【Move to another page】
Quote
https://ift.tt/2Pz9CDR
Beraha constants
MarkZusab: spacing
The '''Beraha constants''' are a series of [[Mathematical constant|mathematical constants]] by which the <math>nth</math> Beraha constant is given by
<math>B (n) = 2 + 2 \cos \left ( \frac{2\pi}{n} \right )</math>.
Notable examples of Beraha constants include <math>B (5)</math>is <math>\phi + 1</math>, where <math>\phi</math> is the [[golden ratio]], <math>B (7)</math>is the [[Silver Constant|silver constant]] (also know as the silver root and Tutte–Beraha constant), and <math>B (10) = \phi + 2</math>.
The following table summarizes the first ten Beraha constants.
{| class="wikitable"
!<math>n</math>
!<math>B(n)</math>
!'''approximately'''
|-
|1
|4
|
|-
|2
|0
|
|-
|3
|1
|
|-
|4
|2
|
|-
|5
|<math>\frac{1}{2}(3+\sqrt{5})</math>
|2.618
|-
|6
|3
|
|-
|7
|<math>2 + 2 \cos (\tfrac{2}{7}\pi)</math>
|3.247
|-
|8
|<math>2 + \sqrt{2}</math>
|3.414
|-
|9
|<math>2 + 2 \cos (\tfrac{2}{9}\pi)</math>
|3.532
|-
|10
|<math>\frac{1}{2}(5+\sqrt{5})</math>
|3.618
|}
== See also ==
* [[Chromatic polynomial]]
== References ==
<ref name=":0">Liquid error: wrong number of arguments (1 for 2)</ref><ref name=":q">Liquid error: wrong number of arguments (1 for 2)</ref><ref>Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.</ref><ref>Le Lionnais, F. ''Les nombres remarquables.'' Paris: Hermann, p. 143, 1983.</ref><ref>Saaty, T. L. and Kainen, P. C. ''The Four-Color Problem: Assaults and Conquest.'' New York: Dover, pp. 160-163, 1986.</ref><ref>Tutte, W. T. "Chromials." University of Waterloo, 1971.</ref><ref>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.</ref><ref>Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case <math>\lambda = 1</math>," Research Report COPR 72-7, University of Waterloo, 1972a.</ref><ref>Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case <math>\lambda = \infty</math>." Research Report COPR 72-4, University of Waterloo, 1972b.</ref>
<math>B (n) = 2 + 2 \cos \left ( \frac{2\pi}{n} \right )</math>.
Notable examples of Beraha constants include <math>B (5)</math>is <math>\phi + 1</math>, where <math>\phi</math> is the [[golden ratio]], <math>B (7)</math>is the [[Silver Constant|silver constant]] (also know as the silver root and Tutte–Beraha constant), and <math>B (10) = \phi + 2</math>.
The following table summarizes the first ten Beraha constants.
{| class="wikitable"
!<math>n</math>
!<math>B(n)</math>
!'''approximately'''
|-
|1
|4
|
|-
|2
|0
|
|-
|3
|1
|
|-
|4
|2
|
|-
|5
|<math>\frac{1}{2}(3+\sqrt{5})</math>
|2.618
|-
|6
|3
|
|-
|7
|<math>2 + 2 \cos (\tfrac{2}{7}\pi)</math>
|3.247
|-
|8
|<math>2 + \sqrt{2}</math>
|3.414
|-
|9
|<math>2 + 2 \cos (\tfrac{2}{9}\pi)</math>
|3.532
|-
|10
|<math>\frac{1}{2}(5+\sqrt{5})</math>
|3.618
|}
== See also ==
* [[Chromatic polynomial]]
== References ==
<ref name=":0">Liquid error: wrong number of arguments (1 for 2)</ref><ref name=":q">Liquid error: wrong number of arguments (1 for 2)</ref><ref>Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.</ref><ref>Le Lionnais, F. ''Les nombres remarquables.'' Paris: Hermann, p. 143, 1983.</ref><ref>Saaty, T. L. and Kainen, P. C. ''The Four-Color Problem: Assaults and Conquest.'' New York: Dover, pp. 160-163, 1986.</ref><ref>Tutte, W. T. "Chromials." University of Waterloo, 1971.</ref><ref>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.</ref><ref>Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case <math>\lambda = 1</math>," Research Report COPR 72-7, University of Waterloo, 1972a.</ref><ref>Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case <math>\lambda = \infty</math>." Research Report COPR 72-4, University of Waterloo, 1972b.</ref>
November 04, 2018 at 12:55PM
