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Richard Walker

Squares Covering Circles

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Edited by Richard Walker, Thursday, 10 July 2025, 23:35

There is a class of geometric problems that ask about covering shapes with other shapes. Some have practical applications but mainly they are studied for interest, and because they can be easy to state but unexpectedly hard to solve.

One type of problems concerns covering a circle with 1 x 1 squares. If we use 1, 2, 3... squares, what is the biggest circle we can completely cover in each case?

Here are the optiomal arrangements for 1, 2, and 3 squares

sketch%20%281%29.png

I constructed the first two in GeoGebra but the third is a sketch based on a diagram by Erich Friedman.

These have been proved to be optimal. Here are even sketchier pictures of the other two cases where the best possible arrangement is known with certainty.

sketch%20%289%29.png

sketch%20%2810%29.png

You may see a pattern here, something like "when the number of square is a square number 12 =1, 22 = 4, 32 = 9 etc, the best arrangement is just a square grid. Obvious really.

But it's not true! When we get to 16 the pattern breaks down and the surprising arrangement below turns out better. I didn't say best, because this is just the best known, and it might be possible to improve on it. By this point I'd so many diagrams I decided to cheat and simply copy Eric's amazing diagram.

scc16.gif

Kind of crazy but it does have a line of symmetry. Erich Friedman's github page here gives best known solutions up to 18 squares and they are all symmetrical about a line, but this not a given; there may be improved solutions with now symmetry, it's perfectly possible.

Here's a puzzle to end with; if we go back to the case of two squares, whar is the radius of the circle? You can look it up on Erich's page of course, but can you work it out?

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Richard Walker

Sweating like a pig

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Here's a little proverb you surely ought to know,
Horses sweat and men perspire but ladies only glow.

—Traditional

I heard someone say they were "sweating like a pig" and wondered where this expression came from. Pigs don't sweat you see; according to Wikipedia not many kinds of animal do. Dogs and cats can only sweat through their paws but rely a lot on panting to keep cool, especially dogs.

This set me wondering about birds too. The RSPB lists a number of ways they can thermoregulate, including

  1. Staying in the shade
  2. Panting (I didn't know birds pant but they do)
  3. Using birds-baths and fountains
  4. Restricting activity to the cooler parts of the day

There have been suggestions that some birds keep cool by flying higher, where the air is cooler, but this is uncertain. There is also something with the exotic name of "gular fluttering", practised by herons and cormorants (for example), which involves the birds fluttering the insides of their throats accompanied by rapid breathing.

And what of sweating like a pig? According to this website the phrase appeared in the Morning Post from Wednesday 10 November 1824, applied to a boxer, but this doesn't cast any light on the origins of the phrase. Many sources seem to link it to pig-iron, which the OED says is

"Cast iron as first obtained from a smelting furnace, in the form of oblong blocks"

and this quote shows why the term pig is applied

  1. 1806
    The lateral moulds or channels are called pigs; and hence cast-iron receives the appellation of pig-iron.

The main channel is the sow you see and the side channels the pigs. This seems fairly convincing, but where does sweat come in? The suggestion is that as the iron cooled down condensation formed on it and resembled sweat, but this awakes the sceptic in me, I don't find it very believable. Why would the hot iron cool below the temperature of the surrounding air and condensation form?

I tend to go with the website referred to above, which suggests pigs are often considered dirty (this is, ironically, because they wallow in mud to keep cool) and from that pig has become a pejorative term, which was then attached to something else (sweat) considered disagreeable or nasty.

 

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Richard Walker

What I'm reading – "Problems To Sharpen The Young"

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Propositiones ad acuendos juvenes, "Problems to sharpen the young", is an early collection of recreational maths problems, generally attributed to Alcuin of York (roughly 735 – 804), although the authorship is not certain.

In it we find some familiar problems, such as the one about the man who has to transport a wolf, a goat, and a cabbage across a river. The wolf can't be left alone with the goat nor the goat with the cabbage.

"De lupo et capra etfasciculo cauli - A wolf, a goat and a bunch cabbages. A man had to take a wolf, a goat and a bunch of cabbages across a The only boat he could find could only take two of them at a time. he had been ordered to transfer all of these to the other side in condition. How could this be done."

