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Richard Walker

Coincidence or what?

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This is something I found on Quora 


but no proof was given there, so here is mine..

Above we see three regular polygons, the equilateral triangle, square and regular pentagon. Each has a side length of 2 units. Associated with any regular polygon are two circles: its incircle, which touches its sides, and its circumcircle, which passes through its vertices.

Surprisingly, the area between these two circles, shaded pink in the figures above, always has an area of π/4 square units, however many sides the regular polygon has. How do we know this?

Here we see two consecutive vertices of the polygon, painy=ts A and B, with M being the midpoint of AB and O the common centre of the incentre and the circumcentre.

The side length is 1, so MB is ½, and because AB is tangent to the incentre and OM a radius of the circle angle OMB is a right angle. Let the radiiusof the circumcentre be R and that of the incircle r, so OB = R and OM = r.

Then using Pythagoras in the irght-angled triangle OMB we have

OM2 + (½)2 = OM2, so r2 + ¼ = R2, or R2 - r2 = ¼.

But now considering the ares of the two circles we have

Area of circumcircle - Area incircle = πR2 - πr2 = π/4.

Notice we haven't used the number of sides anywhere in this! So the area is always π/4 irrespective of the number of sides, as claimed.

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Richard Walker

At the Celebrity Ball

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Ladies, and gentlemen, please put your hands together for the famous Russian ballerina, Sheila Tripova.

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Richard Walker

Fruity Pine Martens

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Edited by Richard Walker, Wednesday, 31 Jan 2024, 00:08

I always wrongly suppose Pine Martens, animals related to weasels and stoats, are exclusively carnivorous  , like their cousins. But it seems now that they also eat fruit and other stuff and are particularly partial to raspberry jam sandwiches. 

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Richard Walker

Daffynitions [1]

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Edited by Richard Walker, Monday, 29 Jan 2024, 21:39

Phobia -  imitation ale

[1] https://en.m.wikipedia.org/wiki/Daffynition#:~:text=A%20daffynition%20(a%20portmanteau%20blend,(or%20group%20of%20words).//en.m.wikipedia.org/wiki/Daffynition#:~:text=A%20daffynition%20(a%20portmanteau%20blend,(or%20group%20of%20words).

Nicked from 3D in today’s Times Quick Cryptic
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Richard Walker

Mr Seal

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Edited by Richard Walker, Monday, 29 Jan 2024, 20:27

A friend spotter this seal on the beach at Aldeburgh. I can’t embed the video on this page but here’s a link to a YouTube playloop:

https://youtube.com/shorts/DNM5P53iUfw?si=NUuBLHTgLb_ABzQk


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Richard Walker

Going Up!

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Richard Walker

Heritage?

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Richard Walker

Meanings you didn't know #1

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From the OED

1889
lobster
One who bowls ‘lobs’ at cricket.

Next up  whisky


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Richard Walker

Wind Haiku

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All night long

The wind was roaring like a lion

About to gobble us up.

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Richard Walker

Record Breaker

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Edited by Richard Walker, Sunday, 21 Jan 2024, 23:47

Yesterday I read in Scientific American that the record for the hottest chilli pepper has been officially broken. The new champion is Pepper X and is listed in the Guinness Book of Records. The plant breeder was the appropriately named Ed Curry (see nominative determinism), who also produced the previous record holder, the Carolina Reaper.

The Scorville rating of the new pepper is 2,693,000 SCU. Here is a chart (courtesy of Wikipedia) that compares this with more familiar varieties of chilli.


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Richard Walker

Unique rhyme?

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I’ve heard it said that there is no word in the English language that rhymes with ‘oblige’, which, as far as I can see is true, unless you count proper names, such as ‘Nige’.

So what about words that only have one other word that rhymes with them? Well, I thought of wombat and combat; but are there any other words that rhyme with these two? If you find any do write in the comments.

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Richard Walker

Sad News

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Sad news, my steeplejack friend had a fall and now he’s expired.. 

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Richard Walker

One Liner

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Edited by Richard Walker, Thursday, 18 Jan 2024, 00:06

Being overcharged for a pencil eraser. That’s daylight rubbery.

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Richard Walker

Words with triple letters

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Edited by Richard Walker, Tuesday, 16 Jan 2024, 01:04

In the Oxford English Dictionary I found the following words where a letter occurs three times in succession.

A list was posted on Quora and all I've done is check which ones the OED lists, plus I thought of oooh.. So there could be others not discovered. Apart from brrr, oooh and yayyy my favourite is frillless, which makes perfect sense, although you don't hear it very often.

brrr

duchessship

frillless

hostessship

oooh

uuula (I guess this is 'uvula' spelt before u and v were regarded as different letters)

vertuuus (which means more or less the same as 'virtuous')

yayyy (!)

Other -ship words are hyphenated, e.g. countess-ship, and I think the OED is just being inconsistent with the two examples above.


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Richard Walker

Down in the Cabbage Patch

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Richard Walker

Dad Joke

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It’s only when you have finger problems that you realise how much you count on them.

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Richard Walker

Helvellyn from Whiteside

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Edited by Richard Walker, Saturday, 13 Jan 2024, 23:41








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Richard Walker

O'er the hills and far away

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Tom, he was a Piper's Son, And fell in love when he was young; But the only Tune that he could play, Was o'er the Hills, and far away.

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Richard Walker

Dad Joke

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Edited by Richard Walker, Wednesday, 10 Jan 2024, 10:43

I keep missing the first letter off words. The doctor says it's rain damage.

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Richard Walker

A Room with a View

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Richard Walker

Night noises

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Why did the cat flap bang just then?
Was it a ghost,
or only the wind.
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Richard Walker

A Lakeland Scene

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Richard Walker

Cave Bears

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This is a tooth from a cave bear. My brother gave it to me some years ago and I keep it by my bedside, as a sort of link to the past and the world in which our ancestors lived. 

The coin is to give an indication of scale. It's a Roman denarius, about 1 cm across.

The tooth looks pretty formidable, and I thought about what sort of a bite it could give. I imagined prehistoric  people competing for cave space with what I though would have been carnivorous animals. However, to my surprise, studies of their teeth suggest they may have been substantially herbivorous [1].

Why did they die out? Some possibilities suggest themselves:

  • The bears' diet was too specialised (think about giant pandas), the plants they eat were affected by climate change and there wasn't enough food
  • Humans hunted them to extinction
  • Humans out-competed them.
It might have ben a combination of all these factors, or something entirely different. Further research may help us learn more about what happened to these fascinating beasts.

[1] Cave Bear: A Vegetarian Carnivorehttps://www.senckenberg.de/en/pressemeldungen/cave-bear-a-vegetarian-carnivore/#:~:text=Together%20with%20an%20international%20team,had%20an%20exclusively%20vegetarian%20diet.

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Richard Walker

The Elephant

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The Elephant is a curious bird

It flits from twig to twig.

It builds its nest in a rhubarb bush

And whistles like a pig.

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Richard Walker

Chicken Joke

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