I’ve had a few thoughts and discussions since I wrote this article about drawing stars. I thought it was worth sharing.
Stars within stars
The first observation is that stars, drawn in the way I described, contain other stars, nested within one another like a set of Russian dolls. Recall that we use the term ‘n-k star‘ to indicate a star with n points such that, if we draw a circle through all the points and consider the boundary of that circle as split up into n curve segments bounded by the points, then the straight line from one point to another traverses k of those curve segments. Like this, for a 7-3 star:
Yuriy made an interesting observation about the stars in my last article, that the sum of the angles of all the points of an n-k star is 180∘ × (n-2k). In the course of thinking of ways to prove that formula true, I came upon the realisation that a n-k star contains a n-(k-1)star, which contains a n-(k-2) star and so on down to the n-1 star, which is a regular, n-sided polygon.
Here’s a picture that shows this for an 11-5 star.
The red, outer 11-5 star contains a green 11-4 star, which contains a red 11-3 star, which contains a green 11-2 star, which contains a red 11-1 star (polygon). The points of each inner star are the innermost vertices of the star that immediately contains it. Since we will be referring to those vertices again later, let’s make up a name for them. We’ll call such a vertex a ‘tniop‘, since it is in a sense the opposite of a point. The above diagram shows a point and a tniop. We’ll call the stars inside a star ‘sub-stars‘.
We saw in my last essay that, when a star cannot be drawn without taking the pencil off the paper, it is made of a number of ‘component stars’, that are rotated copies of one another. Here is a picture of a 16-6 star, which uses different colours to highlight the two component stars. We have two 8-3 stars, one light blue and one red.
And here is a picture that uses colour variation to show the sub-stars of each of the two components.
The 8-3 light blue star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured blue and pink. Similarly, the 8-3 red star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured yellow and green.
The 8-2 stars each contain an 8-1 star (octagon) as the intersection of two squares -pink and dark blue for one octagon and yellow and green for the other.
Finally, those two octagons between them, bound a hexadecagon (16-sided polygon or 16-1 star). So altogether, in the one picture, we have:
- one 16-6 star (red and light blue)
- one 16-5 star (also red and light blue) inside that
- one 16-4 star (pink, green, dark blue and yellow) inside that
- one 16-3 star (also pink, green, dark blue and yellow) inside that
- one 16-2 star (also pink, green, dark blue and yellow) inside that
- one 16-1 star (also pink, green, dark blue and yellow) inside that
- two 8-3 stars (one red, one blue) making up the 16-6 star
- two 8-2 stars (one pink and dark blue, one yellow and green), one inside each of the 8-3 stars
- two 8-1 stars (octagon: one pink and dark blue, one yellow and green), one inside each of the 8-2 stars
- four 4-1 stars (squares: coloured pink, dark blue, yellow and green) which, in pairs, make up the 8-2 stars.
That’s sixteen stars altogether. What a lot of stars in one drawing! Can you see them all?
Here’s a different colouring that makes it easy to see all five 16-point stars:
Although the stars get smaller as k reduces, they do not shrink away to nearly nothing. In fact they get closer together as they go inwards, as if they are asymptotically approaching a circle of some fixed, minimum size.
To investigate this, I drew a 101-50 star:
You’ve probably noticed by now that I’m no longer drawing these by hand. My drawing is much too wobbly to capture the intricacies of stars-within-stars. So I wrote a computer program to draw them for me. I’ll try to remember to attach it at the end of the article, so that those of you who like mucking about with computers can muck about with it.
Anyway, that 101-50 star pretty well killed my hypothesis that the inner stars can’t get very small. In this one they almost disappear out of sight. I like the swirly patterns. I haven’t yet worked out whether they are really features of this very spiky, very complex, star, or whether they are just artefacts of the crudeness introduced by the computer’s need to pixillate.
Here’s a zoomed-in image of the interior of that star. Cool, eh?
This is a low-resolution image. I have saved a moderately high-resolution image of this star here. Zooming in and out is fun. It seems almost fractal as more patterns emerge from the inside when we zoom in. Also, the stars give the illusion that they are rotating as we zoom. To get the best effect you need to download the file (a .png image file) and then open it up, so that zooming is not limited by your internet connection’s speed.