After more than 1,000 years this puzzle remains seems to have retained its novelty and Googling "wolf, goat, cabbage" gets over a million hits.

Here's another I liked

De duobus hominibus boves ducentibus - Two men leading oxen. Two men were leading oxen along a road, and one said to the other: "Give me two oxen and I'll have as many as you have". Then the other said: "Now you give me two oxen and I'll have double the number you have." How many oxen were there, and how many did each have?"

The translations I'm reading are from Hadley and Singmaster [1] and a recent book by Marcel Danesi [2].

[1] Hadley, J. and Singmaster, D.  Problems to Sharpen the Young. The Mathematical Gazette , Mar., 1992, Vol. 76, No. 475.

[2] Danesi, Marcel. (2024) Alcuin's Recreational Mathematics: River Crossings and other Timeless Puzzles.

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Richard Walker

Manky Salad

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Edited by Richard Walker, Wednesday, 2 July 2025, 22:57

I left some salad leaves out and they went off a bit. My friend said they were "manky" and put them in the recycling. 

It's a common word but where did "manky" come from? Looking it up, there's a whole raft of theories, with a lot of interconnections between them, and although there's probably no single path we can trace with certainty, the majority view is that it probably derives from Latin mancus = maimed, which is also the root of French manqué = failed.

Manqué is a strong contender to be the immediate source — we can easily imagine someone pronouncing it "manky" — but it could also have come from cognate words found in other languages, such as Italian mancare or Scots mank.

Wiktionary on the other hand suggests an Old English word *mancian with cognates in other Germanic languages and which could have connections with Latin mancus, and it's even possible all these words come from an ancient Proto-Indo-European root *mank, again with the sense of maimed, damaged, deficient.

It's also been suggested that Polari may have played a part in the word becoming more common from about 1960 onwards. Here's the Google ngram

sketch%20%282%29.png

There are also words such as "mangey" and "mangled" which may be connected with manky in some way, or what have an influence on its use. But in the end it is, as the OED puts it. a word of uncertain origin.

My friend was right about the salad though. It was dead manky.

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Richard Walker

Allyl isothiocyanate - it's mustard!

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Edited by Richard Walker, Monday, 30 June 2025, 23:47

I've always been fond of hot food - hot in the sense that chillies are hot -  so I thought I'd list all the hot spices and vegetables I could think of, and see what was in them that made them hot. I was expecting some degree of overlap, but it was more than I had thought.

Essentially there are just two sort of "hot" food substance. The first shares a family of active ingredients that are chemically related

Spice Active ingredient
Black pepper Piperine
Chilli Capsaicin
Ginger Gingerol
Szechuan pepper Sanshools

All the other "hot' things I thought of (Horseradish, Mustard, Radish, Wasabi) belong to the cabbage family and what makes them hot is Allyl isothiocyanate (aka mustard oil). Here's a model of the molecule, from Wikipedia:

sketch.png

Grey atoms are Oxygen Hydrogen, the black Carbon the blue Nitrogen, and the yellow Sulphur.

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Richard Walker

"The Three Ranger Stations" revisisted

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This puzzle was about situating three ranger stations in a wildlife park with a square boundary in a way that is in some sense optimal.

One solution partitions the park into three regions and positions the ranger stations so that (1) each station has oversight of the same area, (2) each point on the park is overseen by its nearest station, and the maximum the distance between any station and a point in its region of responsibility is smaller than if we simply divided the park into three rectangular strips in the obvious way.

However if we are prepared to compromise over either (1) or (2) the maximum distance of travel can be further reduced.

sketch.png

In the arrangement above the side of the square is s and the other distance are as shown. The dotted lines are diameters of the regions and are all of length sqrt(65)×s/8. The ranger stations are at the midpoints of the diameters (blue dots) and the further distance of travel ever needed is sqrt(65)×s/16≈ 0.504s.

By suitably choosing the distance HP we can equalise the areas, or we can have every point overseen by the nearest station, but we cannot achieve both these objectives simultaneously. Although this problem is obviously highly idealised, it occurred to me in real life situating strategic resources must often involve this type of compromise. 

