2. Mathematical Difficulties in Defining Rotation Numbers

Differential geometry studies the geometric properties of differential manifolds. It is a mainstream research direction in modern mathematics and the basis of general relativity. It is closely related to topology, algebraic geometry and theoretical physics. Finding intrinsic invariants that describe the geometric properties of differential manifolds is the main research content of differential geometry.

Regarding spatial smooth curves, people usually use a vector function r (t) that depends on a certain parameter t to describe the coordinates of each point on the curve. 

(1)

 r'(t), r''(t) and r’'(t)  represent the first to third order derivatives of the vector function r(t) with respect to parameter  t . The unit tangent vector T, unit principal normal vector N and unit binormal vector B of curve (1) can be defined by the following formulas:      

(2)

(3)

and

(4)

The operator × in formula (4) represents the vector product between vectors.

French mathematician, Jean Frédéric Frenet, discovered an important formula describing the intrinsic properties of space curves in his doctoral thesis in 1847, namely Frenet's formula

(5)

This formula uses the arc length s of the curve as a parameter (called the natural parameter of the curve) and describes the change of the unit tangent vector T, the unit principal normal vector N and the unit binormal vector B with s. The scalar functions κ(s) and τ(s) are the curvature and torsion at corresponding points on the curve. Differential geometry has proved that no matter what parameter t is used to calculate these two scalars, they are local invariants. The calculation formulae are respectively

(6)

and

(7)

The numerator of equation (7) represents the mixed product of three vectors.

The basic theorem of differential geometry about space curves, which has been rigorously proven, shows that, apart from differences in position and direction, a curve can be uniquely determined (i.e., " calculated " or " created”) based on its curvature and torsion (references: [1], [2] or [3]). This shows that the curvature and torsion of a curve determine all the topological properties of the space curve. Just like genes in biology, the genetic structure basically determines the growth process and characteristics of the relevant life individuals under appropriate environments.

Interestingly, if the distribution of curvature and torsion of a space curve is known, how to determine the overall topological characteristics of the curve?

Practice has shown that, except for special cases, this is an extremely difficult mathematical problem. For example, judging whether a space curve is closed based on curvature and torsion, and judging the type and number of kinks in a space closed curve, are all difficult mathematical problems. Unless the curve is directly manufactured based on basic theorems, direct observation and research can only lead to specific conclusions.

Similarly in biology, it is extremely difficult to tell whether a person has single eyelids or double eyelids based on their genetic structure (obviously determined by genes), but it is clear briefly by just looking at the person.

 Now let's study whether the number of rotations (turns) n of a given closed spiral (as shown in Figure 7) can be directly calculated (determined) using the two invariants of the curvature and torsion of the curve.

First, let's study the corresponding straight cylindrical spiral segment with two endpoints (Figure 6). According to formula (1), it can be expressed as:

( 8 )

Where a is the radius of rotation of the spiral, 2πb is the height of the spiral, and n is the number of turns of the spiral. According to formulas (6) and (7), the curvature and torsion of the curve are calculated. It is not difficult to find that they are constants that do not depend on the parameter t. 

(9)

(10)

If the curvature and torsion are integrated along the arc length of the curve, the two integral values ​​can be called the total curvature and total torsion of the curve and are represented by  Tκ  and  Tτ respectively. By calculation, we can get

(11)

(12)

Obviously, when both a and b are positive numbers, the curvature and torsion and their algebraic sum, the total curvature and torsion and their algebraic sum, these invariants are generally not equal to the integer n. In general, the number of rotations n of the curve will not be directly reflected by the value of curvature and torsion. Note: As a special and simple case, according to any known value of the above four invariants, such as  of (9), and the known values ​​of a and b, the positive integer n can indeed be calculated by solving the corresponding radical equation (9).

What needs attention is: if the value of  b  is fixed (that is, the height of the spiral is fixed ) , and the rotation radius  a of the spiral  is tended to zero (that is, when the length of the curve tends to the height of the spiral), the curvature of the spiral tends to the rotation number  n , the total curvature tends to  2nπ , and both the curvature and the total curvature tend to zero; if the rotation radius  a  is fixed and the value of  b  tends to zero (that is, the height of the spiral tends to zero), the curvature of the curve tends to the rotation number  n , the total curvature tends to  2nπ , and both the curvature and the total curvature tend to zero.

Let us go back to the calculation of the curvature and torsion of the closed spiral that is connected end to end (as shown in Figure 7). The expression of the curve is:

(13)

The plane circle shown in yellow in Figure 7 is the axis around which the spiral spirals, and the radius of the circle is equal to b. The circle can be expressed as

(14)

Obviously, the circumference of the plane circle is equal to 2πb, which is exactly equal to the height of the original cylindrical helical segment. To avoid the intersection of the individual turns in Figure 7, it is necessary to limit the radius a of the helical line to be smaller than the radius b of the plane circle (14). This is because if a = b, the turns of the n -turn helical line will intersect at the center of the circle, as shown in Figure 9.     

