It’s a busy time, the end of choir practice. It’s 9:05pm and I haven’t had my dinner. I need to put away chairs, don my cycling safety gear, unlock the bike and whiz back home to look for something to eat. Busy, busy, busy. So one is distracted, right?
And I found myself singing that famous line from the Hallelujah chorus
‘and She shall reign for ever and ever’
Why were you singing that, Andrew? I hear you ask.
Well the chorus had been the last thing we were practising and, you have to admit (if you’ve ever heard it) that it’s very catchy. No wonder that GF Handel was the Andrew Lloyd Webber of his day.
No, not that, you respond – I mean, why ‘She’? Don’t you know that the official lyric says ‘He’?
Well actually yes, I do know that, which is why I was a little surprised to find that my subconscious mind, after deciding to make me sing that song, had also decided to make me sing ‘She’. I don’t know why it did. I think it may be because there’s a lovely alliteration in ‘She shall….’ that you don’t get with ‘He’.
But then I thought to myself, as I strode in a purposeful and manly manner towards my bicycle, why not She? Where does it say that God has a sex, and that it is masculine?
Now I know what you’re thinking: the Bible and the Quran are both full of He this, He that, Father this and Lord the other. That’s true, but you ask any theologically sophisticated Christian or Muslim whether God has gonads and I’m pretty sure they’ll say ‘Of course not!’ God is much too big and impressive, not to mention invulnerable, to have a collection of soft, funny-looking, easily damaged organs dangling annoyingly between his legs.
I think there are two reasons why male pronouns and nouns are used to refer to God in the scriptures of Middle-Eastern religions (Christianity, Islam, Judaism), both of which are to do with cultural traditions and have no theological basis.
The first is that ancient Middle-Eastern cultures, like most all other old cultures, including English and American, are patriarchal and use masculine pronouns in all cases except where the person being referred to is definitely female. All sorts of interesting reasons for this can be discussed but, whatever the reason, we cannot doubt that that is the practice. In a sense, ‘He’ is just the way of saying ‘She or He’ in that language tradition. In the modern, progressive parts of the world, we are working to undo those traditions, because of their toxic effect on sexual equality. But that’s a modern phenomenon that occurred centuries after the King James bible, let alone the original versions written in the period 950BCE – 150CE (600-900CE for the Quran).
The second reason is more specific. In those patriarchal cultures, it was assumed that a figure of authority must be male. Yahweh / Allah was the ultimate Boss, so It was described as male, as the notion of a female boss would have just been too incomprehensible – and unacceptable – to consumers of the stories.
Neither of these reasons retain any validity in modern, Western society, so there is no reason to perpetuate the implicitation of masculinity that was adopted at the time of writing. In fact, there are good reasons to actively overturn that implication, as just another undesirable plank in the ugly edifice of male dominance.
There is one other reason that was suggested to me by a Roman Catholic friend, that is more concrete. That is that Jesus was a man. Let’s accept for now the biblical narrative that there was a single man called Jesus of Nazareth, on whom the gospel stories are based, and whose body housed the incarnate spirit of God. Then the worldly container for the spirit of God did indeed have an XY chromosomal pattern, testicles and a penis. But why should that make us think that the immaterial spirit that pre-existed that body, and survived it, also has those things. We are told that Jesus had a beard. Does that mean that the spirit also has a beard?
If God’s plan was to incarnate as a human and preach an important message, It had three options for a body in which to incarnate: as a man, as a woman, or as a human of indeterminate sex. In Palestine CE30, only one of them had any chance of success. Nobody would have taken a woman seriously, and someone of indeterminate sex would likely have been put to death as a perceived infraction of God’s laws. So the choice of Christ (the part of God’s spirit that is said to have incarnated as Jesus) to incarnate as a man was simply an expedient, and says nothing about the sex of Christ.
Christians pray to Christ – the spirit – rather than to Jesus, even though they may say Jesus because it sounds more friendly. Jesus was the incarnated man, and he only existed for about thirty years. It is Christ that the religion says is eternally in heaven, and to whom a Christian prays. And there is nothing to credibly suggest that Christ has a sex.
Are there any other reasons why God should have a sex?
I can’t think of any that aren’t completely silly. One that immediately comes to mind is that God is The Boss, and bosses are more often than not male (although personally I have been fortunate to have had at least as many female as male bosses in my work career, and there is no doubt about who wields the power in the reasonably-happy home I inhabit). We’ve already dealt with that.
