A new proof of Honeycomb Conjecture by fractal geometry methods

Abstract

Based on fractal geometry, we put forward a concise and straightforward method to prove Honeycomb Conjecture—a classical mathematic problem. Hexagon wins the most efficient covering unit in the two- dimensional space, compared with the other two covering units—triangle and square. From this point of view, honeycomb is treated as a hierarchical fractal structure that fully fills the plane. Therefore, the total side length and area are easily calculated and from the results, the covering efficiency of each possible unit is provided quantitatively.

Keywords

Honeycomb Conjecture / fractal geometry / hierarchical fractal structure

References

[1] Hales T C. (<day>8</day><month>Jun</month>1999). “The Honeycomb Conjecture”. Discrete and Computational Geometry 25: 1-22 (2001). arXiv:math/9906042

[2] Hales T C. Cannonballs and Honeycombs. Notices of the American Mathematical Society, 2000, 47: 440-449

[3] Fejes Tóth L. What the bees know and what they do not know. Bulletin of the American Mathematical Society, 1964, 70(4): 468-481

Acknowledgements

The authors thank National Natural Science Foundation of China (No. 10602028).

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg