Spirallohedra and Space Filling. A Tribute to Russell Towle.

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Spirallohedra and Space Filling. A Tribute to Russell

Towle.

Michel Petitjean

To cite this version:

Michel Petitjean. Spirallohedra and Space Filling. A Tribute to Russell Towle.. Symmetry: Culture and Science, 2008, 19 (1), pp.5-8. �hal-01941511�

Vol. 19, No. 1, 5-8, 2008

EDITORIAL TO THE V

. 19 (2008)

SPIRALLOHEDRA AND SPACE FILLING

A TRIBUTE TO RUSSELL TOWLE

Michel Petitjean

Address: CEA Saclay/DSV/iBiTec-S/SB2SM (CNRS URA 2096), 91191 Gif-sur-Yvette Cedex, France. E-mails: michel.petitjean@cea.fr, petitjean.chiral@gmail.com.

http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html

The cover images of this volume are works by Russell Towle (1949-2008).

The k-armed spirallohedra are non-convex objects bounded by k portions of helicoids, where k is an integer greater than 2 (see a rigorous mathematical description in the appendix). The spirallohedra are chiral with a symmetry axis of order k.

M. PETITJEAN

6

The k-armed rhombic spirallohedra are non convex polyhedral objects composed of rhombohedra, discovered by the mathematician Russell Towle (2000a) while studying polar zonohedra.

See the Figure 1, the k-armed rhombic spirallohedron projected on a plane orthogonal to its symmetry axis, for k=3,4,5,6, and then the same ones, projected parallel to the symmetry axis.

When the number of rhombohedra tends to infinity, the bounding polyhedral surfaces become smooth, and we get the k-armed spirallohedra. Although similar figures were known as decorative motifs in the Hungarian folk art (Szilassi and Szojka, 2001), it was unknown (and hard to believe at first glance) that the 3-armed and the 4-armed spirallohedra tile the space by translation: they fill the three-dimensional space without gaps or overlaps. This remarkable discovery, done by Russell Towle in 2000, has been confirmed by Gévay and Szilassi (2006, 2007), who also demonstrated that the 3-armed and the 6-armed spirallohedra together tile the space.

Russell Towle has continued to study zonotopes and tilings (2002), and undoubtly he had more in mind. On October 3, 2006, he wrote to me:

I have an idea that these rhombic spirallohedra are members of an infinite set of space-fillers related to hypercubes. I have named this hypothetical set the Bitten Zonotopes. In three dimensions, the Bitten Zonohedra are the spirallohedra. In two dimensions, one can construct regular 2n-gons tiled with rhombs, and then strip away two or three regions of rhombs from the periphery (the "bites"), leaving a non-convex polygon tiled with rhombs; and both the 2-bite and 3-bite bitten zonogons tile the plane. In fact, the 2-bite zonogons are related to 3-armed spirallohedra, and the 3-bite zonogons, to 4-armed spirallohedra.

But we have probably to wait for a long time until an other genial geometer would be able to describe some high-dimensional space fillers analogs of spirallohedra.

Russell Towle was not only a geometer. He was also interested in geology and natural history, music, and language. His work cannot be summarized here. The reader is invited to read the cited papers (Towle 1996, 2002) and to look at the electronic resources on the web, such as the ones at the Wolfram site (Towle 1997, 2000b, 2003, 2005, 2008). At the end of his seminal 2000 paper, Russell Towle wrote that the volume of the spirallohedra would require further study. The calculation is done hereafter, and we dedicate it to his memory.

APPENDIX

We consider the cylinder of revolution of radius R and height H, with central axis passing through the points (R,0,0) and (R,0,H). We define the helical arc

[

]

0 1 cos( )

x =R − t , y₀= −Rsin( )t , z0=tH/ 2π , with t varying between 0 and 2π, such that the end points are on the z axis, lying respectively at (0,0,0) and (0,0,H). Then we define the k−1 images of this arc via the rotations of angles j(2π/k), j=1..k−1 around the z axis and joining the end points of the helical arcs. The k helical arcs (j = 0 ... k−1) are symmetricaly located around the axis passing through the points (0,0,0) and (0,0,H). This axis is the symmetry axis of the k-armed spirallohedra. The square of the distance r from any point on some of the helical arcs above to its projection on the symmetry axis is 2 2

[

]

2 2 cos( )

r =R − t , i.e., r = 2Rsin(zπ/H).