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Richard Walker

Red Poppies

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sketch%20%282%29.png

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Richard Walker

A Nice Geometry Problem

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Edited by Richard Walker, Tuesday, 24 June 2025, 12:30

While I was having my breakfast I found this Geometry Proof on Instagram.sketch.png

On two adjacent side of rectangle ABCD draw equilateral triangles CFD and BEC as shown. Prove the triangle AEF is equilateral.

Presh Talwalkar solved this by constructing some right-angled triangles and applying Pythagoras and algebra to show all three sides have equal length. 

I approached the proof differently, without Pythagoras or any significant algebra. I'll put my solution in the Comment later on today.

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Richard Walker

The Peach-Leaved Bellflower

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Edited by Richard Walker, Monday, 23 June 2025, 08:23

I found this peach-leaved bellflower (Campanula persicifolia) at the side of the road. It's a British wild flower but the species is also grown in gardens, so this may have been an escape.

sketch.png

Looking the plant up, I found its blue colouration and the blue and violet colour found in all 550+ known Campanula species is due to a compound called Violdelphin, one of a group of compounds, called flavonoids, which plants manufacture, and which benefit plants in a variety of ways. 

In the UK six or more other bellflowers grow wild, including harebells and clustered bellflowers.

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Richard Walker

12 Anglo-Saxon Insects

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Edited by Richard Walker, Friday, 20 June 2025, 19:34

Here are the Old English names of 12 common insects. Can you identify them? Most are still quite close to the modern English equivalents.

1 fleoge
2 wifel
3 bitela
4 gnætt
5 moþþe
6 buterfleoge
7 mycg
8 leafwyrm
9 treowwyrm
10 beo
11 æmete
12 græshoppa

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Richard Walker

Etymology of 'Rollmop'

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I had rollmops for lunch today and decided to look into the origin of the name. A rollmop as you probably know is a pickled herring fillet, rolled up and usually held together with a short wooden skewer. I got the roll bit but the mops was a mystery.

I looked it up in the OED and it turns out it's a 19c borrowing from German Rollmops, which is Roll + mops, meaning 'pug', so Rollmops literally means 'roll pug'. The plural is Rollmöpse, Rollmops being the singular, and English rollmop a back formation.

This is a bit like 'pea', which is a back formation from 'pease', as in pease-pudding. 'Pease' sounded like a plural and so people assumed one of them would be a pea.

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Richard Walker

Solution to "The Three Ranger Stations"

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Edited by Richard Walker, Wednesday, 18 June 2025, 08:24

This puzzle concerned three park rangers who are responsible for overseeing a park in the shape of a square of side 100 km. The puzzle is to partition the park between the rangers so each is responsible for the same area, and to station them so the greatest distance any ranger may need to travel to reach a point in their allocated regions is less than it would be with the simple approach of dividing the park into three identical rectangular strips as shown below..

sketch.png

This problem was posed in the November 1952 edition of the American Mathematical Monthly [1] and a solution provided that also met the further condition that each point in the park be allocated to the ranger stationed nearest to it. The solution given is below.

sketch%20%283%29.png

The oblique lines pass through points side/3 and side/4 from the edges of the square, as indicated by the dotted segments, and are at 120° to one another. One ranger station is at the intersection of two long diagonals of the pentagon at right, and the other two are found by reflecting this point in the boundary lines. The three ranger stations are then at the vertices of an equilateral triangle.

This satisfies all three requirements: equal area, reduced maximum travel distance compared with the three rectangle solution, and allocation of every point to the geographically nearest station.

However if we relax the last condition I think it is possible to reduce the maximum travel still further. Can you see how this could be done?

[1] Ogilvy, C. S., & Bankoff, L. (1952). E1001. The American Mathematical Monthly59(9), 634–635. https://doi.org/10.2307/2306773

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Richard Walker

I, Computer

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Edited by Richard Walker, Tuesday, 17 June 2025, 00:31

Mr Paradock: A comptometer then. What does it matter? Tell me to do something. Go on. Feed me some data.

From the play One Way Pendulum by N F Simpson (1959)

You will know that the earliest Computers were not machines but people. For a while I was one of those human computers. Working for the London Brick Company I was paid 16s an hour, good money in those days. 

What did I do? Computed of course. There were scores of us in a big room, performing various tasks, but mine was drawing up a weekly report on brick sales. It went something like this.

