*Forma,* Vol. 14 (No. 3), pp. 221-237, 1999

*Original Paper*

## On a General Method to Calculate Vertices of *N*-dimensional Product-Regular Polytopes

Motonaga Ishii

Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatu-cho, Sakyo-ku, Kyoto 606-8501, Japan

(Received February 5, 1999; Accepted August 18, 1999)

**Keywords: **
*N*-dimensional Regular Polytope, *N*-dimensional Semi-Regular Polytope, *N*-dimensional Product-Regular Polytope

**Abstract. **
A set of *N*-dimensional product-regular polytopes (*N*-dimensional PRP-set) can be produced from a regular polytope by taking the product space of N solid models which are derived from the boundary figures (vertices, edges, faces, and so on) of the regular polytope.

I found a method to calculate all vertices of the *N*(
2)-dimensional PRP-set derived from an *N*-dimensional regular polytope.

The PRP-set derived from a regular polytope is similar to the set of semi-regular polytopes derived from the regular polytope in point of that both include simple semi-regular polytopes which have the same rotational symmetry with the regular polytope. But the PRP-set differs from the set of semi-regular polytopes because that the PRP-set includes the regular polytope itself, and it doesn't include the snub-type semi-regular polytopes, regular prisms, nor regular anti-prisms, all of which have no same rational symmetry with regular polytopes.

In this paper, I calculated and displayed only 2 to 6-dimensional PRP-sets by using computer.

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