A magazine of martial and movement arts, with a focus on the internal style of Tai Chi Chuan
Mathematical and Mechanical Systems
that Exhibit Tai Chi Behavior
by Christopher Dow
Tai chi chuan is rooted in physics on both a physical level and an energetic one, so it should come as no surprise that its physical and energetic movements can be seen within mathematical and mechanical systems. Undoubtedly, there are many such systems of which I am unaware, but I’ve found a number of them that illustrate my point and shed some light on tai chi.
Perhaps one of the easiest parallels to observe is noted in the statement from the Tai Chi Classics: “Seek the straight in the curved and the curved in the straight.” This statement describes the idea that straight lines can drop into curves and that curves can throw off tangents. An excellent example can be seen in the action of the driving wheel of a railroad locomotive. Like tai chi, a driving wheel uses an offset of linear force in relation to an axis of rotation. (Figure 1) Normally, the wheel is spun using the linear force of the thrusting and counter-thrusting piston to create spin. But it also is possible to spin the wheel to drive the piston back and forth. The wheel’s axis of rotation is like central equilibrium, and the piston is like an arm that is thrust forward or backward by the rotation of the wheel around its axis.
And there are a number of other mechanical and mathematical systems that exhibit tai chi concepts, behavior, and dynamics. Please be aware that my descriptions of these systems are shallow. I am not a scientist, mathematician, or engineer, and the majority of the technical information on these systems comes straight from the Wikipedia entries on them—as do most of the illustrations. But I do know a little about tai chi and how particular types of movements can empower or be empowered by chi flow as well as by particular sorts of physical impulses and movements. What I’m getting at in the following examples is not an understanding of the systems, per se, but how these systems reflect and are reflected by tai chi principles and movements. In short, this might be a “gee-whiz” article, but I think that these systems help demonstrate how closely tai chi adheres to the mechanisms that underlie reality.
We’ll start with the lemniscate and its variants. (Figures 2 & 3) The term comes from Latin and means “decorated with ribbons.” In algebraic geometry, a lemniscate is any of several figure-eight or ∞-shaped curves. Lemniscate are considered to be cross-sections of a torus. (Figure 4) If a torus is bisected by a plane parallel to the axis of the torus, the result in most cases is two circles or ovals. (Figure 5) However, when the plane is tangent to the inner surface of a torus—like slicing a donut right at the inner edge of its hole—the cross-section takes on a figure-eight shape, or, a lemniscate. (1) The illustration of the Cassini Oval clearly demonstrates the torus shape in profile. (Figure 6)
The variations between the several types of lemniscate are of interest primarily to mathematicians and scientists, but the tai chi chuanist will readily recognize the basic shape as being the essence of the doubled tai chi symbol. (Figure 7) I’ve written extensively about how force and energy cycle through the doubled taijitu in my book, Circling the Square: Observations on Tai Chi Dynamics, and elsewhere in Taijitu Magazine (Here and Here), so I won’t reiterate all that here except to say that tai chi chuan is so named because its movements, both physical and energetic, rely for their power on following the path—or some portion of it—of the curves—particularly the figure-eight in the middle—found in the doubled taijitu. (Figure 8)
This fact becomes even more interesting when one further considers the torus itself, which is the structure of which lemniscate are subsets. “A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.” (2) Looking at a picture, it’s easy to conceive of a torus as a static donut shape, but while inner tubes, bagels, O-rings, and even apples and red blood corpuscles are toroidal shapes, they are not true tori, but are solid torus shapes. A solid torus is formed by rotating a disc around an axis and is the torus plus the volume of matter inside the torus. A true torus is formed by rotating a ring around the axis, and the resulting form is empty yet energetic.
A true torus is anything but static. It is in constant motion, but that motion isn’t a simple rotation in one direction, like the rolling of a wheel. Instead, the torus is in constant movement in two dimensions, which anchors it within three dimensions. Two examples of energy tori are the magnetic fields that surround Earth and many other celestial bodies and the biofield that surrounds the human body.
The implications for tai chi are that the movements of tai chi do not simply orbit through the figure-eight in one direction or the other, but that the entire figure-eight also constantly and simultaneously spins toward and away from the onlooker. These complex yet unified movements allow a person who has developed the skill of sensing and riding these currents the ability to rotate, spin, or otherwise avoid incoming force along any curve or tangent and then to cause the force to come back upon itself or to become enveloped in the tai chi practitioner’s energy.
The animation of a punctured torus is a case in point. (Figure 9) Consider the torus to be the tai chi exponent’s field and the point of puncture to be the opponent’s incoming energy. As soon as the incoming energy penetrates the torus field, the exponent melts away then rolls and folds back from the point of entry. At all times, the torus exhibits emptiness to the force, yet the torus remains a torus. Another interesting case is how a torus in four dimensions—the three spatial ones over a span of time—almost magically turns force that is moving in one direction into force that moves in a tangential direction. (Figure 10)
The interesting thing about a torus is that, like ovoids, it seems to have multiple sides—front and back, at least—but at all times, it actually presents only one face to the outside. In essence, it can eternally cycle around itself without stopping or reaching an end. One example of this is the Möbius Strip (Figure 11), which is a toroidal section that has only one side. (3) If an ant were to crawl steadily along a Möbius Strip, it would make two revolutions and end up just where it started. In tai chi, some hand movements mimic the twist of the Möbius Strip or some portion of it. (You can create a Möbius Strip by cutting a strip of paper, giving the strip one twist, and taping the ends together. Presto! You have a created a two-dimensional structure that has finite width and infinite length within three-dimensional reality.)
Figure 1 The driving wheel of a locomotive illustrates the way the body converts linear force into circular movement and vice versa.
Figures 2 & 3 Top: Spiric sections are included in the family of toric sections. Above: the Lemniscate of Bernoulli is a specialized toric section.
Figure 4 A torus is the product of two moving circles.
Figure 5 Bottom halves and cross-sections of the three classes of torus. From left: ring torus, horn torus, and spindle torus.
Figure 6 Some Cassini ovals.
Figure 7 The doubled taijitu forms a figure-eight at its heart.
Figure 8 Energy cycles through the taijitu by moving around the periphery then threading through the figure-eight. The energy can cycle in either direction and thread through the center starting at either the top or the bottom.
Figure 9 A punctured torus in motion. (Click the image to view the animation.)
Figure 10 A stereographic projection of a Clifford Torus in four dimensions performing a simple rotation through the x-z plane. (Click the image to view the animation.)
Figure 11 A Möbius Strip