Figure 9

If the value of a is larger, as shown in Figure 10, the yellow circle will pass through the spiral, retreat to the center of the figure and appear very small, and the spirals will appear to cross each other more often.

Figure 10 .

Using the symbolic calculation of <Mathematica>, we can obtain the expressions for the curvature and torsion of the closed spiral (13 ), which are all expressed as complex radicals of trigonometric functions:

  ( 15 )

          (16)

I have simplified the above formula as much as possible using the software <Mathematica>. Although it is possible to simplify it further by hand calculation, my energy is limited, and I can only simplify it to this extent.

The calculation of total curvature and total torsion will encounter complex situations that cannot be expressed by elementary functions. For example, in the extremely simple case of b = 0, the calculation of the curve length L will encounter the second kind of elliptic integral

         (17)

The calculation of total curvature and total torsion is even more difficult.

The expressions of total curvature and total torsion in this example are not shown here because they are too complicated. Interested readers can request them from me via email.

Therefore, it is not possible to calculate the number of revolutions of a closed spiral (13) from the curvature, torsion, total curvature or total torsion of the curve using precise expressions. Only by accurately constructing the curve based on the curvature and torsion can the number of revolutions be counted intuitively.

When b is a constant (positive number), the radius a of the ring is reasonably reduced and tends to zero. By using numerical calculations, we can find the limit values ​​of the length, curvature, total curvature, torsion and total torsion of the curve.

Taking b = 3 and n = 10 as an example, Figure 7 is just the curve when a = 1. Figures 11 and 12 are the curves of curvature and torsion with parameter t.

Figure 11.

Figure 12.

The calculated values ​​of the length, total curvature and total torsion of the curve are: 65.7363, 9.70061 and 2.86853 respectively.

When b = 3, n = 10   and a = 0.1, the graph of the curve, the images of the curvature and the torsion are:

Figure 13.

Figure 14.

and

Figure 15.

The calculated values ​​of the length, total curvature and total torsion of the curve are: 19.8697, 3.26048 and 9.49087 respectively.

When b = 3, n = 10   and a = 0.01, the graph, curvature and torsion images of the curve are:

Figure 16.

Figure 17.

and

Figure 18.

The calculated values ​​of the length, total curvature and total torsion of the curve are: 18.86, 1.0277 and -0.00549556 respectively.

Based on the above three specific examples, we can preliminarily draw the following conclusion that is consistent with our imagination: that is, as a becomes smaller, the length of the curve will shrink to the length of the plane circular curve ( 14 ) (the yellow circle) it surrounds, 6π (~ 18.8496 ), and the total curvature will approach the total curvature of the plane yellow circle, 1 ; the torsion and total torsion will also approach the torsion and total torsion of the yellow circle, which are both equal to zero.

The above facts show that even in the most ideal, standard, and simplest case of a closed spiral with  n  rotation circles ( 13 ), it is almost impossible to deduce the number of circles n , which is an obvious invariant, using only intrinsic quantities, that is, the values ​​of local curvature, local torsion, or total curvature and total torsion  (unless the corresponding curve is specifically manufactured using the values ​​of local curvature and torsion to visually count). In particular, when the radius of the spiral coil approaches zero, this intrinsic number n almost disappears.   

The above situation also makes it a huge problem to define the rotation (turn) number n using the intrinsic quantity of the space curve, the local curvature, the local torsion, or the total curvature and the total torsion in pure mathematics. In many cases, even if the curve is created, it is difficult to see its rotation number. For example, if the rotation axis (yellow circle) in Figure 13 is removed, the three-dimensional graph of the curve is

Figure 22.

It is difficult to tell the number of rotations of this curve from the graph.

After deep thought, the author found that there is still hope for mathematical theory on the issue of invariants, because the above discussion has not exhausted the full potential of "differential" geometry or "differential" topology.

In the first section of this article, the author uses a movable coordinate frame attached to the curve to define a thin tube with a radius equal to the unit length, which is attached to the curve. As the parameters change continuously, the end arrows of the unit principal normal N and the unit binormal B of the curve leave their own continuous tracks (two curves) on the wall of the thin tube. Similarly, the unit principal normal line segment and the unit binormal line segment also leave their own strip tracks with a width equal to 1. Both strip tracks can be regarded as generalized Möbius strips accompanying the space curve. Since the curve is closed, these two strips must be closed and oriented, and they are both called generalized Möbius strips accompanying the space curve.

Since the strip constructed in this way has two boundaries, one of which must be the space curve itself, and the other of which is the trace left on the thin tube wall by the corresponding unit (major or minor) normal end arrow, the generalized Möbius strip accompanying the space curve is orientable.

