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.

Given a circle centre O and a point P in its interior, the midpoints of all the chords passing through P lie on a circle, which also passes through P and through O.

My title describes something most of us have known about all our lives without realising it.

Why do 'hip hop' and 'ding dong' sound right, but 'hop hip' and 'dong ding' sound wrong somehow? It's not just familiarity, there is a rule at work, as we shall seen. If I invented a new expression 'squop-squip' it wouldn't sound as satisfactory as the alternative 'squip-squop', because the first version doesn't follow the rule, whereas the second does.

Reduplication is when a word is repeated two or more times. In English there are several kinds of reduplication, for example it can be used to amplify the sense of a word, so 'very very' is stronger than simple 'very' and 'big big' is stronger than just 'big'.

Rhyming reduplication is also very common: higgledy piggledy, silly billy, willy-nilly, Humpty Dumpty, raggle-taggle, easy-peasy (lemon squeezy), teeny-weeny, fuddy-duddy and so on.

Ablaut reduplication is a form of reduplication following a pattern called ablaut, in which semantically related words coming from the same root keep the same consonants but vary the vowel sounds in a predictable way. A classical example in English is sing, sang, sung. Notice that if you say these words aloud, you say the first at the front of the mouth, the second further back, and the third at the back of the mouth.

We have many expressions like; tip-top, tick tock, flim-flam, pitter-patter, jim-jam, and they pretty well all follow the ablaut pattern, with the first part being a higher vowel from front-of-mouth, and the second a lower vowel from further back. We even have some examples with three elements, such as tic-tac-toe and bish bash bosh.

This ablaut pattern is thought to be inherited from the ancient and lost language that is the ancestor of moist European and West Asian languages today. This language had no written form that we know of, and no longer exists, but has been extensively reconstructed from its more recent descendants.

A friend photographed this astonishing piece of street art outside Paddington Station. With the aid of Google Lens I found it is "Wild Table of Love", by Gillie and Marc.

"There's a nail in that tyre", Tom said flatly.

It's interesting to consider that back in the day, we'd never heard the expression "Back in the day".

The latest video in the excellent series Words Unravelled, with RobWords and Jess Zafarris, is devoted to "contronyms and other 'onyms'". Most people are familiar with words such as synonym, antonym, pseudonym, acronym, but I hadn't met contronym, which means a word with two meanings that are in some way opposite. For example I might sanction someone, which would mark my disapproval, but I might also sanction a course of action, which would mean I gave it my approval. 

The suffix -onyms is from the Greek for name, and there are quite a few -onyms apart from the ones about. Rob and Jess discuss a whole range of other other interesting -onym words.

I thought I'd explore a bit further and was intrigued to find a number that describe names for physical features of the landscape, each formed by taking an Ancient Greek word for the feature in question and adding -onym. These all refer to proper names, e.g. Helvellyn is an oronym, The Grimpen Mire is a helonym, The Black Forest a drymonym, and so on. Here are the words I managed to find.

agronym – field

anemonym – hurricane

drymonym – forest

helonym – swamp

hydronym – sea

limnonym – lake

oronym – mountain

spelonym – cave