Splendid!

Have you noticed the strange feature of the English language that most of our words that mean ‘really good’ imply disbelief or some other concept that we may not wish to imply?

List of words for ‘really good’ that imply disbelief:

  • unbelievable, incredible, fabulous, fantastic.

For the first two we can easily see the disbelief. For the next two we need to dig a little into etymology. Fabulous relates to fables – things that did not happen. Fantastic relates to fantasy – things that we only imagine.

So let’s not say that somebody has done a fabulous job, since that may imply we don’t believe they really have done a good job. Perhaps it’s not as good as it seems, or maybe they are taking credit for somebody else’s work.

Better to just say they have done a really good job.

Other alternatives imply surprise:

  • marvellous, wonderful, amazing, astonishing, stunning, gobsmacking

Surprise suggests less scepticism than does disbelief, but the scepticism is hinted nonetheless. Even if not scepticism, it could be taken to imply that the person usually does a less good job. Ewan, you got it right for once! Marvellous! You usually make such a hash of things.

Again, I’d stick to ‘Good job, Ewan!

Other praise words serve as superlatives or comparisons:

  • excellent, outstanding, exceptional, superb, superlative, remarkable, unparalleled, unsurpassed, first-class, first-rate

Even if these don’t imply that Ewan usually fails, they could imply that most of his friends do, ie Ewan has excelled relative to his classmates, if not relative to his usual low performance level. Comparisons between people seem unkind to me. To describe Ewan or his work as ‘outstanding’ hints to his sister Eithne that hers is not.

When one does excellently, one must, by definition, have excelled others, from which it follows that those others have achieved less than best practice. Let us call such unfortunates the excellees, contrasting them with Ewan who in this case we can call the exceller.

If we don’t identify, implicitly or explicitly, specific excellees (Hah! Losers!) we identify the human race in general as a bunch of excellees. With so many beople being excelled (surpassed, outperformed, beaten), perhaps they needn’t feel bad about it. But doesn’t it generate a pessimistic feeling about people and about life? The same concern applies to the disbelieving words. If I describe a donation somebody made to a humanitarian charity as ‘incredible’, am I taking a bleak view of human nature – saying that most humans are mean? Some might say that’s just being realistic, and we should not kid ourselves. Perhaps.

Some praise words just mean big:

  • colossal, huge (colloquial)

No harm I suppose. But they only work for a small class of types of work. I wouldn’t say that somebody’s really good, intricate needlepoint work was colossal.

‘Terrific’ sounds nice. Until we look up the etymology and see that it comes from the latin word for terror and means ‘frightful’.

No. None of the above say what’s needed. That’s why, next time Ewan does a great job, I’m going to say ‘Ewan, that’s really splendid!

‘Splendid’, and its posher fellow travellers ‘splendorous’ and ‘resplendent’, means ‘looks really nice’. The website etymonline.com says:

Splendid: 1620s, “marked by grandeur,” probably a shortening of earlier splendidious (early 15c.), from Latin splendidus “bright, shining, glittering; sumptuous, gorgeous, grand; illustrious, distinguished, noble; showy, fine, specious,” from splendere “be bright, shine, gleam, glisten,” from PIE *splnd- “to be manifest” (source also of Lithuanian splendžiu “I shine,” Middle Irish lainn “bright”). An earlier form was splendent (late 15c.). From 1640s as “brilliant, dazzling;” 1640s as “conspicuous, illustrious; very fine, excellent.” Ironic use (as in splendid isolation, 1843) is attested from 17c.

Other good ones that have a similar feel to ‘splendid’:

  • exquisite, admirable, exemplary, sterling, magnificent, sublime, gorgeous, brilliant, inspirational, elegant

I still wonder slightly about ‘exemplary’. Sounds a tad comparative. ‘This is what you SHOULD be doing, instead of the dunder-headed, pointless way you’re going about things at present.’ But let’s leave it there for now, if only to provide variety.

Splendid has such a nice sound. It brings to mind a jolly hockey teacher at an English boarding school, with an unshakably positive mindset that she is doing her best to communicate  her students.

So let’s not use ‘incredible’ or ‘unbelievable’ to describe acts of great kindness or courage, and especially not simple demonstrations of competence. Let’s not imply that humans, or specific individuals, are innately callous, cowardly or incompetent. Let’s acknowledge splendour wherever we see it. That would be splendid.

Andrew Kirk

Bondi Junction, November 2019

I hope the woman whose photo I used at the top of this does not mind – it was just sitting there on the open internet. I just felt the photo captured so well the concept of pleasure at a job well done. Congratulations to that woman on the splendid work that led to her degree!


Dogma, in religions and other places

Most people are familiar with the dogmas promoted by powerful religious institutions such as the Roman Catholic church, evangelical protestant churches and some branches of Islam. The institutions claim they have sole possession of the truth, direct from God, and that anybody that does not agree is a heretic, someone to be avoided, and who may be punished.

