Personal Blogs

I found this striking fungus in my garden today. I believe it is Armillaria tabescens, the ringless honey fungus, but it's hard to be certain. Rather beautiful.

In one of the Inspector Morse books, Lewis repeats to himself the mnemonic "Five 'esses' in 'possess'". I can't trace the quote, but I think Lewis must have needed to use the word in a report and felt nervous in case he spelt it wrongly. (The first book in the series was published in 1975, pre-spellcheckers). 

I thought I would test an AI on this, so I asked Copilot. It said four esses. So I tried to correct it and it said rather patronisingly:

• • P-O-S-S-E-S-S-E-S

1. 1. P-O-S-S-E-S-S-E-S
   2. P-O-S-S-E-S-S-E-S
   3. P-O-S-S-E-S-S-E-S
   4. P-O-S-S-E-S-S-E-S

• • Assesses
  • Assassins
  • Assessors
  • Possesses
  • Sassiness
  • ...

Something that has fascinate me for many years is the fact that, if a disordered systems is larges enough, it will tend to contain pockets of unavoidable regularity. Order will emerge from disorder.

For example, imagine we colour every point of the 2D plane in one of two colours, red or blue say. Then there will inevitably be equilateral triangles all three of whose vertices are the same colour; either all red or all blue. Here's how we can prove this.

Imagine we set out to colour the plane in a way that avoids a monochromatic equilateral triangle. We shall find that this is impossible. Consider the following steps in our attempt.

We start with two points which are coloured red (Step 1).This must be possible; if not, there would no more than one red point in the entire plane, so either all equilateral triangles would be blue monochromatic, or all the ones that avoided the single red point would be. Either way our attempt to avoid monochromatic equilateral triangles would have failed spectacularly.

At Step 2 both the ringed points will have to be coloured blue, otherwise one or both the triangles marked will be red monochromatic.

At Step 3 the ringed point will have to be coloured red, otherwise the triangle marked will be blue monochromatic.

At Step 4 the ringed point will have to be coloured blue, otherwise the triangle marked will be red monochromatic.

At Step 5 a monochromatic triangle can no longer be avoided; whether we choose red or blue for the uncoloured point a monochromatic triangle with be created.

There really is such a word, it's in the the Oxford English Dictionary.

First recorded in 1881, it is the only words I have come across in the OED that lists a citation from Twitter:

Do you know a Hrothgar? I bet you do…

Hrothgar was a semi-mythical Danish king who appears in several places in Old English and Old Norse literature. His name is dithematic – it consists of two elements hroth + gar, meaning ‘famous’ and ‘spear’.

Many Germanic names follow this dithematic pattern and there are hundreds, possibly thousands of examples attested. Many survive in modified form to the present day and are very common and familiar given names, for instance

William - will + helm = ‘wish helmet’

Mathilda -  maht + hild = ‘mighty battle’

Rosamund - hros + mund = ‘horse guardian

Robert -  hroth +  beraht = ‘famous bright’

Several of these elements are easy to recognise – will, maht, hros, beraht - and helm survives as a poetical word for helmet. mund was still found in Middle English but is now obsolete, and hild seems to have disappeared earlier.

As for Hrothgar, it has become Roger, a fact I only realised a couple of days ago. Hroth is an element in several other names: Wikipedia quotes Rudolph, Roderick, and Roland; but seems extinct now except in names.

But gar = ‘pike’ i.e.a long, pointed spear, hangs on. This is a garfish, courtesy of Wikipedia.

My daughter sent me a list of "groaners" she found somewhere on the web. 

Here are three I rather liked:

I felt it my duty to contribute to this genre, so here are some I came up with:

There is an Only Fools and Horses episode in which the road sweeper Trigger announces that he has won an award for keeping the same broom for 20 years, and adds that the broom has had 17 new heads and 14 new handles.

I was thinking about this and remembered that all the atoms in our bodies get replaced on a surprisingly small time scale. Reading up on it, I found that, according to Time magazine, the turnover is about 98% per year. To put it differently, only about 1 in 50 of the atoms in your body will still be there in a year's time.

In 20 years that would be 1 in 100,000 billion billion billion, which is much more than the estimated number of atoms in a human body. So more or less all the matter that went to make up your body will have been replaced, and you will resemble Trigger's broom in this respect.

I've always been intrigued by this paradox; if all the constituent parts of something are replace, is its identity still the same? It's an important philosophical question, but until now I didn't realise it goes by the name of Theseus's Paradox. 

Plutarch wrote

"The ship wherein Theseus and the youth of Athens returned from Crete [...] was preserved by the Athenians [...], for they took away the old planks as they decayed, putting in new and strong timber in their places."

I found this geometry problem posted a few days ago maths YouTuber Michael Penn. It was originally published in 2002 in the Mathematical Gazette. It caught my attention because Penn's solution and the one originally published seemed to use trigonometry and calculation of distances involving square roots, and I felt it should be possible to use a purely geometric approach. 

After a lot of thought, by stepping outside the problem I think I've managed to solve it. Below left is the problem and at right the idea that set me on the right road. Suppose I drew in a square and an equilateral triangle to provide a context for the 45° and 30° angles, would that provide any insight?

