Hallelujah! I have found a solution to the most difficult dilemma faced by those of us on the anti-Fascist fringe of 21st-century politics. I am referring, of course, to the problem of how to accurately criticise the new US president without either falling foul of Godwin’s Law on the one hand, or understating the harm he is causing on the other.
In case you don’t know, Godwin’s Law states that ‘As an online discussion grows longer, the probability of a comparison involving Hitler approaches 1‘, meaning somebody is bound to make the comparison sooner or later. The Corollary of this law is that, if you have to resort to comparing somebody to Hitler, you have lost the argument.
I try to assiduously observe this rule by never making that comparison myself, as a consequence of which the comparisons I make for the most unpleasant of the world’s leaders and policies cover a wide range of alternatives, starting with the perhaps too-obvious Stalin, running through Mao and Pol Pot, Augusto Pinochet, Vladimir Putin, Robert Mugabe and then, just for variety, taking in some serial killers like Fred and Rosemary West.
But the following tweet from some Trump toady, in response to the President summarily banning travellers from a range of countries with Muslim majority populations, causing heartbreak, despair and airport chaos, had me stumped. Here it is, from someone called Kellyanne Conway:
Get used to it. @POTUS is a man of action and impact.
It is just screaming for somebody to respond ‘So was <He with whom, out of deference to Mike Godwin, we will not make comparisons>’
The idea that being a person of action is a virtue, regardless of the morality of that action, is so gob-smackingly asinine that it is beyond parody (which leads to another, lesser-known, internet law: Poe’s Law, which says that it is impossible to tell the difference between a parody of the US fundamentalist right, and the real thing).
But how can one respond, without invoking that famous Austrian, and thereby implicitly losing the argument?
Fortunately, I have found the answer, and it is…………..
That’s right, all you need say in reply to such thuggish ejaculations as Conway’s above is
So was Voldemort
There is even a scriptural basis for this comment. In ‘Harry Potter and the Philosopher’s Stone‘ the wand-maker Ollivander sells Harry his first wand, and is fascinated to find that the wand – which chose Harry, rather than Harry choosing it (don’t ask) – is the ‘brother’ (don’t ask about that either) of the wand that Ollivander had sold to Voldemort as his first wand, decades earlier. Ollivander says:
‘we must expect great things from you, Mr Potter …. After All, He Who Must Not Be Named did great things – terrible, yes, but great‘.
So there you are: when apparatchiks or groupies of the US government confuse – deliberately or otherwise – decisiveness with virtue, or power with goodness, you can now skewer their idiocy without impaling yourself on the spike of Godwin. Just remind them about Voldemort, and all the great, powerful things he did.
By the way, just as an aside, in the course of researching this painstakingly-researched and highly-detailed essay, the internet has made me aware that, contrary to my belief and that of most other people, apparently Mussolini didn’t make the trains run on time.
Post Script: Just prior to posting this I searched for images of Voldemort to decorate it. To my bemusement, I found that many of them have the new president’s face photo-shopped over that of Voldemort, and there was even an article explaining why he really is Voldemort. I also came across reports of Joanne Rowling’s tweets about Trump and Voldemort, saying facetiously that the comparison was unfair to Voldemort. Well, bother! I had previously been aware of those tweets, but had utterly forgotten them. Apparently my subconscious hadn’t though. Well, never mind. If something’s worth saying, perhaps some things are worth saying more than once, by different people, as long as they don’t deliberately copy. I’ll put a nice, unphoto-shopped photo of He Who Must Not Be Named on this anyway, and I hope you’ll forgive my subconscious plagiarism.
It has become apparent to me that the world needs another instalment in my series of suggestions for Adult Amusements. There have been complaints. Some are from pedants, who insist that a single monograph about standing on one leg does not constitute a series. Others, more gravely, have expressed concern about the occupational health and safety implications of people trying to balance on one leg while their mind is distracted by other things, like budgets, work-shopping and brain-storming, not to mention trying to be Pro-Active, Customer-focused, Agile, Continuously Improving and Outside the Box all at the same time.
