Beraha constants explained

The Beraha constants are a series of mathematical constants by which the

n th Beraha constant is given byB (n) = 2 + 2 \cos \left ( (2 \pi)/(n) \right ).

Notable examples of Beraha constants include

B ( 5 ) is \varphi + 1 , where \varphi is the golden ratio, B ( 7 ) is the silver constant^[1] (also known as the silver root),^[2] and B ( 10 ) = \varphi + 2 .

The following table summarizes the first ten Beraha constants.

n	B(n)	Approximately
1	4
2	0
3	1
4	2
5	(1)/(2) ( 3+ \sqrt{ 5 })	2.618
6	3
7	2 + 2 \cos (\tfrac{ 2 }{ 7 }\pi)	3.247
8	2 + \sqrt{ 2 }	3.414
9	2 + 2 \cos (\tfrac{ 2 }{ 9 }\pi)	3.532
10	(1)/(2) ( 5+ \sqrt{ 5 })	3.618

See also

• Chromatic polynomial

References

• Web site: Beraha Constants. Weisstein. Eric W.. Wolfram MathWorld. 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 ," Research Report COPR 72–7, University of Waterloo, 1972a.

• Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case

λ = inf ty ." Research Report COPR 72–4, University of Waterloo, 1972b.

Notes and References

1. Web site: Silver Constant. Weisstein. Eric W.. Wolfram MathWorld. November 3, 2018.
2. Web site: Silver Root. Weisstein. Eric W.. Wolfram MathWorld. May 5, 2020.