Frontiers of Mechanical Engineering >

2013 , Vol. 8 >Issue 4: 367 - 370

DOI: https://doi.org/10.1007/s11465-013-0273-7

RESEARCH ARTICLE

A new proof of Honeycomb Conjecture by fractal geometry methods

  • Tong ZHANG , 1 ,
  • Kai DING 2
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  • 1. Research Center for Solid Mechanics, Beihang University, Beijing 100191, China; School of Engineering, Brown University, Providence, RI 02912, USA
  • 2. School of Instrument Science and Optical Engineering, Beihang University, Beijing 100191, China

Received date: 19 Apr 2013

Accepted date: 05 Jun 2013

Published date: 05 Dec 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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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.

Key words： Honeycomb Conjecture; fractal geometry; hierarchical fractal structure

Cite this article

Tong ZHANG , Kai DING . A new proof of Honeycomb Conjecture by fractal geometry methods[J]. Frontiers of Mechanical Engineering, 2013 , 8(4) : 367 -370 . DOI: 10.1007/s11465-013-0273-7

Acknowledgements

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

References

Publishing order | Descend order by publishing year | Descend order by cited within
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

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