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.
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!
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.
Bondi Junction, April 2019
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
‘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