Voronoi Phyllotaxis Tiling on the Fermat Spiral

Year: 2014 Authors: Akio Hizume; Takamichi Sushida; Yoshikazu Yamagishi

Core claim

Continued fractions let Voronoi phyllotaxis tilings on the Fermat Spiral be generated for the golden ratio and other real numbers.

Topics

phyllotaxis, Voronoi tiling, Fermat Spiral, continued fractions

Domains

geometry, number theory, Voronoi diagrams, quasi-periodic patterns, architecture, product design, 3D printing, installation concepts

Methods

continued fraction convergents, Voronoi cell construction, numerical cell labeling, 3D prototyping

Media

3D printer, physical prototypes, diagram figures, Fermat Spiral plots

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture

Voronoi Phyllotaxis Tiling on the Fermat Spiral

Akio Hizume, Takamichi Sushida, Yoshikazu Yamagishi Department of Applied Mathematics and Informatics Ryukoku University Seta, Otsu, Shiga, Japan

E-mail: akio@starcage.org

Abstract

We developed Voronoi Tiling on the Fermat Spiral using continued fractions and applied this system to architecture and product design. We produced physical prototypes of them using a 3D printer.

Phyllotaxis on the Fermat Spiral

Figure 1: Plot generatrix on the Fermat Spiral.

It is well known that the pattern of seeds in a sunflower is based on the Fermat Spiral, as expressed in the polar equation $r^2=a\ast \theta$ : see [1][2][3]. Figure1 shows the principle for determining the center of seeds as a generatrix. We place numbers on the seeds one by one from the center towards the outside. When $\theta$ is $2\pi (2-\tau )$ (where $\tau$ denotes the Golden Ratio), the arrangement of seeds becomes the ideal phyllotaxis.

Our eye can’t help tracing many spirals. Every spiral consists of congruent numbers modulo the Fibonacci Number. The principal convergents of the Golden Ratio consist only of the Fibonacci Numbers. The even-order convergent minus $\tau$ is always opposite in sign to the odd-order convergent minus $\tau$ . That is why people say that the number of clockwise and counter-clockwise spirals is always any two consecutive terms in the Fibonacci Sequence.

Voronoi Phyllotaxis Tiling on the Fermat Spiral

An attempt at Voronoi Tiling based on the generatrix was already published by Sadoc, Rivier, and Charvolin (2012): see [4]. We derived it independently in 2014 and are able to replace the phyllotaxis angle of $2\pi (2-\tau )$ with any arbitrary angle. We can determine the closest cells using the denominators of convergents calculated by a continued fraction. In the case of the Golden Ratio, it’s enough for us to consider just the seven closest cells, but in case of a different real number, we should consider more than seven cells. So it is fair to say that the Golden Phyllotaxis is the simplest system of all real numbers.

Let us take a look at the sample drawings. Figure 2 is a Voronoi Tiling on the Fermat Spiral based on the Golden Ratio. Cell numbers range are from 1 to 999. The tiling consists of pentagons, hexagons and heptagons. There are no quadrilaterals and octagons.

Hizume, Sushida and Yamagishi

Figure 2: Voronoi Phyllotaxis Tiling on the Fermat Spiral.

Figure 3: 1D Quasi-Periodic Pattern. A sample: from #1509 to #1597

One of the author (Hizume) found in 1986 that a phyllotactic division of a circumference made a one-dimensional quasi-periodic pattern, that is, the Fibonacci Lattice: see [5]. Similarly, when we chose suitable series of cells on the same circumference, we achieved the same lattice,

873,1017,1072,983,839, #928 is 5 gon 874,1018,1073,984,840, #929 is 5 gon 875,1019,1074,985,841, #930 is 5 gon 876,1020,1075,986,842, #931 is 5 gon 877,1021,1076,987,843, #932 is 5 gon 878,1022,1077,988,844, #933 is 5 gon 879,1023,1078,989,845, #934 is 5 gon 880,1024,1079,990,846, #935 is 5 gon 881,1025,1080,991,847, #936 is 5 gon

as shown in Figure 3. We can produce any long sequence from this Voronoi Tiling in such a way.

Figure 5 shows an enlarged view of a portion of tiling. We can identify the numerical order and center of each cell. (After #1,000, the number indicates the lower three digits.) Figure 4 indicates the cell’s kind of polygon and cells surrounding it.

For example, you may confirm the configuration of crll #932 in Figures 4 and 5.

Figure 4: A Data List. Figure 5: Enlarged view of a portion.

Voronoi Phyllotaxis Tiling on Fermat Spiral

We have developed various proposals for applying the Voronoi Phyllotaxis Tiling on the Fermat Spiral as shown in Figures 6 through 9.

Figure 6: Fibonacci Colosseum. Everybody can watch the central stage.

Figure 7: Mt. Fibonacci for playground or solar generator.

Figure 8: Fibonacci Fountain or Terraced House.

Figure 9: Fibonacci Honeycomb.

Figure 10: Prototypes produced by a 3D printer.

Extension into Other Real Numbers

We extended the above attempts into other real numbers in the same way that the authors extended the “Fibonacci Tornado [6]” to become the “Real Tornado [7][8]“. In this endeavor, we made full use of principal and intermediate convergents generated by the continued fraction of the decimal part of a real number. Let us look at some examples.

Hizume, Sushida and Yamagishi

$\theta =2\pi (sqrt(2)-1)$

$\theta =2\pi (exp(1)-2)$

$\theta =2\pi (7/12)$

Figure 11: Various Voronoi Tiling on Fermat Spiral.

Figure 12: 3D printer models.

Acknowledgement

We wish to thank Yotsutani Laboratory and Digital Creation Hall of Ryukoku Univ. for their support. Mr. Peter Engel proofread for us in English.

References

[1] K. Azukawa, Sunflower seeds, Sugaku Seminar 24(7), 34-42, July 1985 (in Japanese). [2] K. Azukawa, and Tsuyoshi Yuzawa, A remark on the continued fraction expansion of conjugates of the golden section, Mathematics journal of Toyama University 13, 165-176, 1990. [3] H. Vogel, A Better Way to Construct the Sunflower Head, Mathematical Biosciences 44, 179-189, 1979. [4] J. -F. Sadoc, N. Rivier and J. Charvolin, Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry, Acta Crystallographica A 68, 470-483, 2012. [5] A. Hizume, Life and Architecture, private press, 1990. [6] A. Hizume, Fibonacci Tornado, MANIFOLD #11, 2005. [7] A. Hizume, Real Tornado, MANIFOLD #17, 2008. [8] A. Hizume, Y. Yamagishi, Real Tornado, Proceedings of the 12th Annual Bridges Conference, 239 - 242, 2009. [9] T. Sushida, Geometry of phyllotactic spiral tilings, Thesis, Ryukoku University, 2014.