Characteristics of Regular 4-Dimensional Polytopes

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Theorem

The $4$-dimensional regular polytopes have the following characteristics:

Name No. of cells No. of faces No. of edges No. of vertices Dual
Pentatope $5$ $10$ $10$ $5$ Self-dual
Tesseract $8$ $24$ $32$ $16$ $16$-cell
$16$-cell $16$ $32$ $24$ $8$ Tesseract
$24$-cell $24$ $96$ $96$ $24$ Self-dual
$120$-cell $120$ $720$ $1200$ $600$ $600$-cell
$600$-cell $600$ $1200$ $720$ $120$ $120$-cell


Proof


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