The Beraha constants are a series of mathematical constants by which the n th Beraha constant is given by
\( {\displaystyle B(n)=2+2\cos \left({\frac {2\pi }{n}}\right).} \)
Notable examples of Beraha constants include \( {\displaystyle B(5)} \) is \( {\displaystyle \varphi +1} \), where \( \varphi \) is the golden ratio,\( {\displaystyle B(7)} \) is the silver constant[1] (also known as the silver root),[2] and \( {\displaystyle B(10)=\varphi +2} \).
The following table summarizes the first ten Beraha constants.
| n | {\displaystyle B(n)} \) | Approximately |
|---|---|---|
| 1 | 4 | |
| 2 | 0 | |
| 3 | 1 | |
| 4 | 2 | |
| 5 | {\displaystyle {\frac {1}{2}}(3+{\sqrt {5}})} \) | 2.618 |
| 6 | 3 | |
| 7 | \( {\displaystyle 2+2\cos({\tfrac {2}{7}}\pi )} \) > | 3.247 |
| 8 | \( {\displaystyle 2+{\sqrt {2}}} \) | 3.414 |
| 9 | \( {\displaystyle 2+2\cos({\tfrac {2}{9}}\pi )} \) | 3.532 |
| 10 | \( {\displaystyle {\frac {1}{2}}(5+{\sqrt {5}})} \) | 3.618 |
See also
Chromatic polynomial
Notes
Weisstein, Eric W. "Silver Constant". Wolfram MathWorld. Retrieved November 3, 2018.
Weisstein, Eric W. "Silver Root". Wolfram MathWorld. Retrieved May 5, 2020.
References
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 1," Research Report COPR 72-7, University of Waterloo, 1972a.
Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case λ = ∞ ." Research Report COPR 72-4, University of Waterloo, 1972b.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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