So, belatedly, here it is. I hope that this will be considered less dangerous, being a mostly sedentary activity.
When in business meetings that do not hold us riveted with fascination, we should draw stars!
But not just any old stars. Special stars. Mathematical ones. Stars with prime numbers in them.
It is the dearest wish of every little child, after that of being a firefighter or an astronaut, to draw excellent stars in their pictures. But a wish is one thing, and its fulfilment is another. When as a child I tried to draw stars, the only technique I could think of was to draw a spiky circle. Start anywhere, and draw a perimeter that goes around an imaginary centre, that is a series of spikes. Maybe this works OK for others, but for me it typically produced a result like this (Figure 1):
It invariably goes wonky, because it’s hard to keep track of where the centre is supposed to be, and to make the points point away from that centre. Mine looks like a confused kookaburra.
When one gets a little bit older and more sophisticated, one learns – by instruction or by observation of others – the two standard techniques for drawing stars. These are the six-pointed star, which is made by drawing an upside-down triangle slightly above a right-way up one (Figure 2):
and the five-pointed Pentacle, which requires a little more coordination, but can be done without taking the pencil off the paper (which I call a ‘single pencil stroke’), by following the arrows as shown (Figure 3):
Learning to draw either of these stars is on a par with learning to ride a bike, in terms of the sense of achievement, wonder and progress. All of a sudden, one can construct an image of symmetry and elegance with the stroke of a pencil – or two strokes, in the case of the six-pointer.
I was very happy with this advance in technology for a long time, but then came the day when I hankered after drawing more bristly stars, with seven, ten or twelve points. I tried, but found I was just reverting back to the method of figure 1, and my bent stars just did not satisfy me.
One could of course take out a protractor and compass and, with a bit of preliminary calculation, measure out the exact angles needed for each point, and draw the star using that. But firstly that’s cheating, and against the Spirit of Doodling, and second it might cause others to notice that one is not paying attention to whatever the meeting is discussing.
I thought I was destined to be forever that object of public ridicule – the man with the two-star repertoire. But just as I was starting to come to terms with this being my fate, a discovery came to me in a blinding flash: instead of trying to draw spikes in a circle, I needed to generalise the methods used for the five and six-pointer. Well, to cut a long story short, I tried that, it worked, and now I can draw stars with any number of points up to about fifty.
Here is the method that generalises the way we draw five-pointed stars:
Drawing a star with a single pencil stroke
- Step 1: pick the number of points N, and draw that number of points, as evenly spaced as you can, around the perimeter of an imaginary circle. If there is a large number of points it’s best to first draw points at the 12, 3, 6 and 9 o’clock locations and then put one quarter of the remaining points into each of the four quadrants. To be precise, divide N by 4 to get a quotient Q and a remainder R. Then draw Q points in each of R quadrants of the circle, and Q-1 points in the other quadrants. Ideally, if R=2, adjacent quadrants should not contain the same number of points, but it doesn’t matter very much if that is forgotten.
- Step 2: pick a number K, greater than 1, that has no common factors with N. To make the spikiest possible star (ie with the thinnest spikes), choose K as the largest whole number less than N/2 that has no common factors with N. For instance if N=12 that number is 5. If N=13 it is 6. If N=6, 4 or 3 there is no possible K, and this method cannot be used. I’m pretty sure that, for any N greater than 6, there is at least one K for which this method will work, but I have not proved that yet.
- Step 3, choose your favourite direction in which you want to draw. Unless you are a pan-dimensional creature drawing on paper with three or more dimensions, your only possible choices are clockwise or anti-clockwise.
- Step 4 starting at any point, draw a straight line from that point to the point that is K steps away from it, hopping from point to point around the circumference in the chosen direction. We can call K the ‘side length’, since it is the length of the line that connects one point to another.
- Step 5: repeat step 4 until you get back to the starting point.
If this process is executed carefully, you will have drawn a star that has a point at every one of the points you drew in step 1. And, if you want, you can do all the actual line drawing in steps 4 and 5 without ever taking your pencil off the paper.