Ratio of Outer to Inner radius
Let me briefly pick up on that idea above about whether there is some minimum inner radius for these stars. For each n, the spikiest n-k star is where k is the largest integer less than n/2, and this contains another k-1 stars, nested one within the other, down to the innermost, which is a n-sided regular polygon. We can work out the ratios of each star to the one immediately inside it, and use that to work out the ratio of the outermost, n-k, star, to the innermost, n-1, star. The attached computer program contains trigonometric formulas to do that. Here are the ratios of the radii of the innermost to the outermost star for each n from 1 to 109:
We observe that the ratios generally go down as n increases, but the decline is not steady. It bumps up and down. I have highlighted the prime numbers with asterisks. Notice how the ratio for those is always lower than for the numbers around them. The two drivers of the ratio seem to be:
- the size of n: the ratio generally declines as n increases; and
- the number of different factors n has. Note how 16 (divisible by 2, 4, 8) has a higher ratio than 15 (divisible by 3, 5) and 56 (divisible by 2, 4, 7, 8, 14) has a higher ratio than 55 (divisible by 5, 11).
This would be interesting to look into further, and to see if there is some neat, sweet, compact formula for the ratio that highlights the relationship to size and number of factors (if there really is one). But I have to stop thinking about that now or I’ll never post this essay.
The most general form of symmetrical stars
We can make an awful lot of stars using the above approach. For an integer n the number of different n-pointed stars is the biggest integer less than n/2.
But in fact there is an infinite number of different n-pointed stars, without having to loosen our standards by allowing asymmetry. After a bit of thought, I realised that the most general form of n-pointed star can be specified by a single number, which is the ratio of its inner radius to its outer radius. The outer radius is the radius of the circle on which the points sit. The inner radius is the radius of the circle on which all the tniops sit. Given n and that ratio – call it θ, we can draw a star as follows:
- Draw two concentric circles with ratio of the inner to the outer radius being θ.
- Mark n equally-spaced dots around the outer circle and draw faint lines connecting each of these to the centre. These will be the points of our star.
- Mark a dot on the inner circle halfway between each of the faint, radial lines drawn in the previous step. These dots will the the tniops of our star.
- Working consistently in one direction around the circle, draw a zig-zag line fro point to tniop to point to tniop and so on, always connecting to the nearest dot.
We will describe such a star as a n/θ star. We use a slash rather than a dash in order not to mix it up with the former type of star. Here is a sequence of six-point stars, with the ratio of the inner to outer radius going from 0.2 up to 0.8:
It is nice that this gives us more options for stars – infinitely many different kinds of star for each n in fact. But they are not as much fun to draw as the n-k stars, and it is harder to make them come out right without geometric instruments – which rules them out as an effective doodling pastime.
Note how, unlike with the n-k stars, the line leading away from a point does not intersect any other point. Nor do we get any inner stars for free. The price of gaining more variety is a loss of structure. The inner structure of a n-k star provides a great richness by enforcing all sorts of relationships between the vertices.
Only very specific values of the radius ratio θ give us n-k stars. I worked out the formula for the ratio by the way, with a bit of trigonometry. The ratio of the tniop radius to the point radius for a n-k star is:
2 sin(π(k-1)/n) sin(πk/n) sin(π(1-(2k-1)/n)) / (sin(2π/2)+sin(π(1-(2k-1)/n)) )
– sin(π((2k-1)/n – ½) )
For a n/θ star, if there are no integers n and k that give values of that formula equal to θ, the star will not have the array of inner stars that a n-k star has.
Computer Program to draw pretty stars
This whole diversion started as an exercise in drawing stars by hand. But there’s a limit to how intricate those stars can get without getting too messy with smeared ink or graphite. For those that like to look at pretty, intricate geometrical pictures, here’s my computer program in R that can draw stars of any of the types discussed.
Sums of angles
For those that like mathematical proofs, there are outlines of proofs here of Yuriy’s observation about the sum of the internal angles at points of a n-k star.
Bondi Junction, April 2017
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.