The intersection of any plane orthogonal to this symmetry axis at z, 0 ≤ z≤ H, describes a regular polygon of circumradius r. Its area is A

k(z)=k(d/2)rcos(π/k), i.e. 2 ( ) sin( / ) cos( / ) k A z =kr π k π k , or 2

[

]

k(z) for z going from 0 to H

gives us the volume V

k of the spirallohedron.

2

sin(2 / ) k

V =kHR π k

From Cavalieri’s principle (see eq. 17 in Gual-Arnau, 2005), the spirallohedron has the same volume than a sinusoid of revolution around the z axis with circular cross section

A

k(z) with radius

[

]

1/ 2 2 sin(R zπ/H) ( / 2 ) sin(2 / )k π π k .

When k increases to infinity, the limit shape of the regular polygon is a circle of radius

r=2Rsin(zπ/H), and the limit shape of the spirallohedron is a solid of revolution, with

volume V_∞=2πR2H.

ACKNOWLEDGEMENTS

I am very grateful to Gay Wiseman for having provided several nice spirallohedra pictures and I thank very much Mark Newbold for having generated the high resolution images in figure 1.

M. PETITJEAN

8

REFERENCES

Gévay G., Szilassi L. (2006) The Spirallohedron (A Space-Filling Body), Symmetry Festival, Budapest, 12-18 August 2006. http://symmetry.hu/festival2006.html

Gévay G., Szilassi L. (2007) Spirallohedron: A Space-Filling Body, Symmetry: Culture and Science, 18, 1, 75-88.

Gual-Arnau X. (2005) On the Volume from Planar Sections through a Curve. Image Anal. Stereol, 24, 1, 35-40.

Szilassi L., Szojka E. (2001) From Corndemon to Geometry of the Corn-Baby, Symmetry: Culture and

Science 12, 1-2, 25-37.

Towle R. (1996) Graphics Gallery: Polar Zonohedra,The Mathematica Journal 6[2], 8-12. http://library.wolfram.com/infocenter/Articles/3335/

Towle R. (1997) Wolfram Library Archive, MathSource: Packages and Programs, http://library.wolfram.com/infocenter/MathSource/536,

http://library.wolfram.com/infocenter/MathSource/610/, http://library.wolfram.com/infocenter/MathSource/617/

Towle R. (2000a) Rhombic Spirallohedra, Symmetry: Culture and Science, 11, 1-4, 296-306. Towle R. (2000b) Wolfram Library Archive, MathSource: Packages and Programs,

http://library.wolfram.com/infocenter/MathSource/545/, http://library.wolfram.com/infocenter/MathSource/554/, http://library.wolfram.com/infocenter/MathSource/593/, http://library.wolfram.com/infocenter/MathSource/611/.

Towle R. (2002) Stellation and Zonotopal Tilings, Symmetry: Culture and Science, 13, 1-2, 121-128. Towle R. (2003) Wolfram Library Archive,. MathSource: Packages and Programs,

http://library.wolfram.com/infocenter/MathSource/613/.

Towle R. (2005) Wolfram Library Archive, MathSource: Packages and Programs, http://library.wolfram.com/infocenter/MathSource/1197/.

Towle R. (2008) Wolfram Demonstrations Project, http://demonstrations.wolfram.com/author.html~

author=Russell+Towle, http://demonstrations.wolfram.com/RhombicSpirallohedra/,

http://demonstrations.wolfram.com/PolyhedraSpheresAndCylinders/, http://demonstrations.wolfram.com/PolarZonohedra/.

Watanabe Y. (1982) A Method for Volume Estimation by Using Vector Areas and Centroids of Serial Cross Sections, IEEE Trans. Biomed. Eng. BME-29, 3, 202-205.