Today's post continues my step-by-step exploration of my poi paper for easier searching. Yesterday featured my introduction and the basics of periodic math. Today we will apply these concepts to modeling flowers and third-order motions.
To be able to see how these functions relate to poi, it is helpful to be able to graph them to visualize the interactions of the different variables. This can be done with any graphing calculator or personal computer programs for graphing, including Grapher for Mac (found under Applications => Utilities) or Microsoft Mathematics for PC (downloadable here). Most diagrams included from here on out will be screen captures of graphs produced in Grapher and will include the equation used to produce the graph as a visual aid. The first such graph can be found below:
Above we have the graph of the equations we arrived at in the previous section: x=sint, y=cost where t is equivalent to all values between 0 and 2pi. First, ensure that if your calculator has one, that it is set to parametric mode. When we plug these equations into our graphing calculator, we should get the result shown above: a circle with a radius of 1. It is the first poi pattern we will model in this paper as it can be considered the pattern that is created by a single poi performing a static spin, or rotating around an unmoving hand. The hand and/or handle may be considered an infinitely small point at the very center of this graph. In real life, our hands do move a bit when performing static spin but the graph is meant to indicate an ideal state for this move--what we are hoping to achieve whether we are physically able to or not.
Now let’s try a slightly more complex pattern, graphing x=sint+sint, y=cost+cost:
At first it may look as though this is the same graph, but looks can be deceiving. I have zoomed out of the graph to ensure it would completely fit on the screen. As a result, it may be easy to miss the fact that the radius of this circle is now 2 instead of 1, meaning that the circle is twice as far across as well as twice the circumference as the previous circle. So what does this have to do with poi patterns? Did the poi get twice as long? Not quite. Let me show you the same graph with one small addition:
Now you will note a smaller circle inside the bigger circle with the smaller circle being modeled with the same equation as the one we used in our first diagram. Here, the relationship between the two is easier to see--the bigger circle is intended to be the pattern of the poi as it traces an extension around a handpath that is one poi length in radius. In other words, we can think of the first pair of terms in the graph as describing the movement of the handpath and the second pair describing the movement of the poi as shown to the right:
With this relationship in mind, we can now model a more complex poi path--perhaps a flower using the equations x=sint+sin(2t), y=cost+cos(2t):
In this case, we can see we have graphed out a 1-petal inspin flower with a handpath that is still 1 poi length in radius--what changed? As you can see from the equation, the terms that govern the movement of the poi have been adjusted such that we are now multiplying t by 2 before we find the sine value of it. This means that the terms governing the movement of the poi are now completing 2 circles for every 1 circle the handpath terms are completing. But how can we be completing 2 circles while getting 1 petal? The method for graphing we are using is based upon Cartesian coordinates, wherein there is an “up” and a “down” (I know I am overgeneralizing this and I apologize). The poi in this pattern completes two “down” beats for every one “down” beat that the hand completes, thus giving rise to the pattern.
Let’s see what happens if the poi terms incorporate a -2 instead of a 2, resulting in the equations x=sint+sin(-2t), y=cost+cos(-2t):
This results in a vastly different looking pattern, but still one wherein the poi has 2 “down” beats for every “down” beat the hand completes. You can see this clearly outlined below:
We would identify the pattern on the right as being a 3-petalled antispin flower, or colloquially a triquetra. A pattern should already be apparent from these two graphs: when t is multiplied by a positive number in the poi terms, the resulting graph will show an inspin pattern. When t is multiplied by a negative number in the poi terms, the resulting graph will show an antispin pattern. Let’s verify this prediction by graphing x=sint+sin(3t), y=cost+cos(3t):
And x=sint+sin(-3t), y=cost+cos(-3t):
The prediction has been verified for these two cases! When we multiply t by 3 in the poi terms, the resulting graph is the pattern for a 2-petal inspin flower while multiplying t by -3 in the poi terms results in the pattern for a 4-petal antispin flower. You may have noticed that if we replace the number we multiply t by in the poi terms with the variable d (“down” beats), resulting in the following equation: x=sint+sin(dt), y=cost+cos(dt), we can then predict the number of petals p that will result with the following equation: p=|1-d|. For d=-2, we can see that p=3 and for d=3 we can see that p=2. The equation works!
But what about if we want to model patterns that fit into the poi unit circle family--that is, poi patterns in which the handpath is half a poi length in radius rather than a full poi length in radius? We will model one such pattern, an isolation, with the following equation: x=1/2sint-sint, y=1/2cost-cost.
I’ve opted not to zoom into this graph to provide a proper sense of scale. For this equation, we do indeed find that now the radius of the circle produced is ½ the poi length. Even more interesting, the handpath and poi path graphs are now the same graph because the poi terms have been set to the opposite end of the circle by making both the poi terms negative terms instead of positive terms. We can switch this to the graph of a unit circle extension by switching the poi terms to positive, making the equations x=1/2sint+sint, y=1/2cost+cost:
Now we can see that the graph of poi terms describes a circle 1½ poi lengths in radius with a handpath in the center that has a ½ poi length radius. Now what about if we wanted to include cateyes? Since cateyes are antispin patterns, we will have to multiply t in the poi terms by a negative number: -1. The equation for a cateye, then, will be x=1/2sint+sin(-t), y=1/2cost+cos(-t). The graph for this pattern is below:
If you are curious, you can also model a horizontal cateye with the following equation: x=1/2sint-sin(-t), y=1/2cost-cos(-t).