‘Partial differentiation’ is an important mathematical technique which, although I have used it for decades, always confused me until a few years ago. When I finally had the blinding insight that de-confused me, I vowed to share that insight so that others could be spared the same trouble (or was it just me that was confused?). It took a while to get around to it, but here it is:
My daughter Eleanor make a drawing for it, of a maths monster (or partial differentiation monster, to be specific) terrorising a hapless student. The picture only displays in a small frame at the linked site, so I’m reproducing it in all its glory here.
Bondi Junction, October 2016
Today is international polysyllablitis awareness day. I hope you can spread the word so that people will better understand this debilitating condition and try to support those that suffer from it.
Polysyllablitis is a communication disability that primarily affects people that read too many fancy books. The main symptom is a swollen vocabulary, leading to frequent difficulty in finding an acceptable word to express a concept they are trying to convey. Such difficulty typically manifests in uncomfortably long pauses mid-sentence, because the speaker was about to say that the proposed expedition to a nightclub would be ‘inimical to his health‘, but didn’t want people to think him a ponce for saying a fancy word like ‘inimical‘, yet the alternatives ‘it would make me feel bad‘ or ‘I’m tired‘ (average syllable count per word = 1.0) refused to present themselves to his desperately searching mind.
For this to happen just occasionally – say every couple of months – is manageable. Many people have such experiences. But people with really serious polysyllablitis (known as PSI to health and remedial vocabulary professionals) can suffer such attacks as often as several times a day. At such frequencies it can become terribly debilitating. Sorry, I mean it makes the person feel really bad.
Chronic sufferers have complained of persistent diffidence (meaning they often feel shy), disorientation (they feel dumb or lost), isolation (they feel lonely) and melancholy (they feel sad).
I have studied this phenomenon (sorry, I mean thing) for many years now. I think there is hope for the sufferers, as long as they don’t get excluded (shut out) from society. That’s why we need this awareness day. If people can keep a look out for others that may be suffering this malady (it makes them ill) they will be able to find ways to help them, reassure them (make them feel good) and put them on the road to rehabilitation (get better).
The best way to help these unfortunates (poor guys) is to include them in your conversations. When they say an unnecessarily fancy word, or get stuck mid-sentence with that look on their face that says they can’t remember the normal-people’s word for ‘lugubrious’*, the best thing to do is to gently correct them, remind them of the normal-person word while making clear that we still love and accept them. (*it’s ‘sad’). Studies have shown that these inclusionary strategies (being nice to them) are in most cases highly efficacious (they work).
However, in my years of study, there is one word for which I have simply never found a way of translating it into normal person speech, and that is the word ‘interlocutor‘ – being ‘the person with whom one is having a conversation‘. I have searched in vain for a simple alternative. The closest I’ve seen is ‘discussant’ but that has the dual problems that (1) it’s ugly and (2) I suspect it’s not a real word.
The next most reasonable alternative seems to be to replace the word with its definition ‘the person with whom one is having a conversation’. But that doesn’t really help much, as that ‘whom’ is bound to raise eyebrows, not to mention the monarchical ‘one’ (sorry – I mean like how the queen would speak). Plus inserting that long string of words into a sentence raises the risk of apparent poseur-ness because of the length of one’s sentences.
‘He’s always interrupting those with whom he is having conversation‘ just doesn’t have the pizazz of ‘He’s always interrupting his interlocutors‘.
I doubt Hemingway would approve.
It wouldn’t matter if it was a useless word, like that silly old ‘antidisestablishmentarianism’ that schoolboys used to quiz each other on, but nobody ever used in a genuine sentence. That was, until the Guinness Book of Records people wanted to get in on the act and invented ‘floccipausinihilipilification’, just so that people would buy their book to find out about the new record-breaking word.
Of course if you want a long word that’s actually used by proper people, it’s supercalifragilisticexpialidocious, which at 34 letters is longer than either of those non-words to boot. Plus it’s used by Mary Poppins, who is cool and not anything like a social reject that got her head stuck in a dictionary, so it must be OK.
But, unlike antidisestablishmentarianism, interlocutor is not a useless word. How can one talk about conversations one had yesterday without using it? More importantly, how can one give counselling and therapy to PSI sufferers if one cannot tell them useful things like ‘try to use the same words that your interlocutors use‘? The word is simply too useful to discard. I find myself needing to use it at least seven times per day on average. I’d be lost without it.
