A word you might say to someone setting out on a journey and I've always imagined it expressed hope for a swift journey. But TIL it has a deeper meaning, because speed is from English sped or speod, which meant 'success, good luck'. So it was wishing that God would grant them a successful journey, rather than a rapid one.
I think we also see this meaning of speed reflected in the traditional phrase Speed the plough or Go. speed the plough, an agrarian blessing, as Wikipedia puts it.
Ultimately speed derives from a Proto-Indo-European root *spe- and this is also the origin of Latin sperare, 'to hope', so speed is related to despair, desperado, desperate, Esperanto, sperate, a rare and archaic legal term referring to a debt which has a reason hope of being repaid — and possibly also prosper, although this is debated.
Pick A Sequence Of Digits. Is There Always A Perfect Square Beginning With That Sequence?
Friday 25 September 2026 at 16:21
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Edited by Richard Walker, Friday 25 September 2026 at 18:34
This question occurred to me and I thought it ought to be true; it holds if we replace 'perfect square' with 'prime number', for example. Of course it is not new and many others have asked it, see [1] for a example proof.
Looking for an informal approach, I enlisted the help of Gemini. Here's, not a proof, but a process, that I think always finds such a square.
As an example here is the number of the Golden Section, , to digits. We can ignore the decimal point because it doesn't make a material difference.
Take the square root
Take the first figures and ignore the decimal point, so we have the integer and also take the integer that is one larger, . Square both these and we should find one of the squares begins with the required digits.
This procedure should let you find a number whose square commences with your birth year, all you need is the calculator on your phone.
Here is a Giant Redwood (AKA as a Giant Sequoia) in a village near where. I often pass it and always stop to admire it. I couldn't get a good angle to take the photo so a bit of another tree is visible at upper left but I hope you can see the Redwood is very impressive.
In their native range, the Sierra Nevada in California, the species is threatened: a combination of historical logging, suboptimal fire policies that led to build-up of combustible material causing the fires that finally occurred to be far more damaging; drought which made trees susceptible to bark beetles; and climate change has reduced the number of trees to around 80k.
However in Victorian times seeds were brought to Europe and in the UK particularly owners of large gardens grew them enthusiastically. And the trees liked it here and have thrived, so now there are about 5k notable specimens and about 500k younger redwood trees. A small minority are Coast Redwoods and Dawn Redwood but the bulk are Giant Redwoods.
Of course most of these are young trees, as Redwoods go. The oldest is a mere 150 years old, compared with the 3,000 years some wild trees have reached. They grow fast here though; the tallest documented is 55 m, compared with around 95 m for wild trees, although the oldest wild specimens have a relatively greater volume.
But paradoxically Giant Redwoods may be commoner in the UK than they are in California (it's not an open and shut case though, because the 80k in California are large trees and we don't have a count of how many smaller trees have survived recent devastating fires and beetle attack). It's good that these magnificent trees have been able to grow so successful on another continent from the one where they evolved and let's hope some of the ones growing here today — perhaps the one picured — will live for 3,000 years.
A swainmote seems in Medieval Britain to have been a sort of tri-annual assembly of the free men living within the bounds of a Royal Forest. I'm not sure how much solid information we have about these assemblies, but the etymology is interesting.
The first element, swain-, is now a rare word meaning a young lover, e.g.
'Who is Silvia? What is she, That all our swains commend her?'
Shakespeare, The Two Gentlemen of Verona
but an earlier meaning was 'country worker', 'young lad', 'servant', 'assistant'. The assistant meaning is seen in e.g. boatswain. Its ultimate origin is an ancient root PIE meaning 'oneself', which developed into the sense of something like 'my assistant'. It has many cognates and is seen in the given name Sven, a fairly common Scandinavian name, and seen it's believed in Swansea, 'Sven's Island' in old Norse, suggesting the settlement was founded by a Viking.
The second element -mote is seen in moothall, a place for community discussions, and a moot point, something debatable. It has cognates in pretty well all Germanic languages.
Swainmote is very rare in modern written English, at 0.01 occurrences per million words. So just by writing this blog post I've used it the number of times we'd expect in three hundred million words.
Can You Find The Ratio Of The Areas? Euclid Would Have *Loved* This Proof
Monday 21 September 2026 at 22:31
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Edited by Richard Walker, Monday 21 September 2026 at 23:06
I found ths on 'Mind Your Decisions' who got it from 'CueMath' and I don't know where 'CueMath' got it. I hadn't seen it before but as is often the case I'd just been missing out, because a search shows it's in lots of places on the web. Anyway, here is my solution, I haven't looked at other answers because I wanted to solve it the way Euclid might have. It's a really nice problem.
We are given a square in which is inscribed a quarter-circle and inside that a semi-crcle tangent to the quarter-circle, as shown. Length PQ = 4. What is the ratio of the portion shaded light purple (the semi-circle) to the area shaded green (the area of the quarter-circle not also in the semi-circle)?
Scroll down for solution
Keep going
The Line of Centres Theorem If two circles are tangent to one another, the line passing through their centres passes the the point at which they are tangent. [1]
So we can draw in line PQR passing through these points.
Thales's theorem If the hypotenuse of a right-angles triangle is the diameter of a circle,
the vertex containing the right angle lies on the circle's circumference. [2]
Since ST is a diameter of the semi-circle and P a right angle, P must lie on the circle through R, S and T. Therefore PQ is a radius of that circle and equal in length to QR which is also a radius. This makes PR, a radius of the quarter-circle, equal to 2QR, a radius of the semi-circle.
So radius of quarter-circle : radius of semi-circle = 2:1
Area of of quarter-circle : area of semi-circle = : = 1 : = 2 : 1.
So the quarter-circle as a whole is exactly twice the semi-circle, making the light purple and green areas the same. The length 4 is irrelevant, a pure red herring!
A Malaphor is when two common idioms or cliches get mixed up, producing a nonsensical and often amusing hybrid. The word is generally ascribed to Lawrence Harrison who wrote a piece 'Searching for Malaphors' in the Washington Post August 6 1976. Malaphors often have a kind of weird surreal sense of their own
A couple of classic examples will give the flavour.
Edited by Richard Walker, Friday 18 September 2026 at 01:40
I've had these olive trees exactly two years and this year they have fruited quite well, as you can see. When I first bought them, I posted here about them and about what may be the world's oldest olive tree.
An Evil Prime — Belphegor's Prime Contains 666 The Number Of The Beast
Wednesday 16 September 2026 at 20:29
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Edited by Richard Walker, Wednesday 16 September 2026 at 23:07
You may not have met this remarkable palindromic number before.
is a prime. Because it contains , the Number of the Beast, this prime is nicknamed Belphegor's prime, after one of the Seven Princes of Hell. Here he is
This prime was discovered by Harvey Dubner, an expert in finding large unusual primes, and is one of a sequence which you can find on the Online Encyclopedia of Integer Sequences (OEIS) here. The next is
The OEIS seems to suggest this is only a 'probable' prime. A probable prime is a number that has passed a long series of tests from which we can conclude the probability it is not prime is very small. We also have exact tests but they are much slower, because they involve much longer calculations. In this case I found that according to SageMathCell, which I think uses an exact test, this is a definite prime, not just a probable one.
The next number after that would be a probable prime with more than ten times as many digits, so I won't try to display it here or prove it is a definite prime.
How did Dubner discover Belphegor's prime? Purely guessing but maybe he spotted the is prime and thought that was neat, so he then tried inserting more zeros and putting the numbers so formed to a probabilistic primality test until he struck gold with thirteen zeros each side of the evil .
I haven't really investigated but I thought interesting palindromic primes might not be all that rare. If there are seven demons in hell, what if we change to in recognition of the fact. Is
prime? Can we prove it? YES WE CAN! How spooky is that?
