Pure and Applied Mathematics Journal
Volume 7, Issue 3, June 2018, Pages: 37-44
Received: Aug. 24, 2018;
Accepted: Sep. 7, 2018;
Published: Sep. 28, 2018
Views 595 Downloads 61
Robert Luckett Shuler Jr., NASA, Johnson Space Center, Houston, USA
The problem of how many colors are required for a planar map has been used as a focal point for discussions of the limits of human direct understanding vs. automated methods. It is important to continue to investigate until it is convincingly proved map coloration is an exemplary irreducible problem or until it is reduced. Meanwhile a new way of thinking about surfaces which hide N-dimensional volumes has arisen in physics employing entropy and the holographic principle. In this paper we define coloration entropy or flexibility as a count of the possible distinct colorations of a map (planar graph), and show how a guaranteed minimum coloration flexibility changes based on additions at a boundary of the map. The map is 4-colorable as long as the flexibility is positive, even though the proof method does not construct a coloration. This demonstration is successful, resulting in a compact and easily comprehended proof of the four color theorem. The use of an entropy-like method suggests comparisons and applications to issues in physics such as black holes. Therefore in conclusion some comments are offered on the relation to physics and the relation of plane-section color-ability to higher dimensional spaces. Future directions of research are suggested which may connect the concepts to not only time and distance and thus entropic gravity but also momentum.
Robert Luckett Shuler Jr.,
Entropy-Like State Counting Leads to Human Readable Four Color Map Theorem Proof, Pure and Applied Mathematics Journal.
Vol. 7, No. 3,
2018, pp. 37-44.
R. Wilson, Four Colors Suffice, Princeton University Press, Princeton 2002, pp 16-19.
K. Appel, and W. Haken, “Every Planar Map is Four Colorable. Part I: Discharging,” Illinois Journal of Mathematics, 1977, vol 21 (3) pp 429-490.
Appel, K. and Haken, W. Every Planar Map is Four Colorable. Part II: Reducibility, Illinois Journal of Mathematics, 1977, vol 21 (3) pp 491-567.
E. R. Swart, “The philosophical implications of the four-color problem,” American Mathematical Monthly, Mathematical Association of America, 1980, vol 87 (9) pp 697-702.
N. Robertson, D. P. Sanders, P. Seymour, and R. Thomas, “The Four-Colour Theorem,” J. Combin. Theory Ser. B, 1997, vol 70 (1) pp 2-44.
G. Gonthier, “Formal Proof – The Four-color Theorem,” Notices of the American Mathematical Society, 2008, vol 55 (11) pp 1382-1393.
A. Wiles, Modular elliptic curves and Fermat’s Last Theorem,” Annals of Mathematics, 1995, vol 141 (3) pp 448-551.
X. Hong, “A Model-Based Approach for The Demonstration of Fermat’s Last Theorem,” Pure and Applied Mathematics J., 2017, vol 6 (5) pp 144-147. https://doi.org/ 10.11648/j.pamj.20170605.12.
R. L. Shuler, “A Family of Metric Gravities,” Eur. Phys. J. Plus, 2018, vol 133 pp 158.
S. W. Hawking, “The chronology protection conjecture,” Phys. Rev. D, 1992, vol 46 pp 603-611.
L. Susskind, “The World as a Hologram,” J. Mathematical Phys., 1995, vol 36 (11) pp 6377-6396.
E. Verlinde, “On the origin of gravity and the laws of Newton,” Journal of High Energy Physics, 2011, vol 29. https://doi.org/10.1007/JHEP04 (2011) 029.
V. Christianto and F. Smarandache, “How many points are there in a line segment? – A new answer from Discrete-Cellular Space viewpoint,” J Pur Appl Math., 2018, vol 2 (1) pp 1-4.
A. S. Eddington, The Nature of the Physical World, The Macmillan Co., Basingstoke UK, 1928.
J. L. Lebowitz, “Time’s arrow and Boltzmann’s entropy,” Scholarpedia, 2008, vol 3 (4) pp 3448.
H. Minkowski, Raum und Zeit [Space and Time], Physikalische Zeitschrift, 10: 75–88, 1908-1909. Translation at https://en.wikisource.org/wiki/Translation:Space_and_Time.
A. Einstein [tr. S. N. Bose], The Foundation of the Generalised Theory of Relativity, Annalen der Physik, Vol. 354 (7) pp 769-822, 1916. http://www.archive.org/details/principleofrelat00eins.
C. Chafin, “Globally Causal Solutions for Gravitational Collapse,” 2014, arXiv 1402.1524v1 [gr-qc].
J. S. Spivey, “Dispelling Black Hole Pathologies Through Theory and Observation,” Progress in Physics, 2015, vol 11 (4) pp 321-329.
T. W. Marshall, “The Shell Collapsar – A Possible Alternative to Black Holes,” Entropy, 2016, vol 18 (10) pp 363. http://dx.doi.org/10.3390/e18100363.