I can only see one way out of this conundrum (tricky thing). That is to make interlocutor an honorary normal person’s word. We could do that by all making an effort to use it at least once a day. Then before long it would seem as normal as ‘but’. There are precedents for this. Normal people use the pentasyllabic ‘qualification’ when talking about who might get into the finals in the footy, and the quadrasyllabic ‘ceremony’ when talking about who earns the right to humiliate themselves in the next round of a reality TV show. So I think, If we all make an effort, we can create some space for ‘interlocutor’ in normal people’s language.
I leave you today with these two requests:
- Please keep an eye out for PSI sufferers, and try to be kind to them (and help them to get better); and
- Try to use interlocutor as often as is consistent with common decency.
Just remember, no matter how strange and scary they seem, every PSI sufferer is somebody’s son or daughter.
Bondi Junction, October 2016
George Orwell said ‘At the age of fifty, everyone has the face s/he deserves’.
I first heard that saying decades ago and for some unknown reason remembered it. I was never very confident about exactly what Orwell meant by it, but I have always interpreted it to mean that if you spend your life being cross you will end up looking like a cranky old wo/man. But if you spend your life smiling kindly, you will look like a kind old person. It goes along with that other old saying, that your face will get stuck with whatever expression you are wearing when the wind changes (or does that rule only apply if you are making a face at somebody?).
If it were true, it’s bad luck for those that suffer a lot of pain or grief in their first few decades. They would end up looking permanently in pain or sad.
There’s not much we can do to avoid pain or grief, but we have at least some control over whether we scowl or smile on those around us.
Orwell’s saying came back to me at around the age of forty. I didn’t remember what the cutoff age was but I remembered that you had to watch out if you didn’t want to end up like Mr Wintergarten or any other fictional old person the neighbourhood children avoided in fear.
I started paying occasional attention to my facial expressions, noting when I smiled. I was somewhat relieved to find that I smiled quite often, partly because my children, who were all below ten years old at the time, often made me laugh or smile at their antics. ‘Thank goodness!’ I thought. I would be safe from ogredom and the neighbourhood children would be free from my future reign of terror.
There are two special occasions when I do my best to smile – they are when riding my bicycle on public roads, and when jogging.
The reason for the jogging smile is that I heard that a famous American public intellectual and wit said something like ‘If I ever see a jogger smiling I might try it‘. For a long time I thought Gertrude Stein said that but now the internet tells me it was actually the comedian Joan Rivers in 1982. I don’t know if others would count Joan Rivers as a public intellectual, but I like to think of Public Intellectuality as a broad church. Anyway, I resented the implication that joggers were a miserable bunch that hated jogging and did it either because, like banging your head against a wall, it feels so good when you stop, or because like an Opus Dei monk wearing a cicatrice, they felt that the pain they were suffering was somehow accumulating points for them in their heavenly bank account.
Fie on you Ms Steinem (yes I know, I get Gertrude Stein and Gloria Steinem mixed up – pathetic isn’t it) I admonished her inside my head. I don’t suffer when I jog. I quite enjoy it most of the time, and sometimes I even love it. But I had to admit she had a point about the smiling. Joggers didn’t tend to smile, perhaps because they were too busy trying to breathe.
So I determined to set the world to rights. I became possibly the world’s first ever smiling jogger. I didn’t smile all the time. It is quite tiring on the facial muscles to maintain a smile for minutes at a time, as any games show barrel girl will attest (some people think that catwalk models look so sulky these days because they are perpetually hungry, but I think it may also be because it is more relaxing to maintain a vacant gaze than a beaming smile). But as soon as a passer-by hove into sight, I lit my face up like a Christmas tree, so they could see just how much fun I was having.
This led to some peculiar looks, and mothers shepherding their children anxiously away from me with worried expressions on their faces.
