SPACE CURVES WITH CONSTANT THICKNESS...

Name: Tohru Ogawa, Katachist and Symmetrist, Physicist, Geometer, Mathematical Scientist, Philosopher (b. Tokyo, Japan, 1936).

Address: University of Tsukuba (Emeritus Professor) and NPO-ISTA (Interdisciplinary Institute of Science, Technology, and Art), 2-106-B-102, Kitahara, Asaka, Saitama, 351-0036, Japan.

E-mail: ogawa-t@koalanet.ne.jp

Fields of interest: Mathematical science mainly geometrical and studies of wide range including culture, education, society, peace, etc. without any boundary in principle, a kind of intellectual gourmet.

Awards: Prize for an Excellent Research Paper (Society for Science on Form, Japan, 1995).

Publications:

Ogawa, T. (1983) Problems in a digital description of a configuration of atoms and some other geometrical topics in physics, In: Yonezawa and Ninomiya, eds., Topological Disorder in Condensed Matter,Springer, 60-77; Electronic publication in: Visual Mathematics, Vol.2, No.1.

Ogawa, T. (1985) On the structure of a quasicrystal: Three-dimensional Penrose transformation, Journal of the Physical Society of Japan, 54, No.9, 3205-3208.

Fujiwara, T. and Ogawa, T., eds. (1990) Quasicrystals, Heidelberg: Springer.

Ogawa, T. (1994) An Aspect of katachi (~form), In: Takaki, R., ed., Research of Pattern Formation, Tokyo: KTK Scientific Publisher, 11-22.

Ogawa, T. (1996) Katachi and symmetry: Towards interdisciplinary and intercultural cooperation, [Keynote Lecture], In: Ogawa, T., Miura, K., Masunari, T., and Nagy, D., eds., Katachi U Symmetry, Tokyo: Springer, 7-26.

Abstract: Aiming a new type of educational attempt to inspire some developing ideas, the author enjoys a kind of designs by the tool of 3D graphics of Mathematica and Maple. He starts the process with giving constant thickness to simple spatial curves. He himself does calculate and draw graphics but these are not the main subject at all for students, who should enjoy the interesting properties, raising some questions and ideas for the next step. He thinks both of the calculation and graphics work as the next task after the students feel necessity for that. The main part of the presentation is to show the interesting properties of the 3D-curves with constant thickness. First of all, we start with the 2D-correspondence of the problem. The concept of helical symmetry is defined that is proper to the present problem.

Nowadays, too many people in Japan regard studying mathematics and physics as for rote leaning of formula. Most of scientists and mathematicians take the situation as serious. We should construct more creative education. It must be changed in order to be more creative one. This work is one of the attempts to this direction and at the same time, he enjoys by himself to study and to images the detailed structure of ideal 3D structures.

A line segment is a straight line of finite length

. It is reasonable to define a line segment with 2-dimensional thickness or width

as a rectangle and the area of a rectangle is width by length

. Similarly, it is reasonable to define a circumference of radius

with width

as a pair of concentric circles whose radii are

and

since the area is

as expected.
It is the most reasonable extension to define the corresponding width in 3D as the circle in the perpendicular plane with its centre at the point on the curve. The definition is acceptable at least for the torus

that is formed by rotating the smaller circle of radius

with bigger radius

, whose volume is given by

. The easiest and the most elegant way to understand the relation is that the sections for any constant value of

where

is a pair of concentric circles by reducing to 2D case.

A straight line is the most homogeneous line with a constant direction but without any curvature and can be in one dimension. A circle is the most homogeneous line with a constant curvature but without any twist (rotation + translation) and is two-dimensional. A circular helix is the simplest essentially three-dimensional curve, being the most homogeneous line with a constant twist, never realize in lower dimensions than three. The curve in Cartesian coordinate is given by

and in cylindrical coordinate

by and

where

stands for the radius of the cylinder on the surface of which the circular helix lies and constant

depends on the pitch. Regard

as time, then it is nothing but pure rotation in a plane. The torsion of the curve (Coxeter, 1961, 1969, pp. 323-325, where

is used) is given by

. A helical surface can be regarded as a set of the circular helix with common value of

and for arbitrary value of

. It is defined as the locus of rotating horizontal straight line around a fixed vertical line. A circular helix is the section of a helical surface and a cylinder of radius

. It is natural to introduce the concept of helical symmetry for the circular helixes belonging to a common helical surface. Reminisce that a step stone of helical staircase is triangular (neglecting the limiting part close to the central axis). The slope of the staircase depends on the distance from the axis; infinite at the axis and tends to zero in infinite remote.

Let us focus on the problem of giving a single circular helix a thickness. Now,

is a constant and an angle

is introduced. Prepare a sheet of paper on which parallel lines of slope angle

are drawn with the periodic separation of

(the corresponding period in intercept is

) as shown in Fig.1. Then, put it on the surface of cylinder so that the horizontal axis just fits around the cylinder of radius

and then the parallel lines make a single slope. For

and then

, a circle is obtained. For

then

, the helix tend to a straight line corresponding to the generating line of the cylinder.

Though the value of

is arbitrary as a line, there exists the minimum value if the finite thickness is given to the helix and then the volume exclusion effect is introduced. Obtaining the minimum value of slope angle is rather complicated. All the circular sections are inclined and there are no diameters parallel to the axis line. It means that the ends of diameters tend to go remote but radial ones. This effect in 3D space causes some confusion. Can you imagine the inner structures and the several planer sections, for example, horizontal and vertical? Another case that can be treated in parallel is double helix. In the cases of more than two helical lines, there are vacant space around the central axis, as easily seen from the fact that there are no point in a plane that three impenetrable circles can share. Therefore it is necessary to start the analysis with how three cylinders symmetrically contact one another.

It is interesting to imagine the hidden fine structures inside. For example, how a circular section contact with another circular sections and how they arrange? Interesting problems arise step by step. From an educational point of view, it is desirable to have, to imagine the situation, and to enjoy such kind of questions even if they cannot find the answer. It will motivate towards many directions such as mathematics, science, designing etc.

The procedure for the author to answer those problems is based on the fact that a helices with thickness is set of helixes with common value of

. All of the helical lines with

that passes in inside of a circle go keeping the whole shape. Then one can trace all of the positions on the trajectories or flow line. See the figures here showing the situation. Some of the figures here are exhibited with some explaining text at gallery. More figures containing some sections will be added in the oral presentation.

Coxeter, H. S. M. (1961) Introduction to Geometry, New York: Wiley; 2nd ed., ibid., 1969.

Ogawa, T. (2004) Geometry of the beauty III: Space curves with constant section, Bulletin of the Society for Science on Form, Vol.19, No.1, p. 32-33 [in Japanese].

Ogawa, T. (2004) Geometry of the Beauty IV: Densest-single helix and double-helix with constant section and some other topics, to be published in Bulletin of the Society for Science on Form Vol.19, No.2 [in Japanese].