Beraha constants
From HandWiki
Short description: Mathematical constants
The Beraha constants are a series of mathematical constants by which the [math]\displaystyle{ n\text{th} }[/math] Beraha constant is given by
- [math]\displaystyle{ B (n) = 2 + 2 \cos \left ( \frac{2\pi}{n} \right ). }[/math]
Notable examples of Beraha constants include [math]\displaystyle{ B (5) }[/math]is [math]\displaystyle{ \varphi + 1 }[/math], where [math]\displaystyle{ \varphi }[/math] is the golden ratio, [math]\displaystyle{ B (7) }[/math]is the silver constant[1] (also known as the silver root),[2] and [math]\displaystyle{ B (10) = \varphi + 2 }[/math].
The following table summarizes the first ten Beraha constants.
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ B(n) }[/math] | Approximately |
|---|---|---|
| 1 | 4 | |
| 2 | 0 | |
| 3 | 1 | |
| 4 | 2 | |
| 5 | [math]\displaystyle{ \frac{1}{2}(3+\sqrt{5}) }[/math] | 2.618 |
| 6 | 3 | |
| 7 | [math]\displaystyle{ 2 + 2 \cos (\tfrac{2}{7}\pi) }[/math] | 3.247 |
| 8 | [math]\displaystyle{ 2 + \sqrt{2} }[/math] | 3.414 |
| 9 | [math]\displaystyle{ 2 + 2 \cos (\tfrac{2}{9}\pi) }[/math] | 3.532 |
| 10 | [math]\displaystyle{ \frac{1}{2}(5+\sqrt{5}) }[/math] | 3.618 |
See also
Notes
- ↑ Weisstein, Eric W.. "Silver Constant". http://mathworld.wolfram.com/SilverConstant.html.
- ↑ Weisstein, Eric W.. "Silver Root". http://mathworld.wolfram.com/SilverRoot.html.
References
- Weisstein, Eric W.. "Beraha Constants". http://mathworld.wolfram.com/BerahaConstants.html.
- Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.
- Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983.
- Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160–163, 1986.
- Tutte, W. T. "Chromials." University of Waterloo, 1971.
- Tutte, W. T. "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.
- Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case [math]\displaystyle{ \lambda = 1 }[/math]," Research Report COPR 72–7, University of Waterloo, 1972a.
- Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case [math]\displaystyle{ \lambda = \infty }[/math]." Research Report COPR 72–4, University of Waterloo, 1972b.
 |  |
