A magazine of martial and movement arts, with a focus on the internal style of Tai Chi Chuan

by Christopher Dow

© 2016

CURVE

Curves can be considered to be sections of circles/spheres, tori, and spirals. Tai chi exponents are acutely aware of the importance of curves in a basic sense, but some curves demonstrate that the form of curves isn’t always simple. We saw above how the taijitu, and particularly its central curvilinear line, forms or is a section of a spiral. Most of us probably think of the figure-eight produced by a reverse doubling of the symbol as being flat, but that isn’t necessarily the case within three-dimensional reality. When moving in three dimensions, one can spiral in one direction then change the angle of the motion to continue spiraling without a break in another direction that is not only tangential, but perpendicular to the first.

This fact is important in tai chi as a way to smoothly alter the direction of incoming force into another path. Generally, the tai chi exponent is thought of as being a sphere that simultaneously backs away from and turns away from an incoming force. This can happen on a simple physical level, but one extremely interesting mathematical curve shows how incoming force can be enveloped by an energetic sphere in preparation for manipulation. This is Viviani’s Curve, which is the result of a cylinder impinging itself upon a sphere.

Not all such impingements will produce the same result. If the cylinder completely penetrates the center of the sphere, the intersections will be two circles, and if it does not fully enter the sphere, the intersection will be something like a bite taken from an apple. But if the cylinder is allowed to penetrate the sphere only as far as its back edge, the result is a Lemniscate of Gerono, a figure-eight that completely wraps the cylinder and can exert spiraling influence on the cylinder from any angle and toward any angle. (Figure 26)

If you map the figure of the Lemniscate of Bernoulli onto Viviani’s Curve (Figure 27), the former seems to warp inward at the ends, making the structure more akin to one of the two leather skins that wrap a baseball, unfolded, than to a flat depiction. Thus, the eyes of Bernoulli’s figure—of the taijitu fishes—seem to be places where a rod or axis of some sort is protruding from opposites sides of the sphere depicted in Viviani’s Curve. This axis also appears to be the central axis of the cylinder, which means that the energy cycling through the loops of Viviani's Curve completely encases the cylinder's one-dimensional axis with three-dimensional energy.

The Watt’s Curve, named for the inventor James Watt, is yet another complex curve that demonstrates tai chi principles of movement. (Figure 28) Thankfully we have animations to show how energy can be cycled through the complex yet basic tai chi circles and figure-eights of these figures to propel energy in certain manners because I don’t think I could possibly describe the movements with words. You can find several ways of looking at these systems that apply to tai chi. For example, when looking at the three moving points, consider the central one to be central equilibrium and the other two to be the hands at the ends of the arms. It can be seen how relatively small yet stable and centered movements can propel much larger swings of energy as one leads the energy through the different structures, particularly the figure-eights.

James Watt gave us another tai chi example in a simple mechanism called the Watt’s Linkage. (Figure 29) This mechanism shows how external force acting on an object with a tai chi center can be diverted by moving through the curve in the middle of the taijitu. Conversely, it shows how the taijitu S-curve can instigate angular momentum. Both are additional examples of finding the straight in the curved and the curved in the straight.

Still another example of the straight and curved is the Trammel of Archimedes, which, when made from wood and sold in roadside joints, is sometimes called the Kentucky Do-Nothing. (Figure 30) It derives this latter name by being a device that is interesting but that has no real function other than to draw or cut ellipses. But from a tai chi perspective, it perfectly illustrates how two linked linear forces can produce not only an ellipsis but can be used to fling an object to some distance or pull it inward. You can look at this device in several ways, say with the center of the cross in the middle as central equilibrium, the two sliders as the hands, and the black square as the external force or object; or the black square can be the tai chi exponent, the farther slider the opponent, and the near slider the pivot point between them. As with all of these systems, you can find your own parallels.

Tai chi teachers emphasize many different aspects of the art, such as relaxed movement, chi development, martial applications, and so forth, but throughout all these aspects runs one unifying concept: naturalness. This naturalness seems to come from the movements, but in truth, the power of the movements stem directly from their perfect adherence to nature, from its most superficial aspects to its deepest reaches.

Notes

All technical information in this article was derived from the Wikipedia entries on the various spirals, curves, and other constructs. The illustrations, except for Figures 1, 7, 8, 22, 23, and 24, also are from the Wikipedia articles on the different structures or systems.

1 Wikipedia entry: "Lemniscate”

https://en.wikipedia.org/wiki/Lemniscate

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2 Wikipedia entry: “Torus”

https://en.wikipedia.org/wiki/Torus

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3 Wikipedia entry: “Möbius Strip”

https://en.wikipedia.org/wiki/M%C3%B6bius_strip

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4 Wikipedia entry: “Archimedean spiral”

https://en.wikipedia.org/wiki/Archimedean_spiral

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5 Wikipedia entry: “Golden Ratio”

http://en.wikipedia.org/wiki/Golden_ratio

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6 Zeising, Adolf, Neue Lehre van den Proportionen des meschlischen Körpers (1854), preface, from Wikipedia entry, “Golden ratio,” http://en.wikipedia.org/wiki/Golden_ratio

Sources

Archimedes Screw

https://en.wikipedia.org/wiki/Archimedes%27_screw

Archimedean Spiral

https://en.wikipedia.org/wiki/Archimedean_spiral

Cassini Oval

https://en.wikipedia.org/wiki/Cassini_oval

Euler Spiral

https://en.wikipedia.org/wiki/Euler_spiral

Fermat’s Spiral

https://en.wikipedia.org/wiki/Fermat%27s_spiral

Golden Ratio

https://en.wikipedia.org/wiki/Golden_ratio

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Golden Spiral

https://en.wikipedia.org/wiki/Golden_spiral

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Hyperbolic Spiral

https://en.wikipedia.org/wiki/Hyperbolic_spiral

Lemniscate

https://en.wikipedia.org/wiki/Lemniscate

Lituus

https://en.wikipedia.org/wiki/Lituus_(mathematics)

Logarithmic Spiral

https://en.wikipedia.org/wiki/Logarithmic_spiral

Poinsot Spirals

https://en.wikipedia.org/wiki/Poinsot%27s_spirals

Spiric Sections

https://en.wikipedia.org/wiki/Spiric_section

Torus

https://en.wikipedia.org/wiki/Torus

Trammel of Archimedes

https://en.wikipedia.org/wiki/Trammel_of_Archimedes

Viviani’s Curve

https://en.wikipedia.org/wiki/Viviani%27s_curve

Watt’s Curve

https://en.wikipedia.org/wiki/Watt%27s_curve

Watt’s Linkage

https://en.wikipedia.org/wiki/Watt%27s_linkage

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Figure 26 The Lemniscate of Gerono results from a sphere wrapping a cylinder exactly to its edge. The figure suggests a three-dimensionality to the taijitu that is not immediately obvious.

Figure 27 When considered as the central figure-eight of the taijitu, the Lemniscate of Bernoulli (above) maps in very interesting ways onto the Lemniscate of Gerono (above left).

Figure 28 Three forms of Watt's Curve. Click on each image to view the animation.

Figure 29 Watt's Linkage shows how linear force can be generated and dissipated by an S-curve. Click on the image to view the animation.

Figure 30 The Trammel of Archimedes, also called the Kentucky Do-Nothing, perfectly illustrates how two linked linear forces can produce not only an ellipsis but can be used to fling an object to some distance or pull it inward. Click on the images to view the animations.