Dogmatism is annoying, anti-social and causes a great deal of misery, both for people growing up under the power of the institution proclaiming the dogma and for some of those that interact with them.

It’s also pretty well recognised. One need only mention religious dogma and heads start to nod. People know what you’re talking about.

Despite the negative connotations the word has for most people, the leadership of the RC church does not object to the term and still uses it as a core part of its teachings. They invented the term, and use it without shame to describe propositions that the church says RCs are obliged to believe. When I was an RC I never thought to ask what happens if one does not believe a dogma. It seemed too impertinent. But now when I research it, the answer that appears fairly consistently across different RC sources is that it is not a sin to disbelieve the dogma, as long as you don’t say so aloud, because that might encourage somebody else to disbelieve it. That would be heresy, which is a grave sin, punishable by an eternity in hellfire. A few centuries ago, the punishment was lighter – a mere burning at the stake.

Although the RC church invented the word ‘dogma’, it is not the only institution to proclaim dogmas. There are plenty of dogmas in evangelical protestantism, and some variants of Islam are heavily dogmatic. Perhaps non-RCs would reject the application of the word ‘dogma’ to their essential beliefs, given the pejorative sense in which the word is mostly used these days. But it would be hard to argue that concepts such as ‘biblical inerrancy’ or ‘justification by faith alone’ are not dogmas for some protestant sects.
It would be a mistake to equate dogma with religion, because most religions are not dogmatic. It is just our misfortune that the three most dominant religions of our world: Roman Catholicism, Evangelical Protestantism and Islam have many adherents that assert an obligation to believe the relevant dogmas.

I am not aware of any pre-Christian religion that had obligatory beliefs. Judaism had many rules, but they were about practices, not beliefs. Even for worship, the injunction was to not worship other gods, or idols in particular. As long as you didn’t bow down or offer sacrifices to golden calves or statues of Ba’al, it didn’t matter whether, in the privacy of your own thoughts, you really believed Yahweh was the greatest god. In fact the Torah says nothing at all about obligatory beliefs, so far as I recall. Other pre-Christian religions, like Buddhism, the many variants of Hinduism, Mithraism, Zoroastrianism and the ancient Greek, Roman and Egyptian religions also appear to set no expectations about their members’ beliefs.

Dogmas appear in places other than religions. Just as some protestants, while abjuring RC dogmas like the Immaculate Conception or Trans-substantiation, insist on their own dogmas, people who are opposed to all religions – the so-called New Atheists – can be as dogmatic as those they criticise. Classic New Atheist dogmas are things like ‘it is wrong to believe anything that cannot be proven to be true’, or ‘for all questions and human challenges, science is the best means to an answer’. For some militant atheists it even seems to be an item of faith that adherence to any religious belief at all must be a sign of stupidity. I know these dogmas because for a while I was a born-again atheist and subscribed to them. I used to listen to podcasts of debates between Christians and atheists about whether God exists, cheering on my side and hoping for the unconditional surrender of the other. Looking back, it seems such an odd thing to do. Neither the debaters nor their supporters in the audience ever changed their views one iota. Each side had their dogmas and stuck steadfastly to them. They may as well have both been shouting into the wind. But really I suppose they were just playing to their supporters. I believe such debates can never get anywhere because it is impossible to prove or disprove the existence of a god, and any attempt to do either relies on presuppositions – usually unstated –  that one side will accept and the other will not.

I have not completely forsaken atheism. I am still atheist on Mondays and alternate Wednesdays. But I have forsaken the dogmatism that accompanies the more aggressive variants of atheism.

Dogmas manifest in wider circles than the theological and anti-theological. Other areas where they crop up are philosophy, politics, economics, psychology and sociology. People debate whether there is such a thing as objective morality, whether equality is more important than liberty, whether wealth really does ‘trickle down’ in a capitalist society, and whether most psychological disorders can be traced back to early childhood experience. Debates between evangelical christians and militant atheists seem mild and friendly compared to the vicious passions unleashed in a debate between a Berkeleyan Idealist and a Materialist acolyte of GE Moore about whether a tree that falls in a forest makes a noise if there is nobody there to hear it.

I’m not suggesting that none of those things matter. It matters very much what political and economic theories are adopted by governments. They affect many people’s lives. Even some sorts of philosophy have huge effects. One can trace the roots of many important social movements to the ideas raised by philosophers, such as the influence of Enlightenment philosophers on the American and French revolutions. It’s hard to see how the ‘actual existence’ or otherwise of impossibly distant galaxies could affect our lives, but other similarly meaningless topics, such as whether the Holy Ghost proceeds from the Father and the Son, or just from the Father, have led to wars, the rise and fall of empires and many burnings of people that had the misfortune of siding with the wrong opinion.

