Recently, the Man in the Lab Coat cracked one of his old questions via simple trial and error, and while it’s an odd question, it’s a good one to which I’d like a mathematical solution, so I hope you’ll indulge me. We all know how to draw a pentagram, a simple five-pointed star constructed of five lines which cross each other. Some of you may already be aware that you can draw an octagram, a similar sort of shape which has eight points and is drawn with eight lines.
The question the Man always wanted answered was: is there a star with more than eight points that can be drawn in the same way? The answer is, obviously in hindsight, yes: one with thirteen points, which I choose to call a triskadecagram (I’m hoping I just coined a phrase, since this shape doesn’t seem to be culturally or mystically significant to anyone else). What’s the secret? Or perhaps, what’s the big deal?
These shapes, I’ve just discovered while looking for pictures, are both examples of regular star polygons. They have radial symmetry, which means you can draw a straight line through the centre of the shape at a number of different angles and find that the halves either side of the line are identical. This makes them very pleasing to look at. But more interestingly, I can instruct you how to draw them easily and at the same time reveal the mathematical progression within.
When you draw a pentagram, you can think of it like this: imagine a circle. Now, imagine five dots drawn on the circle so that they’re evenly spaced. Number them one through five. Put your pen on dot number one, and draw a straight line to dot number three. Now connect dot three to dot five, then dot five to dot two, dot two to dot four, and dot four back to dot one. You’ve just drawn a pentagram. You can think of your progress in terms of “skipping” past a dot to connect to the one after it, continuing to repeat this simple instruction until you get back to the start – dot one – with the fifth line. We can also say we add 2 to the current dot’s number (wrapping around back to 1 when we go over 5) to find the one we want to connect to, giving us the sequence 1, 3, 5, 2, 4, 1.
An octagram follows the same technique, but our circle has eight evenly spaced dots, and you “skip past” two with each line; your first line is from dot one to dot four, then dot four to dot seven, and so on until you get back to the start. In this instance, we add 3 to the current dot’s number, and we “go around” three times before we get back to the starting point. Our numbers are 1, 4, 7, 2, 5, 8, 3, 6, 1.
You can probably guess the progression for my patented triskadecagram. Our circle now has 13 equidistant points along it’s circumference, and we skip three points with each line, or add 4 to the current point’s number to get the number of the next point. This shape’s sequence is 1, 5, 9, 13, 4, 8, 12, 3, 7, 11, 2, 6, 10, 1.
Not counting the ones, it’s a pleasing pattern that all three sequences produce sets of three numbers before “going round again” – overlapping and crossing the first line of the previous set. You can extrapolate backwards, too: an equilatral triange is a regular star polygon with only three points on the ceonceptual circle, and were we don’t skip any dots when drawing our lines – we add 1 to the current number to get the next, giving 1,2,3,1.
I may not be well versed in geometry or maths, but these shapes and their derivation gave me a lot of satisfaction. More so when we try to find a function such that f(x) = n, where n is the number of points, and x is the number added to the current dot to find the end point for a line. How to work this out? Well, f(1) = 3; f(2) = 5; f(3) = 8; and f(4) = 13. From these I have tried to construct a formula, but I don’t have the right mathematical vocabulary; it may even be impossible, since the web sites I have found (MathWorld, for example) discussing these shapes list only how they are named, and make no mention of a function or equation to find bigger regular ones.
Pleasingly, none of these pages list star polygons with more than 12 points, though they do reveal other intermediates I had not thought of – like a 7 pointed star with an x of 3, or a 9 pointed one with x = 4. It may be that a function exists, but I don’t know how to describe the relationship that lists only my shapes, and not these other “interlopers”. x and n don’t seem to have the kind of relationship I imagined – there are plenty of plausible xs for a given n – though the trick to being able to draw the shape without taking pen from paper is that x must not be a factor of n.
So what is the big deal? Well, there can be simple beauty, and complex mathematics, in even a handful of lines. We can all appreciate that, with or without the numbers behind it.