This Bird Was Extinct In Britain For 400 Years. Now It's Breeding Here Again.
Tuesday 15 September 2026 at 22:14
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My brother photographed one last week It's a crane, Grus grus.
Not a great photo but good enough to see clearly that it's a crane. It's thought these imposing birds were once common in Britain but they were hunted for their feathers and to put on the tables of the upper classes. According to Gurney (quoted in [1]) in 1251 Henry III held a Christmas feast at York at which 115 cranes were eaten (amongst other things presumably).
So from overhunting they became rarer and rarer and 1542 was the last time the birds were reconded as breeding in England.
Until 1979, when a couple of young cranes flew in. There had been occasion migrants in the intervening period but no breeding pairs — but these two did.
So some birds flew here of their own accords and there has now been some reintroduction, and no there are an estimated 250 birds in the UK. In 2025 there were 87 pairs documented and 37 chicks were raised.
So a comeback, but it's still a very rare bird in the UK and my brother was lucky to spot this one.
[1] Andrew Stanbury and the UK Crane Working Group, The changing status of the Common Crane in the UK.
From "Cut The Knot" — A Nice Use of the Pigeonhole Principle
Monday 14 September 2026 at 17:23
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Edited by Richard Walker, Monday 14 September 2026 at 18:27
The Pigeonhole Principle says, of course, that if there are more pigeons than pigeonholes then we must be able to find a pigeonhole with more than one occupant.
The Cut The Problem ask for a proof (using the Pigeonhole Principle) that there is a power of whose last three digits are , which sounds quite surprising.
But let's imagine we have pigeonholes numbered . Calculate distinct power of , divide each by and find the remainder, then put that power in the pigeonhole with the same number as the remainder.
There is one more 'pigeons' than pigeonholes, so there must be a pigeonhole with two occupants, that it, two powers that leave the same remainder on division by .
Suppose these are and , being the smaller. Because they leave the same remainder when divided by , must be a multiple of .
can't divide a power of , so it must divide . This means when worked out ends in three zeros , which in turn means ends in .
We can run a computer search quite easily and we find that in fact
fits the bill.
This result can be generalised of course and we can prove that for example there must a power of that ends in a trillion zeros followed by , although given that even the zeros would take up nearly GB the browser is too small to display it.
'That's Put The Kibosh On It' : What Is The Origin Of This Strange Word?
Saturday 12 September 2026 at 00:58
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Edited by Richard Walker, Saturday 12 September 2026 at 00:59
I expect you know the word kibosh; putting the kibosh on something means it has been brought to a sudden stop or been thwarted etc.
I've never thought about its origins until today but tacitly assumed it was from Yiddish — it looks a bit like kibitzer, an onlooker who proffers unwanted advice, which the OED says cautiously is 'perhaps modelled on a Yiddish lexical item'.
The trouble is, it turns out, no-one has found a Yiddish word that kibosh could be derived from, making the Yiddish origin theory unlikely. But plenty of other ideas have been proposed.
From the OED
From Arabic kirbāš, a judicial whip.
From Irish Gaelic aidhp bháis, 'death cap', which might have to do with death sentences or executions or a form of torture or the last garment put on a body before interment... but OED says there is no direct Irish evidence to support this.
Wordorigins.org mentions (but doesn't seem convinced by) another possibility
It is from Hebrew chi + bosh, 'eighteen + pence'. But why on earth would 'eighteen pence' come to men putting a stop to something?
Michael Quinion
In his fascinating book Posh Out, Starboard Home (Spoiler: this is not where POSH) Quinion muses that it could be linked with bosh, 'rubbish'. But the change in meaning seems inexplicable.
OUPBlog
In a 2013 post 'Three recent theories of “kibosh”' we find further candidates.
It is from a French word meaning to behead a stag, which was borrowed into English as cabosh.
It is from a iron tool once used by clogmakers to soften leather.
It was a terms used by builders and sculptors etc. for Portland cement.
