A magazine of martial and movement arts, with a focus on the internal style of Tai Chi Chuan
by Christopher Dow
Spiral. The very word elicits the movement of the line that creates the figure from a specific central point to an arbitrary periphery—arbitrary because, of course, a spiral theoretically spirals outward forever. Spirals abound in nature, and mathematicians have identified a great number of types. All of them illustrate not only the functionality of tai chi but also its universal beauty. We’ll start with the simple basic spiral known as the Archimedean Spiral or arithmetic spiral. (Figure 12) This is a spiral whose successive turns are equidistant from each other in a simple arithmetic progression: 1, 2, 3, 4, etc. In other words, each turn is the same distance from the turns next to it, no matter how large the spiral grows.
Archimedean Spirals can be found in watch balance springs, the grooves of early gramophone records, and products bought in rolls, such as wrapping paper, tape, and vinyl flooring. It also can be used, interestingly enough, in one method of mathematically squaring a circle, and we all know that one of tai chi’s precepts is to find the straight in the curved and the curved in the straight.
A more technical reading of the definition of an Archimedean Spiral is interesting from a tai chi standpoint: Such a spiral corresponds “to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity.” (4) Think of performing a simple Rollback without any sort of flinging or pulling. The tai chi exponent is the fixed point, while the opponent is the one rolling away from the fixed point of the exponent with a constant speed and constant angular velocity.
One interesting application of the Archimedean Spiral can be found in the mechanism of a scroll compressor (Figure 13), which is used to compress gases and liquids. In the animation, the red spiral is stationary because of the device’s function, but it wouldn’t be similarly constrained inside a dynamic system such as the human body. Imagine these two spirals as an energy structure within either the torso or the individual legs, and that you’re looking downward on them, with both spirals revolving: the black spiraling inward (downward) and the red outward (upward). This is very similar to the way that you can spiral chi energy downward then upward in the opposite direction, compressing it in the process, much in the same way that the scroll compressor squeezes gases or liquids. Chi, after all, is a fluid energy.
The energy’s change in direction at the central point—at the Bubbling Well of the foot in tai chi—is well-illustrated by a type of Archimedean Spiral called Fermat’s Spiral. (Figure 14) It’s easy to see in the diagram how energy spiraling inward can smoothly change direction without stopping so that it can spiral outward without losing momentum. You also can see that the curve where the change in direction takes place is like the S-curve running through the taijitu.
Before we leave the well-regulated world of the Archimedean Spiral, let’s look at one more practical application of this structure: the Archimedes Screw. (Figure 15) This is a device for lifting—pumping—water commonly attributed to Archimedes but that probably is older by several hundred years. More recently, the device has found uses in other types of machinery, such as combine harvesters. (Figure 16) The interesting point for tai chi chuanists is the way that screwing or spiraling energy can propel an object along a line perpendicular to the circular action—the circular motion around the shaft propels the object along a line parallel to the shaft. (Figure 17)
Figure 16 A combine harvester is a modern use of Archimedes' Screw.
Archimedean Spirals are nice and regular in a simple sort of way because they exhibit an arithmetic progression, but another class of spiral exhibits a logarithmic progression, unwinding with turns that are wider and wider at a regular mathematical rate, such as 2, 4, 8, 16, etc., or 3, 9, 27, 81, etc. (Figure 18) Such spirals open up faster than they turn. One Archimedean Spiral will look like all other Archimedean Spirals, but the mathematical variety of logarithmic spirals means that such spirals can take a large number of exact forms, though they might have a class-based similarity in appearance.
Logarithmic spirals are widely found in nature: in sea shells, hurricanes, flower petals, spiral galaxies, and much more. It’s even found in the Mandelbrot Set, which mathematically describes the boundary between order and chaos. (Figure 19) This also is the boundary that lies between the tai chi exponent (order) and incoming energy (chaos) that is manipulated by the tai chi exponent by exploiting these spirals.
Logarithmic spirals are frequently employed in engineering. One example is the Euler Spiral (Figure 20), which finds application in railroad engineering to create ideal transitions from straight runs of track as they lead into curves, and vice versa. Appropriate angles of curvature help transit the forward momentum of the train smoothly into and through the curving track so that the train experiences a minimum of tilt. This is yet another example of finding the straight in the curved and the curved in the straight.
One group of logarithmic spirals called the Cote’s Spiral can either spiral outward or inward. (Figure 21) These actions of logarithmic spirals are employed by the tai chi chuanist when Rollback is combined with an outward flinging or an inward pulling.