The common element of dogmatic claims is not their capacity or otherwise to affect our lives, it is their total immunity to proof, disproof, or experimental testing of any kind.

There is no dogma about the law of gravity, no dogma of quantum mechanics or a doctrine of the periodic table. A good biology teacher will not demand that her class believe that cells of mammals have a nucleus containing bundles of DNA and little packets of RNA. A good mathematics teacher will not demand that the class believe that the method being taught for long division works. The teacher is saying: “Here is a method, or an approach to understanding something. Most people find it useful in getting important things done“. The teacher could add – but generally doesn’t bother – “If you don’t like what I’m teaching and want to go and invent your own method of long division (or theory of the elements), be my guest! I’ll still be here to help you learn this method if you change your mind.

It is both ironic and predictable that the claims about which we humans get most dogmatic are those about which it is least possible to be certain. When there is a high level of certainty – as with Newton’s Laws of Motion – there is no need for dogmatism. You can take it or leave it. More fool you if you leave it. But when there is little to no certainty available, as with doctrines of neo-liberal economics (or, to be fair, Marxist economics), doctrines of the nature of the Holy Ghost, or proofs and disproofs of the existence of god(s), people generally ramp up the dogmatism and turn the volume to eleven. They use dogma and noise to make up for their lack of confidence and inability to provide any concrete evidence for the proposition.

This has led to my strongest philosophical position being anti-dogmatism. No matter what proposition somebody makes, be it about religion, ontology, economics or politics, and regardless of whether I sympathise with the belief being promoted or not, I now instinctively react against it and look to debunk it, if it is made dogmatically. That doesn’t mean I don’t hold any opinions on those topics. I have loads. Some of them – mostly the political ones – I hold very strongly and am prepared to march the streets, donate to a cause and publicly argue to try to persuade people over. But I hope I never get to the stage of believing that I am unquestionably right about something and that those who disagree are unquestionably wrong. That seems a poor way to live. I have sometimes been like that in the past, but I think I am not now and hope I won’t be again. For me, unquestioningly accepting a dogma is the coward’s excuse for not thinking for oneself.

That is my opinion, which I acknowledge may be mistaken.

Andrew Kirk

Bondi Junction, April 2019


Obviously, …

When it comes my turn to be king of the world I will ban the word ‘obviously’, together with its fellow travellers ‘clearly’ and ‘evidently’. My challenge to you, the other inhabitants of the kingdom of Earth, is this: find me a single example of a sentence that is improved by the use of the word ‘obviously’!

I assert that, not only is ‘obviously’ never an improvement to a sentence, but it usually degrades a sentence into which it is inserted and renders it foolish, pompous, or just plain false.

The first memory I have of encountering this rebarbative word is in mathematics lectures at university. It was the early 1980s. In those days lectures performed their proofs live on the black board with chalk – a difficult endeavour indeed. As soon as you saw that word on a board, you felt that if you couldn’t instantly see why that line followed logically from the line before, you must be very dim. If you hadn’t seen the connection by the time they finished writing the next line, you started to panic. The only solution was to accept the claim without challenge and try to keep up with what came next. There would be time that evening to go over your notes and try to work out why the claim was ‘obviously’ true.

Sometimes in the evening you could figure it out without difficulty. Sometimes you figured it out but it needed a page or so of closely written reasoning to justify it. Sometimes you couldn’t make it out at all. That’s when you had to summon your courage and challenge the lecturer about it before the next lecture. You’d sidle up to him and say ‘Sorry to bother you but I can’t see how you get line five. Can you please explain it?

In short, it was rarely obvious. Even when it was moderately obvious, there were other lines that were more obvious, for which the tag was not used.

I started to detect a pattern. The word was being used to cover for the fact that the lecturer couldn’t remember, off the top of their head, the justification for the line. By writing ‘obviously’ they made potential hecklers too worried about seeming dumb to challenge the claim on the spot. It was the Emperor’s New Clothes all over again. What was needed was the little boy to blurt out ‘But it’s not obvious at all. In fact I can’t even see it.

I forgive those lecturers, because what they were doing was very difficult. I would feel under a lot of pressure having to perform mathematical derivations on a blackboard in front of a specialist audience.

It is less forgivable when it occurs in text books. In many a mathematics or physics text book I have come across the prefix ‘obviously’ before a line that was the exact opposite. The authors of textbooks do not have the excuse that they have to come up with explanations on the spot, but they are nevertheless under time pressure because, unless a text is chosen as a key text for courses at many major schools or universities, it will not bring in much revenue, so extra time spent writing it makes it even harder to be profitable. Why spend hours deriving a proof of something you are fairly sure is true, but don’t remember why, when you can just write ‘obviously’ in half a second, and move on to the next line?