Wiktionary
In a 1904 quotation 'He was proud to be the Husband of the Lady Ki-Bosh of the Local Knickerbockers' it seems to mean in the (possibly excessive) height of fashion. But that doesn't cast light on the possible orgin.
I guess this list is not exhaustive but what stands out is we don't really know where the word came from. It might be from none of them or it might have come from two or more being confused and ending up merged into one. Or someone might just have coined it with humorous intentions and it caught on. This happened with e.g. discombobulate.
Edited by Richard Walker, Thursday 10 September 2026 at 23:45
Can there be four square numbers in arithmetic progression? Can four numbers that are perfect powers be in arithmetic progression?
The simple answers are no and yes!
First I'll sketch the history of the first question, which developed from a problem studies by Islamic mathematicians more than a thousand years ago.
Then I'll outline how we know the answer to the second question and give an a examplle.
A little background: An arithmetic progression (AP) is when a series of numbers go up (or down) in equal steps, and this size of this step is called the common difference. For example 1, 4, 7, 10 are four numbers in AP, with common difference 3.
The congruum problem asks for triplets of square numbers that are in arithmetic progression, for example 1, 25, 49, whose common difference 24 is called the congruum.
Finding congruums (or congrua)? attracted the interest of medieval and later mathematicians, with several mathematical superstars getting in on the act.
According to Wikipedia the problem was first investigated in the 10th century, by Abu Jafar al-Khazin, who gave example such as the one above and found a general solution.
Things really hotted up in 1225 though, because the Holy Roman Emperor at that time organised a mathematical tournament. Can you imagine such a thing nowadays?
The prize was carried off by one Leonardo of Pisa, better known as Fibonacci, who wrote about the eponymous series 1, 1, 2, 3, 5, 8, ..., and about the Indo-Arabic numerals which replaced Roman numerals and to become the system used today.
Next on the stage is Fermat, lawyer and amateur mathematician who made a claim that got called Fermat's Last Theorem even though it wasn't really a theorem because at the time he never actually proved it (as far as we know). He also made a conjecture about squares in AP — that it is impossible to find a congruum that works for four squares, not just three. Could there be numbers that were in AP?
This challenge was taken up by Euler, another giant famous for many things, including introducing the symbol for the base of natural logarithm. He published a proof but by that time he had low vision and had a helper to tale his dictation, so some writers are not sure the published version is accurate.
Since that time there have been many proofs published but in the literature there are also authors who have commented flaws in earlier papers, and there is active research right up to the present day.
Now for the second problem. Can four numbers that are perfect powers be in arithmetic progression?
Absolutely. Start with if you like. The common difference is , so the next term would be . But, you will say, is not a perfect power. No, but is. So scale everything up by .
Surprisingly we could in principle carry this as far as we like and get a million or a gazillion perfect powers in AP, although as you see the numbers grow dizzyingly fast; we have gone from 24 to 127896 in one step. So I imagine we would soon run out of resources and get to numbers too big to actually write dowm!
I hadn't thought about this until reading something last night with parallel texts in English and German and noticed the German equivalent was KI. Interesting, I thought.
It's short for Künstliche Intelligenz, which is what's called a calque, when individual parts of a compound word are translated literally into the host language.
For example German Regenwald became English rainforest, from Regen, 'rain' and Wald, 'forest'.
Exploring what 'artificial intelligence' translates to in a few other languages, with Gemini doing the heavy lifting, all those I looked into calqued the English phrase, but the abbreviations are overwhelming 'AI', whatever the actual word in a language. For example in Chinese (according to Gemini) artificial intelligence translates as 人工智能 but you could informally write 这个 AI 工具很好用, 'This AI tool is very useful'.
The only exception I found were the Romance languages, which generally put the adjective after the noun, so logically enough we have e.g Spanish inteligencia artificial, IA; Dutch (sometimes) and some Scandinavian languages use KI, like German — but not Swedish where artificial intelligence is artificiell intelligens, AI, more or less the same English).