People brought me records of how many bricks of different varieties (such as Flettons or Wirecuts) had been supplied from a particular factory (such as Stewartby). I merged all these into a big spreadsheet and then worked out the totals. Manually, note; we weren't given mechanical adding machines and pocket calculators didn't exist yet.

We used different coloured inks to distinguish different sorts of brick and pens with very small nibs so we could write very small figures.

Millions of brick were involved but the numbers had to balance, to the exact brick, and to check our working we divided all the numbers in a row or column by 13 using mental arithmetic, added up all the remainders, and checked that the sum agreed with the remainder on dividing the row or column total by 13.

Already it was obvious that the skills this work required would soon be obsolete. In the middle of the office was a small row of Comptometers, like that below, that only specially trained staff could use. sketch%20%282%29.png

They were the latest thing but then in the twinkling of a silicon chip they too were out of date.

  • N F Simpson belonged the movement known as the Theatre of the Absurd. The play concerns the eccentric Mr Groomkirby who wants to build a replica of the Old Bailey in his living room. The Comptometer is introduced in Act II.
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Richard Walker

The Three Ranger Stations

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A new wildlife park has the shape of a square of side 100 km.

The park authorities want to divide the park into three regions, in each of which a ranger station will be built to have oversight of all the points in that region. The regions are to be of equals area, and the ranger stations located so the greatest distance separating any station from a point in its domain is as small as possible.

One idea is to divide the park into three equal strips as shown below, which obviously produces regions of equal area, satisfying the first requirement.

sketch.png

In this arrangement the furthest distance from a station to a point in its region is equal to the length of the dotted line, about 52.7 km. Can you find an alternative way to divide up the park so the furthest distance is less than this?

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Richard Walker

Mondegreen Hearing

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In case you don't know, a mondegreen is when a listener puts the wrong words to what they hear. The term mondegreen was coined in 1954 by Sylvia Wright, whose mother used to read her an old ballad:

Ye Highlands and ye Lowlands,
Oh, where hae ye been?
They hae slain the Earl Amurray,
And Lady Mondegreen.

Who was this mysterious Lady Mondegreen? Sylvia Wright had a vivid mental picture of her and her sad end.

But the was no such person; the last line is actually:

And laid him on the green.

And from Wright's article the word mondegreen made it into our language and can now be found in the OED.

I've posted about modegreens before, they fascinate me , and this morning I came across a nice example. Someone said:

"I think I'm overdosing on drinking tea." 

Makes sense I thought, we all know too much caffeine can be bad for you.

But what they'd actually said was:

"I think I'm overdosing on Vitamin C." 

But they weren't, it's OK.

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Richard Walker

How Much Does it Cost to Feed a Robin?

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I wondered: if I were responsible for meeting the food requirements of a robin, approximately how much would it cost per day?

A robin weighs around 20 grams and the usual estimate is that he or she needs about 40% of body weight per day. Mealworms (a favourite food) are about £6 per kg, so a robin's daily requirements cost about 

20 x 40% x (£6 ÷ 1000) = 8 x 0.6p = 4.8p

or 5p in round figures. 

The figure of 40% (sometimes 50%) of body weight is often quoted (because it's surprising and illuminates how much work small birds must do to survive) but I'm not sure what study or studies it originates from. I tried to find out but didn't really get anywhere. However I did hit on a very interesting paper from 2019, that examined how people feeding wild birds has over the years shaped both the number and variety of birds visiting feeders. The British Trust for Ornithology website gives this overview of the findings:

Newly published research from BTO shows how the popular pastime of feeding the birds is significantly shaping garden bird communities in Britain. The populations of several species of garden birds have grown in number, and the diversity of species visiting feeders has also increased.

The paper was published in Nature and is freely available here.

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Richard Walker

Solution to "The Two Squares"

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Edited by Richard Walker, Monday, 9 June 2025, 00:29
Here is the solution to "The Two Squares", which asked given squares of side 3 and 4 arranged as shown, what is the are of the overlap?
image%20%282%29.png
The answer is 4 square units. Here are two ways to see this.

1. The question doesn't give any information about the orientation of the small square, yet we are asked to find the area in question. This suggests that orientation makes no difference (it's one of those questions where absence of a piece of information is itself a clue). So we can arrange the square however we please and choosing the orientation shown below below tells us the overlap is one quarter of the larger square.


