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Natural Patterns

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 observed 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 alluded to 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. 

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Figure 1 The driving wheel of a locomotive illustrates the way the body converts linear force into circular movement and vice versa. (Mentally rotate the wheel counterclockwise, beginning at the top.)

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 many 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 movements and impulses. What I’m getting at in the following examples is not a deep understanding of the systems, per se, but how these systems reflect and are reflected by Tai Chi principles and movements. This is admittedly a “gee-whiz” sort of 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)

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Figures 2 & 3 Top: Spiric sections are included in the family of toric sections. Above: the Lemniscate of Bernoulli is a specialized toric section.

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Figure 4 A torus is the product of two circles rotating at right angles to one another.

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Figure 5 Bottom halves and cross-sections of the three classes of torus. From left: ring torus, horn torus, and spindle torus. .

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Figure 6 Some Cassini ovals.

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 not only magnets, but Earth and many other celestial bodies. The biofield that surrounds the human body also is a torus.

The implications for Tai Chi are that the movements of the art 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 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.)



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. Or to turn Tai Chi's four principal energies—the Cardinal Energies of Wardoff, Rollback, Press, and Push—into an infinite variety of circular and spiraling movements.


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 at 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 the torso or the individual limbs, and that you’re looking lengthwise at it, with both spirals revolving: the black spiraling inward and the red outward. This is very similar to the way that you can spiral chi energy downward or outward then back 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. (Figure 17)

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Figure 7 The doubled taijitu forms a figure-eight at its heart.

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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.

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Figure 9 A punctured torus in motion. (Click the image to view the animation.)

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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.)

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Figure 11 A Möbius Strip

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Figure 12 Three 360° turnings of one arm of an Archimedean Spiral

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Figure 13 Mechanism of a scroll compressor. (Click on image to see animation.)

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Figure 14 Fermat's Spiral demonstrates how energy spiraling inward can change direction without pause or loss of momentum.

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Figure 15 Archimedes' Screw is a hand-operated device for lifting water from one level to another.

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Figure 16 A combine harvester is a modern use of Archimedes' Screw.

Figure 17 An Archimedes' Screw uses circular motion to propel matter along the axis of rotation. (Click the image to view the animation.)

Archimedean Spirals are nice and regular thanks to their simple 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.


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Figure 18 A logarithmic spiral opens faster than it turns.

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Figure 19 Logarithmic spirals can be found throughout nature, including the Mandelbrot Set (right), which mathematically describes the boundary between order and chaos.

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 minimal 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.”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. The mathematics of the Fibonacci sequence can be graphically depicted in the design of the "Golden Spiral." (Figure 22)


Indeed, the Golden Ratio is called the “divine proportion” because it is ubiquitous 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 others.


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 23), and it has been used more recently by Salvador Dali, Piet Mondrian, and others. One extremely obvious example is Johannes Vermeer's Girl with the Pearl Earring. (Figure 24) 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 the visual representation called the “Golden Spiral.” This 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, 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 25)


Figure 26 illustrates how the taijitu and the Golden Spiral can work together 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 27) 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.

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Figure 20 The Euler Spiral finds application in railroad engineering to create ideal transitions from straight runs of track as they lead into curves.

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Figure 21 The Lituus (above) spirals outward, while the hyperbolic spiral (left) arcs inward.

Figure 22 The Golden Spiral, a foundation of order and beauty in nature, resembles a single taijitu fish.


Figure 23 Leonardo Da Vinci's Vitruvian Man approximates the Golden Spiral.

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Figure 24 Johannes Vermeer's Girl with the Pearl Earring illustrates one way that artists have incorporated the Golden Spiral into their work.

Figure 25 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 26 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. Above, the spiraling produces a corkscrew effect. (Click image to see animation). Right, energy can spiral into and out of a static golden spiral. Compare with Fermat's Spiral, above.

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Figure 27 Poinsot's spiral.



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 tijitu as being flat, but that isn’t necessarily the case within three-dimensional reality. Instead, the taijitu should be visualized as a three-dimensional torus, with an infinity of possible curves and spirals rotating around and through it. 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 lead 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 Viviani's Curve, a figure-eight that also is called the Lemniscate of Gerono 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—are where the cylinder's axis touch the surface of the sphere depicted in Viviani’s Curve. This 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 taijitu circles and figure-eights to propel energy in certain manners because I don’t think I could possibly describe the movements with words. Several ways of looking at these systems 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 curves and junctions, 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 apparent real function, though the Trammel of Archimedes is used as a method 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. This is another device that you can look at 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 opponent; 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 relaxation, continuity, 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.

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Figure 28 Viviani's Curve is the figure-eight space that results from a sphere wrapping a cylinder exactly to its edge. The figure, which produces the Leminscate of Gerono, suggests a three-dimensionality to the taijitu that is not immediately obvious from the two-dimensional symbol.

Figure 29 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).

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Figure 30 Three forms of Watt's Curve. Click on each image to view the animation.

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Figure 31 Watt's Linkage shows how linear force can be generated and dissipated by an S-curve. Click on the image to view the animation.

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Figure 32 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.


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.

Wikipedia entry: "Lemniscate”

Wikipedia entry: “Torus”

Wikipedia entry: “Möbius Strip”

Wikipedia entry: “Archimedean spiral”

Wikipedia entry: “Golden Ratio”

6  Zeising, Adolf, Neue Lehre van den Proportionen des meschlischen Körpers (1854), preface, from Wikipedia entry: "Golden ratio,”


Archimedes Screw

Archimedean Spiral

Cassini Oval

Euler Spiral

Fermat’s Spiral

Golden Ratio

Golden Spiral

Hyperbolic Spiral



Logarithmic Spiral

Poinsot Spirals

Spiric Sections


Trammel of Archimedes

Viviani’s Curve

Watt’s Curve

Watt’s Linkage

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