This was nearly right, but then after more thought I saw  a better approach was to use a square and an equilateral triangle of equal side length, like this.

We can successively compute the angles marked in the diagram is three stages.

1 (green angles marked by single arcs). At C we an internal angle of the equilateral triangle, which is 60 degrees, and two angles which are each half of a right angle, so they are 45 degrees. The line BG passes through H, the midpoint of CF and from the symmetry of the equilateral triangle we see that it makes a right angle with CF, and that the angle CBF is 30 degrees.

2 (blue angles marked by double arcs). The angles at C must sum to 180 degrees, and so angle DCF must be 30 degrees.

CD = CF, because the square and the equilateral triangle have equal side length, and so triangle CDF is isosceles. Hence the two angles marked at D and C are equal and must both be 75 degrees because the angles of CDF must add up to 180 degrees.

3 (red angle marked triple arcs). From consideration of the angles in triangle CGF the angle marked at G must be 30 degrees.

Moreover, triangle CGE has two angles of 75 degrees and so it is isosceles. Because the line through BH bisects the base of this triangle at right angles, the triangle is symmetrical about the line, and the line must pass through point G.

And now look at the kite AGDC. Can you see? - it is symmetrical about the diagonal CG, and therefore θ, the angle we are asked to find, is equal to angle CGD = 30 degrees. Problem solved.

We have the contractions

But there is no amn't, at least not in standard English. Where did it go?

The best theory seems to be that it did once exist but became ain't, (maybe via amn't > an't > ain't) and then the latter lost its connection to the first person singular "I" and became an informal contraction that can be used with any person: I ain't, you ain't, it ain't, we ain't, they ain't. See here. I think was at one time fairly common but now has an antique feel, to me at least.

Thinking about such contractions I notice that in a question we can use aren't which is a standard contraction: I'm going to the ball, aren't I?

Καλημέρα

It's quite surprising - to me at any rate - that if we take any positive number whatsoever, there is always a prime number that starts with the digits of the number we picked. In fact there are an infinite number of such primes.

I wrote a Python program that lets us input a number and searches for the first prime beginning with the number we entered.

Say we take Einstein's birthdate, 14/03/1879. Write this as 14031879 (which is not itself prime, it's divisible by 3).

Entering this into the program yields 140318797, which as you see starts with the famous scientist's birthdate, and has no divisors other than itself and 1, so it's prime.

Here's another example, this time a random number (I suppose) - the National Lottery jackpot for 14/09/2024, which was 21 27 38 47 49 55. Running the program and entering 212738474955 we get 2127384749557.

By tweaking the program we can get a list of the first few primes that start with the lottery number.

2127384749557
21273847495583
21273847495591
212738474955233

Fascinating!

This is the proof of the question I posted at https://learn1.open.ac.uk/mod/oublog/viewpost.php?post=285828

Here's a sketch

Chord AB is a typical chord passing through a fixed point P in the interior of a circle. Point M is the midpoint of AB and I have drawn in (shown dotted) the two line segments joining P and M to O, the centre of the circle.

By symmetry, the line MO joining the midpoint of the chord to the centre must be perpendicular to the chord. So triangle MOP is right angled, and PO is its hypotenuse.

By the converse of Thales' theorem the hypotenuse of a right angled triangle is the diameter of a circle passing through the three vertices of the triangle, and the centre of the circle is the midpoint of the hypotenuse, N in the diagram above.

The diameter and centre of this circle are fixed by the position of P and O and midpoint M must be on this circle for any chord through P, which is what we wanted to prove.

For information about Thales (fl. 626/623  – c. 548/545 BCE), who seems to have pursued many scientific, mathematical and philosophical interests see here

Richards often get called Dick - my father always did - but I never really thought much about how Richard could translate to Dick. But yesterday I was reading about how studying pet names for people can help with etymological research and came across the idea of "rhyming nicknames", the result of a process like this

name -> shortened form -> rhyming nickname

So we get Richard -> Rich -> Dick, it's as simple as that

Obviously I knew about ordinary shortenings and diminutive pet names (e.g. Thomas -> Tom ->Tommy) and I've also been intrigued by the fact that names whose shortened form would end in 'R' are modified to end in 'L', e.g. Terence becomes Tel and Harold becomes Hal (probably because "Ter" and "Har" don't trip off the tongue so easily). Another such is Chas.

We can also shorten names from the other end, so Richard can be Hud and William can become Liam for example.

However I hadn't really twigged about the rhyming nicknames but there are lots of them and they seem to have been popular in Middle English. Here are some common examples:

Richard  (again) -> Rick -> Hick

Edward -> Ed ->Ted (or Ned)

William -> Will -> Bill

Here's a surprising one:

Margaret -> Molly -> Polly

or

Margaret -> Meg -> Peg

Finally we have 

Robert -> Rob (possibly via Robin, a diminutive)  -> Bob (or Hob) and Hob is the first element in Hobgoblin, a kind of mischievous elf. Robin Goodfellow, a kind of goblin and Puck-like character, became Hob the goblin.

Goblin is interesting in itself, it seems to be from the same root as German Kobold,  and became the name of Element 27.  Element 28 another troublemaker.