The above facts are completely existing structures in the theory of differential geometry itself, but few mathematicians or scholars in other fields have noticed it, let alone made full use of it.

Therefore, I guess that if we imagine the generalized Möbius strips accompanying the space curves as being made of matter (materialized), then these two strips are likely to reveal the mystery of the rotation number of the closed space curve. It turns out that my guess is correct. Let's first study the facts revealed by the following four photos:

Figure 23.

Figure 23 shows a blue ribbon, the front side is very smooth, and the back side is obviously less smooth. Wrap the ribbon smoothly around a yellow empty medicine bottle four times, and do not let the ribbon get tangled during the process. Always keep the back side of the ribbon close to the medicine bottle, with the front side facing outwards, as shown in Figure 24.

Figure 25.

After wrapping it 4 times, place it on the table as shown above. You can see that the end of the ribbon near the bottle mouth is facing up, and the other end is naturally attached to the table with the reverse side facing up. If you connect the upper and lower ribbon end smoothly together and keep the reverse side of the ribbon facing the surface of the medicine bottle, you will find that this connection is theoretically possible. I simulated this connection by using a paper clip, see Figure 26.

Figure 26.

At this point, the ribbon can be viewed as a directional closed generalized Möbius strip that rotates (or circles) around the medicine bottle four times. If the medicine bottle is removed, it can be found that if the ribbon is not disconnected, no matter where the ribbon is placed or no destructive deformation is performed, the number of four turns is always the intrinsic invariant of the ribbon, as shown in Figure 28

Figure 28.

or as shown in Figure 29

Figure 29.

Therefore, the twist or rotation number of a closed generalized Möbius strip is its intrinsic topological invariant. Thus, it can be said that the rotation number of a generalized Möbius strip accompanying a closed space curve is the rotation number of the curve, which is a differential topological invariant.

The rotation number of a closed space curve can be visualized using a generalized Möbius strip accompanying the curve.

For example, although the number of rotations of the closed spiral shown in Figure 22 is not visually obvious, the accompanying generalized Möbius strip formed by the trajectory of its unit binormal segments (the red strip in Figure 30) can clearly show the number of twists of the strip, which is exactly equal to the number of rotations of the curve, 10.

Figure 30.

The space closed curves that appear in mathematics at present are basically derived from the space limit cycles of three-dimensional nonlinear differential autonomous systems. Such limit cycles cannot generally be expressed by elementary functions, and only numerical solutions are available. The limit cycles and the accompanying generalized Möbius strips can also only be seen through numerical methods.

For example, in 2013, based on a vibration problem of a flat plate material by a physics professor at Colorado State University (R. Mark Bradley), I derived a related third-order autonomous system

            (18)

System (18) belongs to a special type of system named after the Georgian mathematician Silnikov (or saddle-focusing system), which was rarely studied at the time. The author found a series of interesting properties of this system (references: [4], [5], [6], [7]), especially when the parameters a = 1, b = 1/3, the system has a limit cycle symmetric to the origin and the number of rotations is 13. Figures 31 and 32 show this limit cycle and the accompanying generalized Möbius strip:

Figure 31 .

Figure 32.

Paying attention to Figure 30 and Figure 32, it is not difficult to find that they both have the phenomenon that the Möbius strip suddenly twists its direction in certain places. This often happens when viewing a stereoscopic image at a certain angle, and when the viewing angle changes continuously, the position of the twisting will also change continuously. It occurs naturally.

At the inflection point of a plane curve, the generalized Möbius strip corresponding to the principal unit normal vector also appears to be disconnected, shifting from one side of the curve to the other. For example, the following illustration on page 64 of <Differential Geometry> ^[1], edited by Mei Xiangming and Huang Jingzhi, which I used when I taught in China, shows this phenomenon.

Figure 33.

A similar diagram can be found on page 69 of Wu Daren’s <Differential Geometry Lectures> ^[2], another major textbook in China at the time. When discussing plane curves, these textbooks or lectures (including Su Buqing’s <Differential Geometry> ^[3]) all specifically use the more reasonable concept of relative curvature kr_, which can take negative values, to establish the Frenet formula for plane curves. In fact, the definition of relative curvature kr is almost the same as that of equation (3), except that the absolute value sign in the numerator is removed as a burden. This allows the plane curves described to have inflection points, intersect, and define the rotation number of the plane curve tangent, making the theoretical content more colorful.

In the  second  part of my first blog post on ScienceNet, Starting from Nature and Beauty - In Memory of My Mentor Professor Qin Yuanxun , I easily used relative curvature to make a plane curve similar to one with 17 teeth .     

Figure 34.

Therefore, the author raises a question worth pondering: In the theory of space curves, if the absolute value sign of the numerator in the definition of curvature (6) is removed, will the entire theory be more reasonable and more exciting?

(to be continued)