I don’t begrudge them saving that time, but there are more honest and helpful ways to do it. Other phrases that can be used are “It turns out that…” and “It can be shown that…”. These make it clear that what the author has written is not a full proof, and that the step over which they are glossing is not trivial. When I encounter those I don’t mind very much because they don’t contain the implicit challenge “If you can’t see why this line follows from the last one you must be stupid!”. The most generous excuse of all is “It is beyond the scope of this paper / text / chapter to prove X, so we will take it as read”. That way the reader knows that proof is long and difficult.

It is annoying when academics use the word ‘obviously’ in that way, but at least they use it in relation to a claim that is true. In political argument, that is not the case. People use ‘obviously’ to justify any claim, no matter how dubious, or sometimes just plain wrong. Examples abound, from politicians, shock jocks and reactionary newspaper columnists.

Obviously, decriminalising marijuana use would make the problem worse

Obviously, it makes no difference whether Australia reduces its greenhouse gas emissions, since ours only make up a small part of the world’s total

Obviously, what’s needed to solve our city’s traffic problems is to build bigger roads

Obviously, we have to be cruel to refugees, otherwise many more would come to our country”.

It’s used as an excuse to not even consider any evidence that may be available, to not even entertain rational discussion on a topic. It implies that anybody that does not accept the claim must be stupid or have dishonest intentions. It’s an attempt to shut down inquiry and discussion, lest that lead to an outcome against which the speaker has an entrenched prejudice.

Is anything ever obvious?

Perhaps, but we need to very careful in suggesting that. What is obvious to one may not be at all obvious to another. A high-visibility yellow vest is obvious to normal-sighted people but not to the colour-blind. A person walking across a basketball court in a gorilla suit is not obvious to observers that have been tasked with counting the number of times each player passes the ball.

Further, beliefs in what is obvious are often founded on stereotypes that may be damaging. Is it obvious that boys are better at maths than girls, or that men cannot be trusted to care for other people’s children?

This leads me to wondering whether there is any sentence in which the word ‘obviously’ can play a useful role. I don’t apply the same challenge to ‘obvious’ because it can have observer-dependent roles, as in “It eventually became obvious to Shona that the doorman was not going to let her into the club”. Or we can use it to express relative obviousness, as in “Not wanting to mislay them, he left his keys in the most obvious position he could think of – in the middle of the empty kitchen bench”.

But “obviously”? That adverbial suffix ‘ly’ seems to strip from the adjective any ability to convey subtleties of degree. There seems to be no way of using it that does not imply that anybody who does not agree with the following proposition, and understand why it must be correct, is simply stupid.

No wonder it is used either as a tool of bullying or as a lazy attempt to escape the need to justify one’s claims.

Sometimes it occurs without intent, as a verbal tic. Like most verbal tics, it is rooted in the insecurity of the speaker. Although it sounds like it has an opposite meaning to other tics like ‘if that makes sense’ or ‘if you like’, it serves the same purpose in deflecting attention from the speaker’s insecurity – but in an offensive rather than a defensive way. In both cases the speaker hopes not to be challenged. With ‘if that makes sense’ the hope is that the humility it projects will discourage a listener from saying ‘that doesn’t sound right’, if only out of charity to the speaker. The ‘obviously’ is like the puffed-out frill of a lizard – a pretence at invulnerability intended to discourage attack: ‘Challenge me on this and you’ll end up looking foolish!’. Except that the intent is usually subconscious and, once one has used the phrase many times, it becomes reflexive, devoid of any meaning, or even of subconscious intent.

I vowed quite some time ago never to use the word, or any of its synonyms. I think I have managed to keep the vow. I hope I have. But I cannot be sure. One uses so many words in the course of a week, that it’s hard to keep track of them all.

If something is truly obvious to almost everybody, there should be no need to state that. It will be obvious that it is obvious. If, as is more often the case, it is far from obvious, it is foolish at best, and dishonest at worst, to imply that it is.

Andrew Kirk

Bondi Junction, April 2019


More fascinating facts about stars

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:

star 7-3 with arrows

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.

11-5 with labels

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.16-6 [1,0,0] star

And here is a picture that uses colour variation to show the sub-stars of each of the two components.

16-6 [1,1,0] star

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:

16-6 [0,1,1] star

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:

101-50 (cc_0,cs_1,st_0,bn_0,bg_FALSE) 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?

zoom of big star

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:radius ratios table

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:

  1. the size of n: the ratio generally declines as n increases; and
  2. 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:

  1. Draw two concentric circles with ratio of the inner to the outer radius being θ.
  2. 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.
  3. 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.
  4. 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:

row of 6-0.2 stars

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.

Andrew Kirk

Bondi Junction, April 2017


Mastering maths monsters

‘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:

https://www.physicsforums.com/insights/partial-differentiation-without-tears/

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.

Andrew Kirk

Bondi Junction, October 2016