Another is that God is portrayed as a Father. But again, the intent of this metaphor (metaphor because It’s not really a father – there is no divine sperm involved) is to convey that God has the same loving, guiding, protective relationship to us that a parent typically has to their child. The scripture writers just wrote Father rather than Mother or Parent because of the language conventions mentioned above.
Any more reasons? No, I’m afraid I can’t think of any.
On the other side, there are excellent theological reasons against attributing a sex to God.
According to 1 John 4:8, God is Love. Does love have a specific sex? No.
According to John 1:1 God is The Word. Do words have a sex? No.
According to the influential theologian Paul Tillich, God is the Undifferentiated Ground of Being. Do Grounds of Being have a sex (provided we don’t differentiate them!)? No.
According to St Thomas Aquinas, God is Pure Actuality. If we distill Actuality until it is pure, does it acquire a sex? No.
According to St Augustine, God is Goodness Itself. Does Goodness have a sex? No.
I can tell you don’t want me to go on, so I won’t.
Right, now that we’re all agreed that God has no sex, what are we going to do about the fact that nearly all the words written and spoken about God attribute masculinity to It?
This is my plan. Please listen carefully.
From now on, whether you believe in God or not, in every reference you make to God that is in a context where use of a sexed pronoun is natural, I want you to use the female form.
As you are all intelligent and attentive readers, you naturally understand that this is not because I think God has a sex and that sex is female. Rather it is that, even if this idea went viral, it would have no hope of balancing out the enormous number of references to God as male that are out there. So we’ll keep on at this until God references achieve sexual parity, and then we’ll think about what to do next. This is not, as Alan Jones or Donald Trump might claim, ‘playing with words’ or ‘political correctness gone mad’. It’s just using sensible language that recognises that women and men are equally human and equally capable of anything except for a very few sex-specific activities such as fertilising an ovum or gestating a baby human. It’s a step that subverts the subtle message that only a man can be a person of power and wisdom. It’s a small but meaningful step in the project of gradually dismantling millennia of male dominance and oppression. And who better to lead such a step than the religions that have historically been – and unfortunately in some cases still are – platforms for those that seek to perpetuate that dominance.
So, if you please, it’ll be:
‘Our Mother who art in Heaven….’, in The Dame’s Prayer.
‘And He shall reign for ever and ever….’
Jesus is the Son of Woman (note the preservation of the word Son for Jesus, on account of the real-life testicles on the body used for Christ’s incarnation).
‘And She looked down on Her creation, and saw that it was good’.
The hymns will need reworking too:
‘Hail Redeemer Queen divine’
‘Queen of Queens, and Dame of Dames’
Everything, except specific references to the body of Jesus of Nazareth, has to go, and be replaced by its feminine equivalent.
What nice, friendly, inclusive places churches will become when this is adopted. I would happily visit them and sing along to ‘God rest ye merry (gentle)women’ in a spirit of ecumenical solidarity.
I don’t want to pick unfairly on Middle-Eastern religions, even though, they being by far the most powerful ones, they can take it. So let’s pause to consider the others.
Non-Middle-Eastern religions seem to generally be less patriarchal than the Middle-Eastern ones. There are powerful goddesses in Indian, Egyptian, Native American, Norse, Greek and Roman religions. But in all cases the boss of the gods is male. Apparently there have been, through the twentieth century, groups of scholars that believed that ancient religions such as druidism worshipped an Earth Mother type deity as their main focus, but these beliefs have fallen into disfavour in academia, and start to look more like wishful thinking of survivors of the Peace and Love generation of the sixties, than historically accurate accounts. The only well-known religions – ancient or modern – in which the most powerful being is female are neopagan religions such as Wicca. Well good for them, I say. But they are a very small minority, and the male dominance of the other religions I mentioned at least lets the Middle-Eastern triumvirate that currently dominates the world off the hook a little.
But Andrew, you protest, you are not a practising Christian, or a Muslim, or a Jew, so why should you care what words they use to talk about their gods?
You make a fair point, dear reader. The religions towards which I feel the greatest affinity are Buddhism and Vedanta, neither of which have any connection with the Middle East. But although I am not a Christian, Christianity has a major effect on my daily life and the lives of those around me, through the enormous influence that Christian power-brokers have on our laws and social customs. So it is in my interest, and in the interest of anybody that wishes for a kinder society, for the average Christian, as well as the power hierarchies of the various Christian sects, to become more consultative and compassionate. I think the religion becoming less male-dominated and male-oriented would help in moving along the road towards that goal.
And the same applies to Judaism and Islam. While their influences are minor where I live, there are parts of the world where their influence is intense. The people living in those regions would greatly benefit from those religions shedding some of their patriarchal orientation, and where better to start than to stop pretending that God is a bloke.
