SPACE CURVES WITH CONSTANT THICKNESS:
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.
2 THICKNESS OF A LINE IN 2D and 3D
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
3 THE SIMPLEST SPATIAL CURVE: CIRCULAR HELIXES
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.
3.1 The densest helixes with thickness
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
3.2 How are the fine structures inside?
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].