What is a triquetra?
For most of the past year, triquetra has been synonymous with three-petal antispin flowers and in some cases the hybrids that can be created by combining them with other patterns. Nick Woolsey even posted this video, explaining the concept and the term and its significance to poi spinning in general. After doing the math, however, I've come to the conclusion that what we describe as triquetras don't actually match the visual or mathematical properties of triquetras at all and that a couple of the conclusions we've reached based upon this assumption are false.
There are two assertions made in this video that are closely related and at the heart of the issue. They are:
If you consult the wikipedia entry on triquetras, you see quite a few different diagrams and examples of what triquetras look like. Below are the two most popular options, along with the shape generated by a 3-petal antispin flower (fig 1). The two patterns rendered are either the intersection of 3 vesica piscis between 3 circles (fig 2) or the intersection 3 vesica piscis between 4 circles (fig 3).
A vesica piscis is defined as the overlap of two circles that meet at each other's center points (fig 4). When creating the first type of triquetra shown above, all three circles thus touch each other's centers (fig 5) while in the latter type, three circles all intersect a center circle at its center, but not each other (fig 6). Because of this, the three shapes wind up having vastly different geometric properties.
For the first type of triquetra, if we isolate only those parts of each circle that are inside the triquetra proper, we find that this segment of the circle is half it's total circumference (fig 7). With three such segments we find that the distance traveled by a poi head in this shape would be 1 1/2 times the distance of the extension around this shape--a vast difference. On the other hand, if we take the region within the triquetra of the 4-circle version, we find that the sections overlapped into the center circle are each 1/3 the circumference of each circle and the extension circle around them (fig 8). In this case, the triquetra really does have the same circumference as an extension it would be inlaid within. But if we superimpose the actual path a poi head follows in a 3-petal antispin flower, we find the two don't match up perfectly (fig 9). The poi path is clearly slightly larger, but by how much?
While staying in Africa back in August, I had the good fortune that my host Will Ruddick, in addition to being a poi spinner with a keen interest in international development, was also an accomplished Python programmer. While there, we collaborated on a program that could measure the total distance traveled by a poi head in any shape that could be described with a parametric equation. The method we used is as follows:
Will used a turtle draw program to graph the following parametric equation:
Where R1=radius of hand path circle, R2=length of poi (radius of poi path circle), n1=number of hand path downbeats, n2=number of poi path downbeats, and t=position in circle in radians. The program graphs points along the poi path and measures the distance between these points. To ensure the accuracy of the distances measured, we compared the program's measurements of extensions of different size to their circumference as derived by 2*radius*pi. We found that graphing points at increments of pi/100000 in radians yielded results that were identical to our control. With this result in hand, we proceeded to graph a number of inspin and antispin flowers to measure the total distance traveled by the poi head.
To produce a triquetra given these equations, then, we would set R1=1, R2=1, n1=1, n2=-2.*
The result was that the poi head of a 3-petal antispin flower travels 6% more distance than an extension of the same size (the actual proportion is 0.94025222741786, which will be true of a poi tether/arm of any length). In other words: though a 3-petal antispin flower closely resembles a triquetra rendered by the intersection of 3 vesica piscis and 4 circles, it travels more than 1/20th the circumference of the extension further. A triquetra, then, is not an inverted extension.
Indeed, to make a triquetra given that it is the product of intersecting vesica pisces, one would need to perform 1/3 of an extension, point isolate around the poi head another 1/3 the distance of the extension, rotate the poi head another 1/3 the distance of the extension, etc until the poi head and hand had both performed 3 of these operations which would be difficult if not functionally impossible.
We may already be at a point where the term has saturated spinning culture to the point that it's now inseparable from 3-petal antispin flowers, but the distinction mathematically is an important one. Indeed, when we measure the distance traveled by the poi head we find that the inverse of a 3-petal antispin is actually a 1-petal inspin flower. Both shapes are 2 poi downbeats for every 1 hand downbeat.
If you're interested checking out the Python program, you can download it here. A complete spreadsheet of all the poi head distances will be coming as I write up more of the results.
* Added 1/18/2013 as it was pointed out that though I included the equation used I did not include the variables necessary to generate the pattern in question.