Edited by Richard Walker, Tuesday 8 September 2026 at 23:02
This definition for the Oxford English Dictionary made me smile, 'View entry' seemed very apt.
The word door goes back to early Proto-Indo-European, perhaps as far back as we can trace that reconstructed language. It hasn't changed much in all that time; the original root would have been something like *dwer- and words derived from it appear in nearly all branches of the PIE family, with senses of 'door' or 'gate' or 'outdoors'.
The Italic branch has given us several familiar words: Latin foris, 'door' gave forum, because forums are outside, and because forums are a good place for discussing legal matters we got forensis, which ends up as English forensic. We also have foreign, someone or something from outside, which reached us via Anglo-Norman.
I'll finish appropriately with a Deoch An' Doris, Scots Gaelic 'A Drink for the Door', the same idea as 'One for the Road'. Cheers!
Edited by Richard Walker, Monday 7 September 2026 at 22:29
This is a solution to the problem "A Quick Puzzle — Find x" posted 1/9/26.
Comment Trisecting an angle with straightedge and compasses was a problem posed in antiquity and which many mathematicians attempted to solve over the ages, until Wantzel proved it was impossible in 1837.
So have we achieved the impossible? Well, no. The original problem assumed the use of an unmarked straight edge, and our construction requires one we can mark. Or we could use three rods of equal length.
Our construction formed the basis of an angle trisection method given by Étienne Pascal, father of Blaise, he of the triangle which you will recognise
1 1
1 2 1
1 3 3 1
1 4 6 4 1
However what was essentially the same method was already known to Archimedes over a millennium earlier.
There is an old joke allegedly from a real exam script
Edited by Richard Walker, Sunday 6 September 2026 at 15:51
I'm not talking about some kind of Creature from the Swamps but about a recent (2026) discovery that the tiny water droplets in fog are home to populations of bacteria [1].
The bacteria grow and divide in the water droplets, so it looks as though they like it there, and amazingly they can eat formaldehyde and break it down into carbon dioxide and water; in fact it appears they break down more than they can eat and it's thought that this is because too much of it would harm them. Formaldehyde is harmful to human health [2] and the waterborne bacterial may play a useful part in purifying the air.
That's what made me think about formaldehyde as a domestic pollutant. Friends have Dyson air purifiers and removing formaldehyde is an important part of what these devices do.
So maybe the bacteria in water droplets have an important role in cleansing air. And I wonder if they exist in clouds, which are just fog high up. It may be hard to find out; the researchers had difficulty getting the fog samples they wanted because wind disturbs the fog and getting good samples from clouds thousands of feet up where winds are much stronger would be difficult.
I have to thank science YouTuber Anton Petrov for his video which put me on to this research.
In the Phys.Org article I read, one of the researchers was quoted at the end as cracking a joke, so I'll end with (a rather appropriate) joke too.
Why Do Types of Cloud Have 'Double-Barreled' Names Like 'Cirrus uncinus'?
Friday 4 September 2026 at 22:46
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Edited by Richard Walker, Saturday 5 September 2026 at 20:53
My brother snapped these Mares' Tails, high altitude (20,000 feet and counting) ice crystal clouds. Their meteorological name is Cirrus uncinus, which apparently means 'curly hooks'.
It's interesting that different kinds of cloud have double-barrelled 'binomial' names with a genus and a species, a form generally kept for living things. But (thanks Gemini!) I found that viruses have binomial names too, although whether viruses are alive is debatable.
Karl Linnaeus, whose influence is pne of the main reasons binomial names are used in biology. even explored a similar naming scheme for minerals, considered by him a third 'kingdom' (and we still speak of 'Animal, Vegetable and Mineral') but ultimately it was not adopted by the scientific world.
Clouds are an exception amongst non-animate entities: Luke Howard who at the beginning of the 19th century famously pioneered assigning systematic names to cloud forms imitated Linnaeus in using binomial names and his scheme has survived.
