Polytopes Built of Congruent Bipyramids. Part 1.

Above is a picture of a 3-dimensional cross section of a polytope with 96 cells – all these cells are congruent bipyramids.

Bipyramid is a polyhedron formed of two congruent pyramids glued face-to-face, where this faces are the bases of these pyramids. We’ll assume that bases are congruent regular polygons. We can find pictures of examples of bipyramids on pages of ‘Literka’ Polyhedrons Build of Equilateral Triangles (type 3) and Cross Section of a Polytope (at the top of this page). Let us mention here that a regular octahedron, presented on the page of ‘Literka’ Plato's Polyhedrons is also a bipyramid. Base of this bipyramid is a square.

There is an easy way to construct a polytope with bipyramids. Take a regular polytope W and one of its cells V. Take the vectors from the center of W to all centers of faces of V. Finally, take hyperplanes passing through these centers and perpendicular to the corresponding vectors. These hyperplanes intersect in one point, say A. Take a face P – one of the faces of V. Construct a pyramid with vertex A and a base P. Take a symmetric point B to A with respect to P. We constructed a bipyramid with vertices A, B and vertices of P. All such bipyramids are cells of a convex polytope.


Type 1.
For the case W is a 5-cells regular polytope we’ll receive a polytope with 10 cells. Each cell is a bipyramid based on an equilateral triangle. Main cross section of this polytope looks like this:

This polyhedron is a truncated regular hexahedron - four vertices are cut. It reminds a polyhedron presented on the page of ‘Literka’ Semi-regular Polyhedrons. It is built of 6 hexagons (but not regular hexagons) and 4 equilateral triangles.


Type 2.
If we start with a regular 8-cells polytope (hypercube), we’ll receive 24-cells polytope built of bipyramids. However, this polytope is nothing new, since it is a regular 24-cells polytope built of regular octahedrons. (See pages of ‘Literka’ Cross Sections of Regular Polytopes and Nets of Regular Polytopes ).
To see an applet with more cross sections of polytopes of type 2 click on: Cross sections of a regular 8-cells and 24-cells polytope.



Type 3.
Starting with 16-cells regular polytope, we receive a polytope with 32 cells. Each cell is a bipyramid based on an equilateral triangle. Main cross section is a regular octahedron (see Plato's Polyhedrons ). That is why we present a picture of another central cross section:

It has 16 faces – 8 triangles and 8 deltoids.
To see an applet with more cross sections of polytopes of type 3 click on Cross sections of polytopes built of congruent bipyramids (24 and 32 cells).



Type 4.
Finally, if we start with a regular 24-cells polytope, we’ll receive 96-cells polytope built of bipyramids. Each cell is a bipyramid based on an equilateral triangle. Main cross section of this polytope is shown at the top of this page. It has 32 faces, 8 equilateral triangles and 24 triangles, which are faces of cells of a polytope.
To see an applet with more cross sections of polytopes of type 4 click on: Cross sections of polytopes built of congruent bipyramids (96 cells).





There are two more examples of polytopes built of bipyramids on the page of ‘Literka’ Polytopes Built of Bipyramids. Part 2.

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 list of pages of 'Literka' about polytopes.
Return to the main geometrical page of 'Literka'.
Return to the main page of 'Literka'.