

Let us consider cross sections of 4-dimensional pyramids. In the case a base of pyramid is a regular icosahedron (see Plato’s Polyhedrons), one of its cross sections looks like this:



All vertices of this cross section lie on a surface of some ellipsoid.
A base of a pyramid doesn’t have to be a regular polyhedron. Let us take a pyramid such that its base is a polyhedron with 80 faces, from a page of ‘Literka’ cross sections of a regular 600-cells polytope.
One of its cross sections looks like this:

Since a base reminds a ball, this cross section reminds an ellipsoid.


See pages of 'Literka about:
Polytopes Built of Bipyramids. Part 1.
Polytopes Built of Bipyramids. Part 2.

Applet: Cross sections of 2 polytopes built of congruent bipyramids (24 and 32 cells).
Applet: Cross sections of 2 polytopes built of congruent bipyramids (720 and 1200 cells).
Applet: Cross sections of polytopes built of congruent bipyramids (96 cells).

See new applets of ‘Literka’:
Applet: Cross sections of a regular 8-cells and 24-cells polytope.
Applet: Cross sections of a regular 600-cells polytope.

See pages of ‘Literka’ about cross sections of other regular polytopes:
Hypercube,
16-cells Polytope,
120-cells Polytope,
600-cells Polytope.


See pages about nets of:
5-cells polytope and hypercube,
16-cells and 24-cells polytope.

Return to the main geometrical page of 'Literka'.
Return to the main page of 'Literka'.