A Very Surprising Number Pattern (And A Neat Proof It Goes On For Ever)
Saturday 29 August 2026 at 22:50
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Edited by Richard Walker, Tuesday 1 September 2026 at 00:01
Isn't that neat? It was posted on Quora by Alexander Ferrugia, along with a proof that the pattern goes on forever. His proof was quite long though, and I found one which is a bit shorter.
Something that jumped out at me was the resemblance to recurring decimals. , and and I thought there had to be a connection. So I took the second line (the first doesn't fully show what the pattern is, because 6 doesn't appear) and played with it in the Python Shell, trying to get 16, 50 and 33, the numbers on the left hand side ...
... and now it works. What about the right hand side? If we break it up we see it represents .
To get the equivalent expressions for the -th line we can just replace by everywhere and now we get
, with the left-hand side becoming
We want to prove these two expressions are equal and we can do that very easily. First make them more general by replacing by , giving
and
It is not hard to see that if these were expanded we would obtain cubic polynomials of the form . How can we prove they are the same? We might use paper and pencil to expand them and collect terms in the good old fashioned way. Or we could enter them into reliable computer algebra system and let it do the heavy lifting.
Or much more elegantly we can use this fact
If two cubic polynomials of the form take identical values for four different values of they are equal.
Why is this true? Because four values let us set up four equations and solve them to find the four constants .
But we have the requisite four values! This post began with them and so we have proved the two expressions are equal for any value of and for any value of in particular.
Why Are So Many Berry Names Weird? (And Inexplicable)
Saturday 29 August 2026 at 00:04
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Edited by Richard Walker, Saturday 29 August 2026 at 22:43
Many or most do (have unusual names), you see. A handful are perfectly explicable, such blackberry and blueberry from their colour, and Loganberry because in 1881 James Logan accidentally crossed a blackberry with a raspberry.
But many are inexplicable. What has a gooseberry have to do with geese or a strawberry with straw? A word story commonly advanced for gooseberry is that the first bit is a borrowing from French grosele or Dutch croesel. both ultimately from a Germanic root meaning 'crisp' or 'curly'. But the OED is unconvinced, and why would a gooseberry be described as curly?
As for strawberry, it was already in Old English as streawberigei, but where the straw comes into the mix is uncertain.
But at least these two fruits start with a recognisable element having a meaning in its own right., even if it is hard to link with the fruit concerned. Many names of the form 'something-berry' start with a element that is not a word on its own, isn't found anywhere else in the language and on its own has no meaning for native speakers.
The classic example is cranberry. Whatever is a cran? ¶ What does it mean? Well in this case we think we have an etymology: it was originally a crane-berry, as in the bird, but the 'e' got dropped and the meaning forgotten. (But why it was called a craneberry is still a mystery).
This cran is an example of a unique morpheme. A morpheme is like a sort of atom of language, the smallest unit of language and indivisible, and all words are made up from one or more morphemes. A morpheme may or may not be able to stand on its own. For example bookmark has two morphemes that can be independent words. whereas the un- in unhappy cannot exist alone. But crucially, it carries meaning; the cran in cranberry no longer does and contemporary speakers cannot explain what it means.
And most of the X-berry clan are like this; the X- may have a recognised etymology or it may be uncertain, but either way the X is a unique morpheme, found only in this combination, and whatever meaning it may once have had has evaporated.
Of course there are other examples of unique morphemes that have nothing to do with fruit; often quoted example are luke- in lukewarm, -ept in inept and twi- in twilight. The are often called 'cranberry morphemes' in fact, because that was the example used in early writings about this linguistic oddity. But they are particularly common in the vocabulary of berries.
For more example. think of huckle- (what's a huckle) and mul- and bil- and rasp-, all mysterious. Whortleberry looks at first as though it is not a unique morpheme, because whortle is a distinct word. but alas when we look it up we discover it is simply short for whortleberry, so it hardly counts. And then there is lingon-, borrowed from a Swedish word meaning something like 'heather berry' in Swedish, but now a unique morpheme in English.