My campaign of smiling on a bicycle was for a different reason, and met with rather more success. There are a bunch of nasty ‘shock jocks’ in my city that anathematise anybody that expresses any concern for the environment as a Luddite, anti-democratic communist. They save their most virulent hatred for refugees and bicycle riders, in both cases, apparently because they clog up the roads and thereby interfere with the God-given right of every right-thinking person to drive their Land Cruiser down any street at 60km/h plus, unimpeded.
While most people, fortunately, are not influenced by this outpouring of bile, it does have some spillover effects and it did tend to generally increase the degree of hostility between cyclists and motor-car drivers. I thought that if I smiled at motorists that I encountered (or, at least, at the ones that hadn’t just nearly killed me by turning in front of me, cutting me off, passing too close and fast or just blaring their horn at me so close as to make me nearly fall off in fright) I would be doing my little bit to rebuild cordial relations.
I am pleased to report that this little strategy, unlike campaign Joggers-Can-Smile-Too, met with unexpected success. I received plenty of return smiles, waves and other gracious, heart-warming gestures. So, take that, Alan Jones!
For some reason it is also easier, and feels more natural, to smile when riding than when jogging. It might be because riding is after all more intrinsically fun than jogging, because of the whizzing. We all love to whizz after all, and not many of us are capable of jogging at whizzing speed. I used to be able to, but have not been able to for a long time now. Plus, every time one’s foot hits the ground (which is about eight-three times a minute, in case you were wondering), one’s facial muscles all get wobbled about by the shock-wave, making it more than usually hard work to maintain a smile. If you don’t believe me, look at a slow-motion replay of the 100m race in the Olympics and watch what the faces do. Ignore that famous sideways smile photo of Usain Bolt at the Rio Olympics. That was in a semi-final, so he wasn’t really running very fast (for him).
There’s also the fact that, because the air is rushing towards you quite fast on a bike, you don’t need to open your mouth into a big fat O shape to get enough air in. A sweet smile leaves more than enough opening for enough of the rearward rushing air to find its way to the lungs.
After a while, it just became a habit to smile when I was riding my bicycle, at least, when I wasn’t climbing a difficult hill or negotiating a particularly dangerous traffic situation.
So, in between the child-induced smile, the jogging smile and the bicycling smile, it seemed that my face was probably doing what was necessary in order to meet Mr Orwell’s challenge.
Now I am well past fifty, so I suppose I am out of danger. My face has, I suppose, become set in whatever configuration it is to maintain from here on in. The only expected future changes are ever-increasing numbers of wrinkles, perhaps sun-spots and scars from removed skin lesions and a gradual loss of teeth and hair. But can I be sure of that? After all, while Mr Orwell’s skill as an author is beyond question, his expertise as a gerontologist is comparatively unknown. Could he have been mistaken? What if it is sixty, seventy, or even eighty? One cannot be too careful. Perhaps it is too early to stop smiling.
Which brings me to the topic of this essay (better late than never): adolescent and young-adult offspring just don’t seem to compel beaming, helpless smiles from adults in the same way that two year olds do. Of the positive emotions that adolescents can generate (we’ll not dwell on the negative ones), there are affection, pride, sympathy and a number of others but “Oh my goodness that’s so adorable!” is not usually one of them. I presume this has something to do with evolution. We are programmed to find almost every utterance and action of a two-year old adorable, because they cannot fend for themselves and, if we didn’t find them adorable, we might not be inclined to fend for them – which wouldn’t do at all, not if we want them to grow up to be Prime Ministers. But above the age of about sixteen, the fending skills of the human species appear to be adequate, so evolution decided to ease off on the adorability spell. That may be all very well – after all, many adolescents prefer to spend time in any company other than that of their parents, and parents are easier to shake off if they are not following you around with adoring grins on their faces. But how are we to meet our smiling quota in the absence of such an influence? I have a feeling that now I may spend less than half the amount of time smiling that I did ten years ago. I can put some of that down to my mid-life crisis, but I think the partial maturation of my children has to bear some of the responsibility.
What, then, is to be done? One has to find other things to make one smile. But what? That will have to be the topic of another essay.
Ian Dury knew though. He made a list, in his song Reasons to be Cheerful (Part 3).
Here’s a picture of Ian Dury showing he had not much to smile about, with his grim environment and the after-effects of his childhood polio on display. And yet…….