Here is a depiction of that process for an eleven-pointed star with side length 5:
And here is a depiction of this process for a sixteen-pointed star with side length 5:
Why do we not allow the side length K to be 1? That’s because if we do that, we just get a N-sided shape which, ignoring any irregularities in our drawing, is a regular polygon, like this, for N=12 (a ‘dodecagon’):
Now the thing about stars is that they are not convex, while regular polygons are. Using the word ‘vertex’ for a place where two edges of a shape meet, an N-pointed star has 2N vertices, of which N are points – the outermost part of a peninsula (if we imagine the shape as an island in an ocean) and the other N are the innermost part of a bay. As we go around the vertices of a star they alternate between inlet and bay. So a regular polygon is not a star because it has no bays, and that’s why K must be more than 1.
Stars with more than one pencil stroke
We observed that the above method does not work for N=6. But we know we can draw a six-pointed star, using two pencil strokes to draw two overlapping triangles. We can use the approach taken there to invent many more stars. In fact, for an N-pointed star there are M different types we can draw, where M is (N+1)/2-2, rounded down to a whole number. Each of these shapes corresponds to using a different value of K, from 2 up to the biggest whole number below N/2.
Here is how we do it:
- Step 1, for picking N and drawing the points around an imaginary circle, is the same as above.
- Step 2. We pick any K as any whole number greater than 1 and less than N/2.
- Do steps 4 and 5 from above. This will draw a shape that is either a star or a polygon. Now comes the tricky bit.
If the shape you drew has not touched all the N points around the circle, repeat the process starting on a point that has not been touched yet. I like doing this with a different colour pencil, as it helps me see the pattern and avoid getting confused.
Repeat that process, using a different colour pencil each time, until all points have been touched.
You will now have a N-pointed star, made up of a number of identical overlapping shapes, which are either all polygons or all stars.
For those that like mathsy stuff, the number of overlapping shapes – the number of pencil strokes required – will be the greatest common factor of N and K. It’s fun to try to work out why that is.
The traditional six-pointed star in figure 2 above is what you get under this method when you use N=6 and K=2. Here are a couple of others:
If we are going to draw a lot of different stars, we need names for them. We could call the star drawn with N points and side length K a ‘N-K star’, so that the pentacle is a 5-2 star and the traditional six-pointer is a 6-2 star.
If we wanted to, where N is even, we could let K be N/2. What we get then is this sort of thing:
The shape we have drawn with each pencil stroke is a single line between a point and the point directly opposite it. Strictly speaking, this too is a star, but I mostly leave it out because it’s not as interesting as the others because (1) everybody knows how to draw a star like that; (2) as any five-year old would tell us, that’s not what stars look like in pictures of things in the night; and (3) it has no inside, so we can’t colour it in all yellow (well, actually the one I drew has a tiny little inside in the middle, because it’s not perfectly symmetrical. But a more accurate drawing would have all the lines going exactly through the middle of the circle, so that there’s no inside at all).
So now you know how to do lots of great stars. You need never be bored in a meeting again. Imagine if you started drawing all the possible stars, starting at the smallest number of points and going up in side-lengths and points until the meeting finished. Leaving out the too-easy ‘thin stars’, you would draw the following stars:
5-2, 6-2, 7-2, 7-3, 8-2, 8-3, 9-2, 9-3, 9-4, 10-2, 10-3, 10-4, 11-2, 11-3, 11-4, 11-5, 12-2, and so on.
Just drawing those, given a due amount of tongue-stuck-into-side-of-mouth-concentration, should be enough to get you through at least a half hour of Death By Powerpoint.
But let’s not forget our roots. With a very few exceptions, we all started off drawing stars like Figure 1. There is a touching ingenuousness about such stars, and I think it’s good to draw them as well. Often really interesting shapes arise when we do, looking like monsters or funny animals. And one good thing about that way is that you don’t have to decide how many points it will have before you begin. You just draw spikes around a circle until you get back to the start. I’ll sign off by doing that for a star with LOTS of points (it ended up being 21), and following it up by a series of the nine different stars with the same number of points drawn by the above method.
I think that each has a certain appeal, in a different way.