Beraha constants

Mathematical constants

The Beraha constants are a series of mathematical constants by which the  n th {\displaystyle n{\text{th}}} Beraha constant is given by

B ( n ) = 2 + 2 cos ⁡ ( 2 π n ) . {\displaystyle B(n)=2+2\cos \left({\frac {2\pi }{n}}\right).}

Notable examples of Beraha constants include  B ( 5 ) {\displaystyle B(5)} is  φ + 1 {\displaystyle \varphi +1} , where  φ {\displaystyle \varphi } is the golden ratio,  B ( 7 ) {\displaystyle B(7)} is the silver constant[1] (also known as the silver root),[2] and  B ( 10 ) = φ + 2 {\displaystyle B(10)=\varphi +2} .

The following table summarizes the first ten Beraha constants.

n {\displaystyle n} |  B ( n ) {\displaystyle B(n)} | Approximately

|
|

1  | 4  |
2  | 0  |
3  | 1  |
4  | 2  |
5  |  1 2 ( 3 + 5 ) {\displaystyle {\frac {1}{2}}(3+{\sqrt {5}})} | 2.618
6  | 3  |
7  |  2 + 2 cos ⁡ ( 2 7 π ) {\displaystyle 2+2\cos({\tfrac {2}{7}}\pi )} | 3.247
8  |  2 + 2 {\displaystyle 2+{\sqrt {2}}} | 3.414
9  |  2 + 2 cos ⁡ ( 2 9 π ) {\displaystyle 2+2\cos({\tfrac {2}{9}}\pi )} | 3.532
10  |  1 2 ( 5 + 5 ) {\displaystyle {\frac {1}{2}}(5+{\sqrt {5}})} | 3.618

## See also

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* Chromatic polynomial

## Notes

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1. ^ Weisstein, Eric W. "Silver Constant". Wolfram MathWorld. Retrieved November 3, 2018.
2. ^ Weisstein, Eric W. "Silver Root". Wolfram MathWorld. Retrieved May 5, 2020.

## References

[edit]

* Weisstein, Eric W. "Beraha Constants". Wolfram MathWorld. Retrieved November 3, 2018.
* 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  λ = 1 {\displaystyle \lambda =1} ," Research Report COPR 72–7, University of Waterloo, 1972a.
* Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case  λ = ∞ {\displaystyle \lambda =\infty } ." Research Report COPR 72–4, University of Waterloo, 1972b.

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