2. Alternatively, let's add three copies of the smaller square and now the fact that the overlap is one quarter of the small square, irrespective of the orientation, becomes obvious.



In fact what makes it true that the overlap is one quarter of the smaller shape is not that is a square, it is that it has 4-fold rotational symmetry. The square can be replaced with any arbitrary shape so long as it has this property. Here's a pinwheel, for example:



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Richard Walker

Four Well-Beings

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As you will know, a Well-Being is a Being that dwells in a Well. Here are four kinds:


1. The Mossbound Oracle

Appearance: A humanoid figure made of moss, algae, and stone, with glowing eyes like bioluminescent fungi. Vibe: Ancient and wise, it speaks in riddles and echoes. Habitat: Clings to the walls of the well, emerging only when the moonlight hits the water just right.

2. The Water Wyrm

Appearance: A serpentine creature with translucent scales, fins like lace, and eyes like polished pearls. Vibe: Elusive and graceful, it coils through the water silently. Habitat: Dwells in the deepest part of the well, surfacing only when it senses a pure heart.

3. The Forgotten Child

Appearance: A small, ghostly figure with dripping hair and pale, waterlogged skin, wearing tattered clothes from another era. Vibe: Haunting but not malevolent—more sad than scary. Habitat: Sits at the bottom, whispering stories of the past to those who listen.

4. The Echo Sprite

Appearance: A shimmering, semi-transparent being made of sound and mist, constantly shifting shape. Vibe: Mischievous and curious, it mimics voices and laughter. Habitat: Lives in the acoustics of the well, only visible when someone speaks into it. The above was the rather imaginative response I got from Copilot when I prompted it with:

'imagine I have a "being" living in my well, what could it look like'.

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Richard Walker

The Two Squares

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A square of side length 3 units has a vertex at the centre of a larger square whose side length is 4 units. Find the area of the shaded region.



Faires (2006) First Steps For Math Olympians, MAA.
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Richard Walker

Malaphors

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Yesterday I heard someone say, "I'm an old hat at this" instead of "an old hand", making me wonder if there was a special term for such a usage. Not a mixed metaphor (there's no metaphors, nothing is being liken to something else), or a malapropism (confusing two similar sounding words but the speaker wasn't confusing "hat" and "hand", but rather two whole phrases).

This was something different, and after some research I found the word I was looking for, a malaphor. This seems to have been coined by Lawrence Harrison in 1976, according to Douglas Hofstadter, whose insightful discussion is available here.

Searching on malaphor throws up many examples. Some, such as "It's not rocket surgery" I suspect of being intentional humour, but others have a surreal logic and are probably bona fide. See here for Susie Dent's Top 10, including the magical and inspired "Like lemmings to the slaughter".

The things we say reveal ways in which our minds work and it's interesting that in all the example above the intended meaning is perfectly clear. The speaker groped for a stock phrase that would make what they said more vivid, found two candidates that shared some features (structure, vocabulary, semantic field etc.) and confused them, but it didn't matter, because the substantive information had been conveyed elsewhere, and the general drift ("going back a while", "very hard and technical", "unwitting victims") came across anyway.

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Richard Walker

Cetti's Warbler

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In the bird reserve the evening before last I heard a very striking song, which the BirdNET app reported as "almost certainly" Cetti's warbler. Comparing the song with online recordings confirmed the identification.

Cetti's warblers don't seem to have bred in the UK until the 1970s, when they first appeared in Kent. Since then, possibly as the result of global warming, they have pushed as far as Wales, the north of England, and recently Scotland. There are now perhaps 5,000 or more breeding pairs across the UK.

I didn't see the bird.They are hard to spot; small and inconspicuous and in the words of the RSPB like to skulk in patches of scrub (although "skulk" seems a bit pejorative to my mind).

Traditionally people talked about bird watching but when it comes to birds like warblers bird listening is more appropriate, because warblers recognise potential mates not by conspicuous plumage, but by conspicuous song. I'm not very good at recognising birdsong, but the phone app is, and it's opened up a whole new world for me, where I can observe birs even though I can't see them.

