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Hypercubes and the normal distribution

Published online by Cambridge University Press:  01 August 2016

Martin Griffiths
Martin Griffiths*
Affiliation:
Colchester County High School for Girls, Norman Way, Colchester CO3 3US
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Extract

As demonstrated in [1], it is possible to generalise the familiar 3-dimensional cube to n dimensions for each n > 0. We present here a fascinating link between a geometrical aspect of these 'n-dimensional hypercubes' and the normal distribution.

In what follows we shall identify with n-dimensional space, and use n-D to denote ‘;n-dimensional’. For the sake of mathematical convenience we initially consider n-D hypercubes of edge length 1 unit whose 2n vertices are situated in at all possible n-tuples (c1, c2, ... , cn), where ck is equal either to 0 or 1, . We denote these mathematical objects by Cn, n = 0,1,2,3, .... Our results, however, can easily be generalised to hypercubes of any edge length, and situated anywhere in .

Type
Articles
Information
The Mathematical Gazette , Volume 92 , Issue 524 , July 2008 , pp. 223 - 234
Copyright
Copyright © The Mathematical Association 2008

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References

1. Griffiths, M., n-dimensional enrichment for further mathematicians, Math. Gaz. 89 (November 2005) pp. 409–416.Google Scholar
2. Grimmett, G. and Stirzaker, D., Probability and random processes, Oxford University Press (2001) pp. 113–114.CrossRefGoogle Scholar
3. Robin, A. C., A quick approximation to the normal integral, Math. Gaz. 81 (March 1997) pp. 95–96.CrossRefGoogle Scholar
4. Morland, T., Approximations to the normal distribution function, Math. Gaz. 82 (November 1998) pp. 431–437.Google Scholar