One very special logarithmic spiral is produced from a mathematical concept that is called the “golden ratio” or “golden spiral.” Mathematically, the golden ratio occurs when the ratio of two quantities is the same as the ratio of their sum to the larger of the two quantities. In other words, where a is the larger quantity, there is a golden ratio if a+b is to a as a is to b. (5) Mathematicians since Euclid have studied the properties of the golden ratio. In 1202, the Italian mathematician Fibonacci (Leonardo Pisano Bigollo) published a sequence of numbers—called the Fibonacci Sequence—that approximates the golden ratio, and this sequence has found application in computer algorithms, graphs, and other scientific and mathematical techniques.
Indeed, the golden ratio also is called the “divine proportion” because it is exhibited in natural physical structures, such as the branching of trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, the uncurling of a fern, the arrangement of a pine cone, the veins of leaves, the spiral form of some mollusk shells, and many other instances. Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the skeletons of animals and the branching of their veins and nerves, the proportions of chemical compounds, and the geometry of crystals. The golden ratio’s ubiquitous presence throughout nature prompted him to see it as a universal law of natural structure. In 1854, Zeising wrote that this universal law “contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.” (6)
Naturally, artists as well as scientists and mathematicians have been fascinated with the golden ratio. Leonardo Da Vinci roughly displayed it in his famous Vitruvian Man (Figure 22), and it has been used more recently by Salvador Dali, Piet Mondrian, and others. The Argentinean sculptor, Pablo Tosto, has listed more than 350 works by well-known artists whose canvasses feature the golden ratio or a close approximation. It is found in the architecture of the Great Mosque of Kairouan, and the famous Swiss architect Le Corbusier extensively used the golden ratio in his designs. It also is present in the proportions of Medieval manuscripts, and even in music. Musicologist Roy Howat has observed that the formal boundaries of Claude Debussy’s La Mer correspond exactly to the golden ratio, although it is disputed whether this was deliberate or not.
For tai chi enthusiasts, the golden ratio should have its own special meaning, particularly when it is converted, using the Fibonacci Sequence or other techniques, into a visual representation called the “golden spiral.” The golden spiral is a logarithmic spiral that gets wider by a factor of the golden ratio for every quarter turn it makes. One look at this spiral (Figure 23), and you will instantly see what I mean. A single spiral is similar to one tai chi symbol fish, and when the spiral is doubled, it forms an approximation of the entire tai chi symbol—or, rather, the taijitu approximates a doubled golden spiral.
Figure 24 depicts how the taijitu and the golden spiral can work together to illustrate tai chi’s ability to draw incoming energy into a vortex and then expel it through an unwinding of the vortex, similar to Fermat’s Spiral, mentioned above. It also demonstrates the fact that tai chi, like the taijitu and the golden ratio, is fractal: No matter what its scale, it always takes the same form and operates on the same principles.
Before we leave spirals, let’s look at one more: Poinsot’s Spiral. (Figure 25) This is an altogether more complex logarithmic spiral that spirals back upon itself. Interestingly, the way that chi energy spirals through the major portion of Grasping Bird’s Tail in Northern Wu Style, if viewed from above, almost perfectly mimics the entire Poinsot’s Spiral. And many other tai chi movements utilize various portions of it.
Tai chi movements often are described as circular, but in reality the descriptor should be “curvilinear” or “spiraling,” for the art takes advantage of movements that expand or contract along curving lines, leading to spirals or other non-ending curvilinear structures, such as parabolas and hyperbolas. Over time and with practice, the spirals that begin by being large expressions of physical movement contract, becoming small spirals that exist almost exclusively inside the body, spiraling downward and upward through the torso and legs and outward and inward along the arms.
Figure 24 A combination of the taijitu and the golden spiral demonstrates tai chi's ability to coil and then uncoil energy without halting its movement or momentum. Compare with Fermat's Spiral, above.
Figure 12 Three 360° turnings of one arm of an Archimedean Spiral
Figure 13 Mechanism of a scroll compressor. (Click on image to see animation.)
Figure 14 Fermat's Spiral demonstrates how energy spiraling inward can change direction without pause or loss of momentum.
Figure 15 Archimedes' Screw is a hand-operated device for lifting water from one level to another.
Figure 17 An Archimedes' Screw can use circular motion to propel an object along the axis of rotation. (Click the image to view the animation.)
Figure 18 A logarithmic spiral opens faster than it turns.
Figure 19 Logarithmic spirals can be found throughout nature, including the Mandelbrot Set (left), which mathematically describes the boundary between order and chaos.
Figure 20 The Euler Spiral finds application in railroad engineering to create ideal transitions from straight runs of track as they lead into curves.
Figure 21 The Lituus (above) spirals outward, while the hyperbolic spiral (left) arcs inward.
Figure 22 Leonardo Da Vinci's Vitruvian Man approximates the Golden Spiral.
Figure 23 The golden spiral, a foundation of order and beauty in nature, resembles a single taijitu fish. When the spiral is doubled, it creates a figure resembling the tai chi symbol.
Figure 25 Poinsot's spiral.