Bondi Junction, December 2017
I try hard to be open-minded. I think I succeed at that reasonably well, but I still regularly get surprised at the discovery of a prejudice I didn’t know I had.
I don’t know whether it’s possible to rid one’s self of all prejudice – I suspect it’s not. If so, the best I can aim for is to be on the alert for prejudices, try to rid myself of them when I discover them, and try to always remember that any opinion I have – regardless of how carefully thought out it may seem – may be inextricably tied up with some prejudice I don’t yet realise I have.
The Wikipedia article on Cognitive Biases has a very long list of them. With so many opportunities to go wrong, it’s hard to imagine one can escape all of them.
There’s a popular phrase ‘It’s good to have an open mind, but not so open that your brain falls out‘. I don’t like that phrase at all. It is most commonly used by bigots in an attempt to defend their bigotry while at the same time appearing rational. Nobody’s brain has ever fallen out from being open-minded, either literally or metaphorically. However, the rankest nonsense phrases often have a grain of truth of them, and there is a grain of truth even in that one. It is that, in order to achieve anything with our thoughts, we need a framework within which they can operate, and that framework will be made of rules and suppositions that are accepted without evidence. I agree that we need such a framework, but what is crucial is that we acknowledge the existence of the framework, that it has no supporting evidence, and that we hence have no basis on which to claim it is better than any other framework. That doesn’t mean we should refuse to act on conclusions drawn within our framework. But it does mean that (in my opinion, which was derived within my mental framework!) it is a good idea to regularly examine and challenge our framework, and consider alternatives. Sometimes that may lead to a radical change in worldview, which opens up whole new vistas.
It may lead to a Christian becoming a Buddhist, or vice versa. It may lead to a Socialist becoming a Libertarian, or vice versa. It may even (heaven forfend!) lead to personnel exchanges between Platonism and Existentialism. I may have my own preferences about which of those and other sets of ideas most people aligned themselves with but, regardless of the outcome of any migrations of beliefs, I see it as good that people regularly examine their beliefs, so that belief migration becomes a commonplace possibility. If we know what our prejudices are, we have the power to change them. But we cannot change a prejudice we don’t even know we have.
Here are two of my prejudices. The first is that it is preferable for there to be less suffering in the world. I know it’s a prejudice. I know I can’t prove it. But I’m going to hang onto it, for now at least.
The second prejudice is that if I have observed two phenomena to occur in close conjunction many, many times then, in the absence of strong reasons to the contrary, I should expect them to continue to occur in conjunction in future. Every self-supporting person on Earth has this prejudice. But nobody even realised it was a prejudice until David Hume pointed it out in the eighteenth century – his famous ‘Problem of Induction’. If you don’t believe me, think of how you use language. You speak English to somebody – say it’s Bertha, expecting them to understand it, because they have understood English when you spoke it to them in the past. But why should the fact that Bertha has always understood spoken English in the past indicate anything at all about whether she will understand it in the future? You might object that you know that Bertha learnt English as a child, so you know she knows English. But then you are relying on the association between the events ‘X has learned English‘ and ‘X understands English‘, which has been reliably observed in the past, but why should that tell us anything about whether it will be observed in the future? Whatever objection is raised, I (or rather David Hume) can find an answer to it. But I’m still going to hang on to this prejudice.
Prejudice in Music
I had been thinking over this in the context of musical styles. It’s hard to think of any other human activity, the study of whose history is so riddled with the use of the word ‘shocking‘. The most casual observer probably knows about how Rap was considered shocking when it emerged in the eighties, ditto Punk in the seventies, how Rock n Roll was considered shocking when it emerged in the fifties, and how Jazz was considered shocking in the early twentieth century.
But the history of people being shocked by music goes back much farther than that. The history of classical music in particular is regularly punctuated by shocks when some innovator broke hallowed rules. Working back in time we have Schoenberg, Stravinsky, Debussy, Wagner, Beethoven, Haydn and Monteverdi as major disruptors of established musical conventions.