With these examples, we can then state with confidence that we can describe all flowers and unit circle patterns with the following equation:
Where h is the radius of the handpath, p is the phasing of the poi at the beginning of the pattern, and d is the number of downbeats of the poi relative to the handpath’s downbeats. So given the examples we have above, we could plug the following values in for the variables:
This in and of itself is enough of a model to describe most 2D spinning that poi performers engage in. But it is still limited in that it does not describe either third-order motions or any form of 3D spinning. We can expand our equation, however, to make these two types of movement possible to model. We will start with third-order motions because they can be modeled in a 2D environment as well.
Third-order motions are an extension of flowers that were first named by Damien Boisbouvier after analysis of a number of patterns including Zan’s Diamond. Where flowers consist of two centers of rotation (the center of the handpath and the handle of the poi), third-order motions are generated by adding a third center of rotation. This is usually accomplished in two ways:
Either way, we already possess the tools to model this type of movement by extrapolating what we already know about flowers. Since we now have three centers of rotation, we can add an additional term to our equation to accomplish this task. For example, we can graph Zan’s Diamond with the following equation: x=sint-sin(-3t)+sin(5t), y=cost-cos(-3t)+cos(5t).
Again, I have also graphed the handpath so its relationship with the poi pattern is clear. In this case, the handpath follows the general pattern of a 4-petal antispin flower, with the poi adding two petals for every one petal the handpath leaves. It is important to note that in this update to our equation, the terms governing the handpath and poi movement have changed slightly. While the last set of terms still governs the poi, now the first two sets of terms govern the hand instead of just the first one as outlined in the diagram on the side. We can see another example of this type of pattern with what has been dubbed a “fractal” flower and in graphing it we will see why the term applies. For this graph, we use the equation x=sint+sin(-2t)-sin(4t), y=cost+cos(-2t)-cos(4t).
Here, we can see that the core handpath of the pattern is a Triquetra with a 6-petal pattern overlaid on top of it. If we imagine that the first set of terms multiplies its t variable by 1, then the relationship between the multipliers of the first two t’s comes out to 1:-2 or if we were to divide them, -½. Likewise, because the relationship between the t multipliers in the last term and the second is -2:4, we can divide them to arrive at a proportion of -½. In other words, the number of downbeats added between the first and second term is proportionally the same as that added between the second and third term. Because this pattern demonstrates a limited degree of recursion in its structure, we refer to it as a fractal flower after the concept of a mathematical set that displays self-recursion.
Both Zan’s Diamond as well as the fractal flower above also demonstrate a common phenomenon in performing third-order motions: adding two antispin petals for every petal encountered in the handpath. I will label this family of third-order motions “triquetra expansions” as they can also be thought of as adding one triquetra per petal. I will give formulas for both triquetra expansions as well as fractal flowers at the end of this section.
Thus far, all the patterns we have seen are examples of what would be called antispin-antispin third-order motions. That is, the handpath displays the patterns we saw above in antispin flowers and the direction of the poi is opposite that of the hand. This framework also predicts inspin-antispin, antispin-inspin, and inspin-inspin third-order motions. Though these latter three types are rarely performed, I will nonetheless provide modeled examples of each, starting with inspin-antispin: x=sint+sin(3t)-sin(-5t), y=cost+cos(3t)-cos(-5t).
This pattern grafts a triquetra expansion on a 2-petal inspin handpath. For an antispin-inspin, we will use the equation x=sint+sin(-3t)-sin(-7t), y=cost+cos(-3t)-cos(-7t).
This pattern results in a core handpath pattern of a 3-petal antispin flower with a 1-petal inspin grafted onto each antispin petal. You may note that the third t multiplier is a negative number just like the second t multiplier--this doesn’t mean that we’ve lost the antispin-inspin pattern we were shooting for, only that I had to do a little bit of mathematical wrangling to ensure proper petal placing on the poi pattern. Another approach to doing the same operation would be to add pi to the multiplier of last set of t terms, such that: x=sint+sin(-3t)-sin(7t+pi), y=cost+cos(-3t)-cos(7t+pi)
Finally, inspin-inspin can be executed with the equation x=sint+sin(3t)+sin(5t), y=cost+cos(3t)+cos(5t):
Here, we’ve added one inspin petal for every inspin petal of a 2-petal inspin handpath. With several of these patterns now presented, we can assemble an equation with enough variable assignments to cover everything we can do with third-order motions. That equation is presented below:
Such that h is the radius of the first center of rotation (either the shoulder or imaginary), p is the phasing of the second center of rotation (handpath), d is the number of downbeats the handpath possesses, q is the phasing of the poi pattern, and f is the number of downbeats in the poi pattern. That’s a lot of variables to keep track of! As mentioned above, here are some recipes both for triquetra expansions and fractal flowers:
For triquetra expansions of antispin-antispin third-order motions, p=+-1, d<=-1, q=-p, f=|d|+2
For fractal flowers in antispin-antispin, p=+-1, d<=-1, q=+-1, f=d^2. Please note that values for p and q may need to be tried in different combinations until the desired aesthetic effect is reached for the resulting pattern.
Tomorrow, we will start to dive into the 3D realm by modeling both simple and complex weaves, including body tracers.