Why are these unique morphemes there if they don't mean anything? Well we need them, they have a purpose which is to distinguish between different kinds of berry. So they do not need to carry meaning, they only need to be A. different, and B. easy enough for speakers to remember them.
¶ Scrabble players and word geeks may point out that a cran is a measure of herrings, but that is a different word, not our cran.
Edited by Richard Walker, Wednesday 26 August 2026 at 23:02
Imagine 5 unsociable people live on the surface of a tiny spherical world. Being unsociable they want to arrange themselves so the smallest distance between any two them is as big as possible. How can they do this?
If there were 6 of them they could position themselves at the vertices of a regular octahedron, as in this model [1] of a Sulphur Hexafluoride molecule, where 6 Fluorine atoms surround a single Sulphur atom. Electrostatic repulsion causes the Fluorine atoms to take up this octagonal configuration.
We can measure the distance between two points that lie on a sphere by giving the angle they make at the centre and you can see that in this case all the angles are 90°.
But now suppose one of them jumps in a spaceship and flies off the some other tiny world. Now the remaining 5 have a little more space to spread themselves out more.
Except they don't! 5 points can't do any better than 6 could and the best minimum distance that can be achieved is still 90°. Here's my attempt at a proof.
If there are five points on the surface of a sphere, then no matter how they are arranged at least 4 of them lie in the same hemisphere.
I think this result may have first been published by Martin Gardner, although I don't have the reference.
You can think of this as being like the Northern hemisphere, including the Equator and the North Pole, and 2 or more points must lie on the Equator.
If all 4 points lie on the Equator the greatest possible spacing is 90°, achieved when they are situated at the corners of a square inscribed in the Equator.
Otherwise we must have at least 2 on the Equator and at least 1 in the same hemisphere but not on the Equator. The greatest possible separation between this point and the 2 on the Equator is 90°, achieved when that point lies at the North Pole. So, either way, there must be a pair of points separated by 90° or less.
We'd like the configuration to be as symmetrical as possible, so we can put 3 points at the vertices of an equilateral triangle inscribed in the Equator and the other 2 at the North and South Poles.
At this point you might wonder if there are molecules with 5 Fluorine atoms surrounding a single atom of some other element, and if so are the Fluorines in the arrangement described above. The answers are Yes and Yes. Here's a picture of a Phosphorus Pentafluoride molecule [2]
This problem of optimally spacing points on a sphere, which has always fascinated me, is known as the Tammes problem and the best configuration for 2-14 points and 24 points were found and proved between 1943 and 2015 [3]. It is a hard problem, in that it has to be solved for each number individually. I imagine we can find estimates for the general case but finding exact solutions is challenging.
Edited by Richard Walker, Monday 24 August 2026 at 16:45
Here's a Sun Dog I snapped one evening last week.
Sun Dogs are seen when the sun's light is refracted by flat hexagon ice crystals drifting in the clouds.
In favourable conditions, if the cloud formation is right and no nearby objects get in the way, we may see a pair of Sun Dogs, 22° left and right of the sun, but this is rarer than seeing just the one.
They fall into the class of phenomena you don't notice until something or someone makes you aware and then you realise they are quite common and see them frequently. Sun Dogs are more often visible than rainbows and I spot them often, usually looking West as the sun is setting in a bank of thin high cirrus clouds. You can see them to the East at sunrise too, of course, it's just that I see sunset more often.
These Sun Dogs are just a small part of a whole complex of arcs and circles that have sometimes been observed, although unless we are in the mountains or in Antarctica we are unlikely to witness them. My sketch below only includes the best known halo-related phenomena; at least 119 have been documented.
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Many or most do (have unusual names), you see. A handful are perfectly explicable, such blackberry and blueberry from their colour, and Loganberry because in 1881 James Logan accidentally crossed a blackberry with a raspberry.