The following story from a radio music presenter made a big impression on me. They told of how they had been working in the archives of a classical music operation, listening to, classifying and cataloguing recordings. After having been doing this for a few weeks they walked past a studio in where music was playing over the loudspeaker. Appalled at the terrible, disorganised racket they were hearing, they asked somebody what the noise was. It was JS Bach! [For the non-classical music buff, JS Bach was a genius who lived from 1685 to 1750, in the ‘Baroque’ period, and is as revered a part of the musical establishment as it is possible to be] The reason it sounded so terrible and formless was that the music the presenter had been listening to non-stop for the last few weeks was all pre-Baroque, and hence operated within a framework of rules and norms that Bach’s music ‘broke’. If they had heard it a few weeks earlier they would have likely thought ‘how lovely!‘ or maybe even ‘that’s a bit old-fashioned!‘
I want to pick on Schoenberg, because on the face of it he might seem to go as far as one can go in breaking rules. The Austrian composer Arnold Schoenberg rebelled against the tyranny of tunes having to be in a musical key, like G major or A minor. Although only classical pieces tend to state their key, with names like ‘String Quartet in E flat Major‘, nearly all pieces have one. Perhaps the most famous song of all, Lennon and McCartney’s ‘Yesterday‘, could have been called ‘Sad song in F major‘. Key changes do occur within a piece, but they have a big effect, because we become attached to the key in which the tune is set. That’s why key changes are often used towards the end of a song to build up the levels of excitement and energy towards a final climax.
Schoenberg’s project was to refuse to use any key at all, not even for one phrase at a time. To do that he invented his ‘Twelve-tone system’ in which he set a rule that every one of the twelve notes in a chromatic scale must be used exactly once within each short section of the piece (called a ‘tone row’). By giving every one of the twelve possible notes equal status, he prevented any note gaining prominence as the ‘Tonic’, the home note of a key. Unlike another famous Austrian, Schoenberg was very anti-racist: he wanted the black piano keys to get as much opportunity as the white piano keys in his pieces (note that’s a different use of the word ‘key’).
Here, from YouTube, is a Schoenberg piano piece using his twelve-tone system, for you to enjoy.
As I was musing over whether Schoenberg had achieved the ultimate in open-mindedness, I suddenly realised a hidden prejudice. Sure, he had proclaimed equality between all twelve notes. But a note is defined by a frequency – vibrations per second. So the number of possible notes is infinite, not only twelve per octave. Between any two different notes there are an infinite number of frequencies between them. In Western music, which is descended from Ancient Greek music, the smallest interval between two notes is a semitone, which means the ratio of the two frequencies is 21/12 or about 1.06.
In classical Indian music, the twelve Western notes are used, plus ten others, called ‘nadas’, giving twenty-two altogether, so that the average gap between adjacent permissible notes is just over a quarter-tone. A piece containing nadas sounds to a Western ear, which is not trained to understand those extra notes, like it is being performed on an out-of-tune instrument.
Here is a scale that goes up an octave in twenty-four quarter-tone steps (not exactly the same as an Indian scale, but closer to that than to a Western scale), then walks back down again. What does it sound like to you?
For comparison, here is the same thing using just the twelve notes in the Western scale.
Try to sing or hum along to first the Chromatic scale, and then the quarter tone scale. I’m a reasonably accurate singer and can do the first, but can’t even get started on the second.
But even writing music that includes nadas or quarter tones still involves a prejudice against the in-between notes. It’s just a smaller prejudice than Westerners like me have. I expect a piece involving eighth-tone intervals would sound just as weird to an Indian as one using quarter-tones does to us.
If we want to write music that is free from all prejudice, we need to go beyond Schoenberg, beyond Indian music, beyond even eighth tones, and write music in which each note can be any frequency at all, without limiting the choice to notes that are certain multiples and ratios of others.
I wrote a piece of such music. To be precise, I programmed a computer to randomly generate a series of frequencies and note-lengths and produce notes using those. I then produced another version of it, in which each note was rounded to the nearest semitone, so that only the twelve notes in the Western scale were used.
Can you tell which is which? They are both weird. Both break most of the rules we are used to. But one is a bit weirder, a bit more free, than the other.
Click here to find out which one is which.
The above is a long way from JS Bach, but is it free from any form of musical prejudice (aka structure)? No. For a start I have constrained the notes to be within the audible frequency range, even though it is entirely conceivable that notes that we cannot consciously hear may still have an effect on our body and thereby alter the sensory experience. I have also constrained the notes to not be very short or very long, in order not to frighten or bore the listener. The volume is also constant, rather than varying between notes, or even within notes. The shape of each sound wave is a perfect sine curve, whereas the wave shape could be allowed to change between and within notes too. That would not change the ‘tune’ but it would change the texture (‘timbre’ in musician-speak). I expect there are other prejudices in there that I have not yet realised.
I like most of my prejudices. I prefer Bach to Schoenberg most days of the week. But it’s good to challenge oneself every now and again with a bit of Schoenberg (or its equivalent), and then to occasionally challenge the Schoenberg with something even more radical.
Bondi Junction, July 2017
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