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Dissection of the hypercube into simplexes
| Content Provider | Semantic Scholar |
|---|---|
| Author | Mead, D. G. |
| Copyright Year | 1979 |
| Abstract | A generalization of Sperner's Lemma is proved and, using extensions of p-adic valuations to the real numbers, it is shown that the unit hypercube in n dimensions can be divided into m simplexes all of equal hypervolume if and only if m is a multiple of n!. This extends the corresponding result for n 2 of Paul Monsky. The question as to whether a square can be divided into an odd number of (nonoverlapping) triangles all of the same area was answered in the negative by Thomas (31 (if all the vertices of the triangles are rational numbers), and in general by Monsky [2]. In this note we generalize Monsky's result to n dimensions and prove the following: THEOREM. Let C be the unit hypercube in n dimensions. Then C can be divided into m simplexes all of equal hypervolume if and only if m is a multiple of n!. The proof is divided into two parts. In the first we obtain a slight generalization of Sperner's Lemma while the second employs extensions of p-adic valuations to the real numbers. Let R be an n-polytope in n-space. A simplicial decomposition of R is a division of R into simplexes such that if v is a vertex of some simplex on the boundary of the simplex S, then v is a vertex of S. We consider a simplicial decomposition of R in which each vertex of a simplex is labeledpi for some i, O < i < n, and we call the k-simplex S a complete k-simplex if the vertices of S are labeledpo, p1, . .. * PkLEMMA 1 (SPERNER'S LEMMA). Consider a simplicial decomposition of an n-polytope R in which each vertex is labeled pi, 0 < i < n. Then the number of complete n-simplexes is odd if and only if the number of complete (n 1)simplexes on the boundary of R is odd. PROOF. Note that every complete (n 1)-simplex on the boundary of R occurs in one n-simplex while all other complete (n I)-simplexes occur in two n-simplexes. Also, a complete n-simplex has precisely one complete n dimensional face, while an "incomplete" n-simplex has 0 or 2 such faces. Received by the editors September 25, 1978. AMS (MOS) subject classifications (1970). Primary 50B30; Secondary 12B99. |
| Starting Page | 302 |
| Ending Page | 304 |
| Page Count | 3 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1979-0537093-6 |
| Volume Number | 76 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1979-076-02/S0002-9939-1979-0537093-6/S0002-9939-1979-0537093-6.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1979-0537093-6 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
