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Pascal's Prism

Published online by Cambridge University Press:  23 January 2015

Harlan J. Brothers
Harlan J. Brothers*
Affiliation:
Brothers Technology, LLC, PO Box 1016, Branford, CT 06405-8016, USA, e-mail: harlan@brotherstechnology.com
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Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.

2. From Pascal to Leibniz

In Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.

Specifically, we define the sequence sn ; as follows [6]:

Information

Type
Articles
Information
The Mathematical Gazette , Volume 96 , Issue 536 , July 2012, pp. 213 - 220
Copyright
Copyright © The Mathematical Association 2012

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References

1. Ross, J. F., Pascal's legacy, EMBO reports, 5, Special Issue (2004) pp.7–10.Google Scholar
2. Edwards, A. W. F., Pascal's arithmetical triangle: the story of a mathematical idea, Johns Hopkins University Press, Baltimore (2002) pp. xiii, 1, 27, 34–37.Google Scholar
3. Gullberg, J., Mathematics: from the birth of numbers, W. W. Norton and Company, New York (1997) p. 141.Google Scholar
4. Weisstein, E. W., Pascal's Triangle from MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/PascalsTriangle.html Google Scholar
5. Brothers, H. J., Pascal's triangle: The hidden stor-e , Math. Gaz. (March 2012), pp. 145–148.Google Scholar
6. Sloane, N. J. A., Sequence A001142. http://oeis.org/A001142 Google Scholar
7. Brothers, H. J. and Knox, J. A., New closed-form approximations to the logarithmic constant e , The Mathematical Intelligencer 20 (1998) pp. 25–29.Google Scholar
8. Knox, J. A. and Brothers, H. J., Novel series-based approximations to e , College Mathematics Journal 30 (1999) pp. 269–275.Google Scholar
9. Brothers, H. J., Sequence A168510. http://oeis.org/A168510 Google Scholar
10. Weisstein, E. W., Leibniz Harmonic Triangle from MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html Google Scholar
11. Richardson, L. F. and Gaunt, J. A., The deferred approach to the limit, Philosophical Transactions of the Royal Society of London 226a (1927) pp. 299–361.Google Scholar
12. Brothers, H. J., Sequence A191510. https://oeis.org/A191510 Google Scholar
13. Adamson, G. W., Sequence A132818. http://oeis.org/A132818 Google Scholar
14. http://www.brotherstechnology.com/math/pascals-prism.html Google Scholar
15. Pickover, C. A., Computers, pattern, chaos, and beauty: graphics from an unseen world, St. Martin's Press, New York (1990) pp. 173–185.Google Scholar
16. Guy, R. K., The second strong law of small numbers, Mathematics Magazine 63 (1990), pp. 3–20.CrossRefGoogle Scholar
17. Bardzell, M. and Shannon, K., The PascGalois project: visualizing abstract algebra, Focus 22 (2002) pp. 4–5.Google Scholar
18. Frame, M. L. and Mandelbrot, B. B., Fractals, graphics, and mathematics education, Cambridge University Press (2002).Google Scholar
19. Frame, M. L. and Neger, N., Fractal tetrahedra: What's left in, what's left out, and how to build one in four dimensions, Computers & Graphics 32 (2008) pp. 371–381.Google Scholar
20. Weisstein, E. W., Tetrix from Math World – a Wolfram Web Resource. http://mathworld.wolfram.com/Tetrix.html Google Scholar
21. Frame, M. L. and Neger, N., Dimensions and the probability of finding odd numbers in Pascal's triangle and its relatives, Computers & Graphics 34 (2010) pp. 158–166.Google Scholar