A bad hangover leaves you confused the next day but a worse one lasts until the daze after tomorrow.

## Personal Blogs

Seated in a chair, raise your right foot from the ground and start moving it in a clockwise circle. While continuing to move your right foot, extend your right for finger and draw a figure 6 in the air. What happens?

See here for a YouTube video about this surprising effect.

It appears on lots of web sites, often under the title 'How smart is your right foot?' but I hadn't come across it before.

I’m reading a book about blotting paper. Very absorbing.

I’ve just bought 12 baby polyanthus plants via Amazon. When they flower they will be an assortment of 6 colours and I’m assuming there will be 2 plants of each colour. But at the moment they all look pretty much the same.

I’m planning to give 4 plants to a friend. Given we will be picking the 4 at random…

What is the probability that my friend will receive 4 plants all different colours?

It’s very beautiful and resembles frost crystals, or perhaps some kind of mineral formation (although I suppose ice is a mineral, thinking about it). But what you see here is a fungus, the ‘urchin earthfan). My brother took the photograph.

There’s a good article about thus gpfungus here

https://www.first-nature.com/fungi/thelephora-penicillata.php

What colour leaves you in the dark? Light blew.

Tonight when I paid my bar bill I held the card reader to my right nostril, and my Apple Watch to the left. Everything went through. So I literally paid through the nose.

New TV series about falconry. Britain’s got talons.

What colour leaves you abandoned on a desert island?

Maroon.

"By their very nature bureaucracies have no conscience, no memory, and no mind."

I've been trying to remember this quote. I found it striking when first told it but I'd partly forgotten it so it's taken a while to track down. It’s by the American anthropologist Edward T. Hall, although I still can’t find where he wrote or said it

You can only fight it for so long.

Eventually you can betcha,

The System will getcha.

I read today in the New Scientist about “Miracle Berries”, seen below. Eat one and for twenty minutes or so sour things will taste sweet. Isn’t that remarkable?

A link to the NS article is below, and there is also information on Wikipedia.

Naturally I want to grow some and the seeds are on their way.

https://images.newscientist.com/wp-content/uploads/2022/08/30202824/SEI_120900398.jpg?width=800

The origin is Old Norse *hus-thing*, “house meeting”, and referred to an assembly of the people in the household of a noble man or woman. Over time its meaning has come to mean a platform for politicians to make speeches from.

The *thing* element just meant assembly and survives today in the name of the Icelandic supreme parliament, the Alþingi**.**

Old English also had *folk-moot*, “folk meeting”. Here moot is the same as in the phrase “a moot point” = a point to be discussed and also found in moot hall. Folkmoot survives but mainly in historical contexts.

A delegation of villagers sought an audience with the abbot.

Reverend sir, they asked; when is the best time to plant a tree?

Ten years ago, replied the abbot.

Or sooner if possible.

You have no means of recording the numbers as they are read, and although you are quite good at mental arithmetic you only have an average memory and remembering the entire list of numbers as they were read out is beyond you.

What is the best way to identify the missing number?

A few days ago I wrote about an elegant proof that a regular dodecahedrons has 4 diagonals that all intersect at a point other than the centre of the polygon, see https://learn1.open.ac.uk/mod/oublog/viewpost.php?post=248909

Since then I have found a paper by Poonen and Rubenstein ^{[1]} in which they completely solve the problem of concurrent diagonals in regular polygons. They prove several interesting facts, including:

If the number of sides is odd there can never be 3 concurrent diagonals.

The smallest number of sides that allow 3 diagonals to meet at a point other than the centre is eight, for diagonals twelve sides are needed, and for 5 concurrent diagonals eighteen sides.

To get more than 5 diagonals meeting at a point other than the centre we need to go to thirty sides. The regular *triacontagon* has sets of 6 and 7 concurrent diagonals.

And then suprisingly it stops. No regular polygon, however many sides it has, can have eight or more diagonals intersecting at a point other than the centre.

[1] https://mathproblems123.files.wordpress.com/2011/03/ngon.pdf

This sounded like a portent of doom

“So fear the wren”.

But the speaker was actually talking about a film star.

Looks like all the graphics have disappeared.

Interesting how cliches morph into phrases totally disconnected with the original. Tonight I heard ‘Beats into a cocktail’. The speaker didn’t mean egg flip, just that A was hugely better than B.

But that’s how language changes and all power to it.

I got this message in my Celestial Mailbox

Dear Earthdweller

For maintenance reasons, the Milky Way will be unavailable for up to 10,000 Earth years from next Tuesday.

We apologise for any inconvenience caused.

*The Galactic Team*

It’s hot now and we complain.

But soon the snow will blow.

At the first flake, we’ll shiver and shake.

Wishing summer back again.

*Anon*

Problem 1358 at gogeometry [1] asked for a proof that in a regular 12-sided polygon the four diagonal shown all meet at a point. This is quite surprising; it\'s not hard to find threee diagonals the meet at a single point but four is rarer.

After playing arouind for a while I found a proof which was reasonably neat, but I had to use sines and cosines at one point, and I'd hoped for something simpler; and there was nothing very illuminating about my proof in any case. Stan Fulger came up with something much more insightful. Here is his beautiful answer.

It uses two well-known facts about triangles.

The altitudes of a triangle, i.e. the lines drawn from each vertex at 90° to the opposite side, meet at a point.

The angle bisectors of a triangle, i.e. the lines which divide each angle in half, meet at a point.

For example

Now for a "look and see" proof.

In the picture below three diagonals are angle bisectors in the blue triangle, so they meet at a point. Also three diagonals are altitudes of the orange triangle and therefore meet at a point. Two of the diagonals are both a bisector in one triangle and an altitude in the other. Therefore all four diagonals meet at a point.

I drew the figures above using GeoGebra classic.

[1] Geometry problem 1358 https://gogeometry.com/school-college/4/p1358-dodecagon-regular-concurrency-diagonal-infographic-classes.htm

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