Counting the number of bounded domains separated by hyperplanes
Hiroshima journal of mathematics education Volume 7
Page 55-62
published_at 1999-03
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| File | |
| Title ( eng ) |
Counting the number of bounded domains separated by hyperplanes
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| Creator | |
| Source Title |
Hiroshima journal of mathematics education
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| Volume | 7 |
| Start Page | 55 |
| End Page | 62 |
| Abstract |
When some hyperplanes H_1, ..., H_m of the n-dimensional Euclidean space R^n are given in general position, Schläfli has determined the number of the bounded connected components in R^n - ∪^m_i = _1H_i, the complementary set of the union of the hyperplanes. It is equal to the binomial coefficient ((m-1)/n), which is also equal to the number of vertices which are the intersections of n hyperplanes in H_1, ..., H_<m-1>. Although Schläfli's proof is implicit and intuitive, the fact reflects an interesting aspect concerning configurations of hyperplanes. We clarify how the condition of general position works, and re-prove the fact in all of its details.
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| NDC |
Education [ 370 ]
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| Language |
eng
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| Resource Type | departmental bulletin paper |
| Publisher |
Department of Mathematics Education, Faculty of Education, Hiroshima University
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| Date of Issued | 1999-03 |
| Publish Type | Version of Record |
| Access Rights | open access |
| Source Identifier |
[ISSN] 0919-1720
[NCID] AN10444573
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