And who was Cetti? Francesco Cetti was a Jesuit, mathematician, scientist and naturalist who published three volumes on the natural history of Sardinia in the 1770s. This is the volume on birds,

The bird was collected on Sardinia and described by Alberto della Marmora, who named it Cettia cetti in Cetti's honour.

Fun Fact: The eggs of Cetti's warbler are bright red.

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Richard Walker

❦❦❦ Flowerbeds at Wimpole ❦❦❦

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Flowerbeds in the Walled Garden at Wimpole Hall


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Richard Walker

Equicrural

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Another unusual word I owe to the "Word Watching" item in Times 2.

It means the same as isosceles – "equalled legged"– but is derived from Latin rather than Greek. The second element is from Latin crus = leg.

First attested in 1650, it's now rare or obsolete and has been wholly supplanted by isosceles, which came via Latin from Greek ἰσοσκελής, which is made up of iso- (as in e.g. isobar) + skelos = leg.

Here is a nice quote, courtesy of the OED, from the famous Robert Recorde, writing in 1551.

There is also an other distinction of the names of triangles, according to their sides, whiche other be all equal..other els two sydes bee equall and the thyrd vnequall, which the Greekes call Isosceles, the Latine men æquicurio, and in english tweyleke may they be called.

I suppose "tweyleke" must mean two-alike but this is just a guess.

Recorde was born in Tenby Pembrokeshire and wrote a series of highly influential books on mathematics, and enjoyed the unique distinction of having invented the equals sign as we know it today.

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Richard Walker

Crossword Clue

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Stir up 1009? (3)

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Richard Walker

Conway's Circle Theorem

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An old story tells of a Maths professor, who at a certain point in his lecture told the class, "The proof of this is obvious", but then began to have doubts. After gazing at the blackboard for some time he said, "Hmmm. Perhaps it's not obvious. I'll have think about it and let you know in next week's lecture."

The following week the class assembled, and after a new minutes the professor arrived. Gazing round the lecture theatre, he said "Since our last lecture I have been thinking about that problem in every waking hour, but I simply could not see it. But you will be glad to know that on my way here today I finally saw the answer."

"It is obvious."

Well, I have been thinking for several days about how to prove the Conway Circle Theorem, a rather nice result usually credited to the brilliant and original mathematician John Horton Conway, who sadly died in the recent Covid Pandemic. I could fairly quickly see a proof but it had a rather messy feel to it, it wasn't anything I would want to explain on this blog. It wasn't what I would call elegant. I felt there must be a nicer proof, one that would make the proof of the theorem, well, obvious.

After much chewing the problem over I think I have a proof which does make the theorem fairly obvious. But I'd better tell you what the theorem is before we go any further. Consider the following diagram.



As shown in the diagram, the three sides of triangle ABC have each been extended at both ends, AB by distances a and b, BC by distances b and c, and CA by distance c and a. This gives the six points G, H, I, J, K, L.  Conway's theorem says that (rather surprisingly) these six points lie on a circle, see below:



How to prove this? I'm going to use some well-known properties of isosceles triangles, triangles with two sides equal in length.

In triangle PQR sides PQ and QR are equal. The line bisecting the base PR at right angles, the perpendicular bisector, must pass through the apex Q, and it bisects the angle PQR. Moreover any point, for example Y, that lies on the perpendicular bisector is equidistant from points P and Q, as shown by the dotted lines.

Going back to our original diagram, let's join up the six points to make a hexagon, and then draw in the perpendicular bisectors of its sides, shown dotted:


We see that each side of the hexagon is the base of a triangle, which by the method use to construct the six  points must be isosceles. Each perpendicular bisector passes through a vertex of triangle ABC and bisects the angle there, and is fact the shared perpendicular bisector of a pair of the hexagon's sides lying opposite one another.

It's well known that the three lines bisecting the angles of a triangle meet in a single point, called the incentre. Because the three dotted lines bisect the angles of ABC this mean they meet at its incentre.

Because the incentre is on the perpendicular bisector of HI it must be equidistant from H and I. Similarly because it is on the perpendicular bisector of IJ it must be equidistant from I and J. By the same argument, it must be equidistant from J and K; from K and L; from L and G; and from G and H. Thus it is equidistant from all six points and is the centre of a circle that passes through them all.

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