Here’s a piece I wrote explaining the mathematics behind the peculiar phenomenon of acoustic ‘beats’.
It’s a bit maths-y. But for those that don’t love maths quite as much as I do, it also has some interesting graphics and a few rather strange sound clips.
Bondi Junction, August 2016
Have you ever been in a meeting or other group activity that was just dragging along, keeping you teetering interminably on the edge of profound boredom? It happens to me quite often.
When children are caught in this sort of situation – such as in church or on a long car journey – they can relieve their feelings by complaining to their responsible adult ‘I’M BORED’ or ‘Are we there yet?‘
But we poor adults do not have that excellent outlet available to us. Partly because we have no responsible adult to complain to, and partly because people would judge us if we were to blurt out such phrases.
So I thought it was time that somebody came to the rescue of the wretched responsible adults that have to endure these situations. To that end, I am starting a series devoted to equipping adults with the tools to amuse themselves and stave off boredom, when caught in unexciting, unavoidable group activities.
I don’t know how long the series will be – perhaps not long at all. It is, after all, so much harder for adults to amuse themselves than it is for children, to whom everything is new and exciting (until they reach adolescence, when suddenly everything becomes old and beneath contempt).
Here, then, is my first piece of Useful Advice For Bored Adults.
Stand on one leg!
Start by lifting one foot just a little off the floor, and see how long you can keep it off. If you only lift it a tiny bit, nobody will notice, and it may not affect your balance much. You may find you can do it for ages.
Once you’ve mastered that, which might be straightaway, or might take a little while, start increasing the height to which you raise the foot. The higher it goes, the higher one’s centre of gravity is and the easier it is to overbalance.
Don’t overdo it with the high foot. If you raise your foot above your waist, people might start to look at you funny. But kudos to you if you can do that and remain balanced though. I couldn’t do it to save my life.
I recommend that, once you can sustain the foot at near knee level, you move to the next phase, which I think of as the Aboriginal pose. I think that name springs up in my mind because when I was a wee lad, for some reason the pictures we were shown of traditionally-living Australian Aborigines in the outback often showed them standing like this. I am a little nervous of calling it that in a public blog, lest anybody think it disrespectful. That is certainly not my intent. And, since the ability to sustain the pose is an admirable skill, I am hoping that it is not considered disrespectful. It certainly seems no worse, and probably much better, than saying that somebody gave a ‘Gallic shrug’, which seems a fairly accepted (if somewhat dated) turn of phrase that is by no means complementary to our French cousins.
Here’s what that pose consists of: you lift one leg and bring the foot of that leg to rest with the sole against the side of the knee of the other leg. More advanced practitioners may even rest the foot on the thigh above the knee. Rookies may content themselves with resting the foot against the upper part of the calf.
I can do this pose a bit. I find that I can rest motionless for a while like that – maybe up to twenty seconds – then I start having to make lots of little adjustments with my planted foot to try to remain in balance. These adjustments increase in frequency and amplitude until either I overbalance and have to put the foot down, or – magical relief – I re-attain a stable body position. The latter doesn’t happen very often, but when it does, it’s like winning gold at the Olympics! One looks around in triumph, just a little puzzled as to why the others in the group activity haven’t broken out in rapturous applause.
While engaged in this entertainment, I often overhear myself telling myself that not only am I staving off boredom, but I am burning calories, toning my leg muscles, getting closer to nature (really?) and building a much-needed sense of balance. This is based on a total number of scientific studies that was, at last count, approximately none. But I still feel good about it.
Plus, you get to feel like a four-year old for a while.
That’s all for now. Stay tuned for the next instalment – ‘drawing stars’.
By the way, could it be that the reason for standing on one foot in the outback is to minimise the amount of heat soaked in from the hot sand? If so, that sounds like a very sensible arrangement. But whatever the reason, I remember always thinking that traditionally-living aborigines must have a much better sense of balance than we clumsy Europeans.
Oh, and one last thing. Remember to switch feet from time to time. Otherwise you’ll end up getting all asymmetric, like Arnold Schwarzenegger on one side of your body and Woody Allen on the other.
Which would make it hard to find clothes that fit.
Bondi Junction, April 2016