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Richard Walker

The Square Peg Problem

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Edited by Richard Walker, Monday, 27 May 2024, 23:28

No round holes involved, the Square Peg Problem is a deceptively simple mathematical problem, first posed over a hundred years ago. It asks whether every closed curve has an inscribed square, that is, a square with all four of its vertices on the curve, as in the example below.


I've been fascinated by this problem for nearly 50 years. It's easy to state but hard to solve and although solutions have been found for many particular categories of curve, a completely solution is still lacking and it remains an active research topic. There is a good summary of the history and current state of the problem here:

https://diposit.ub.edu/dspace/bitstream/2445/151918/2/151918.pdf

Recently I remembered a problem someone showed me many years ago. Given an acute triangle can you always inscribe a square in it? The answer is yes and with the right insight it’s not hard to see why.

The idea is to begin at (a), with a small square that has three vertices on the triangle, the point P and two others on the base of the triangle. In (b) we move P along the side it lies on, while maintaining the same square configuration, and at some stage the point Q must meet the third side of the triangle (c), and we have the desired inscribed square. Rather neat.

It occurred to me that this was a special case of the square peg problem, so I went off and caught up on the latest research, and the summary I referred to above reproduces a number of proofs for different classes of curve. Most of them are quite difficult, some very, but I found one that looked as though it ought to be easy, although I still struggled with it a bit. I wanted to write something about the square peg problem and give a simple and intuitive proof for at least one case, but this one seemed to involve too much maths to be generally accessible.

After a couple of days pondering this, I woke up suddenly in the early hours of the morning with a eureka moment. The proof I'd being looking at is really just the same as the triangle problem above. There is really nothing special about the triangle, it could be any path that starts at a base level, has some ups and downs, and eventually end up at the base level again. It might be a section through a hill for example, like this:


As before we draw squares with two vertices at base level, and a third vertex P which lies on the hillside. We start with a small square, as shown in (d) and move P along the surface of the hill as in (e). In (e) point Q just misses being on the surface, but we carry on and eventually there must come a time, as shown in (f), when Q meets the surface, and we have an inscribed square. Of course we would need to tighten this argument up a bit before it became a rigorous proof, but it's basically correct and quite easy to follow.

Staying with idea of the hill section, the inscribed square means there must necessarily be two points on the hillside with the same elevation whose distance apart equals that elevation.
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Richard Walker

Ivy-leaved toadflax

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This attractive little plant likes my front garden wall, where it grows profusely. In the photo it’s intertwined with some actual ivy, at left, and you can see that the leaves of the toadflax really do look like miniature ivy leaves.


This year there is more of the toadflax than I can remember ever seeing before. Perhaps it’s because we have such a lot of rain in the last few months.

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Richard Walker

The Magic Apple Tree

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Richard Walker

Dad Joke

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🍇

If you try to steal my grapes, then you better watch Shiraz.

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Richard Walker

This Perfect Rose

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“Golden Bouquet”

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Richard Walker

The Heart Of The Peony

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Richard Walker

New blog post

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Edited by Richard Walker, Sunday, 5 May 2024, 16:49

Here's a rather neat bit of geometry. If we take a quadrilateral ABCD and join the midpoints of its sides, we get a parallelogram.

This theorem is named after Pierre Varignon (pictured, courtesy Wikipedia), a mathematician of the 17th and 18th century. 

200px-Pierre_Varignon.jpg

Varignon was well-connected; it seems he knew Newton and Leibnitz, for example.

Now why should the theorem be true? Well suppose we concentrate just on EH and FG and draw in the diagonal BD, see below


Now there is a theorem that says if we join the midpoints of two side of a triangle the segment so obtained is parallel to the third side and half its length. Looking at triangles ABD and DBC tells us are half the length of the diagonal and parallel to it. Hence HEFG must be a parallelogram.

We can also see that a half is not special, for example if F, G, H and E had been one-third the way along the sides they lie on, instead of one half, we would still have obtained a parallelogram.

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Richard Walker

Solva

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Last week I visited Solva, a place in South-West Wales. It's a fishing village and harbour. Up on the cliff overlooking the sea there is an (Iron Age?) fort, it was once an significant commrercila port and centre for lime burning, and it probably has some Viking associations. The name might be derioved from Old Norse Sol = sun and Vo/Voe whichh means inlet in English and so may have had a similar meaning in Old Norse. But the orgin of the name does not seem to be attested - there is no early written evidence - so it's hard to know for sure.

Here is a picture of the estuary by Bill Boaden.

File:The estuary at Solva - geograph.org.uk - 4604599.jpg

Here is a photo taken by one of our party, showing what it looks like from the shore with the tide out.


From Solva there is a cliff path that takes you to St David's and the cathedral of the monastery founded by the saint. It was a fine day and I would have liked to have walked it, but I am simply not mobile enough.




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Richard Walker

Another sign of Spring

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In my hedge grown these “pinkbells”, from bulbs I planted many years ago.


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Richard Walker

Midnight cherry

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Richard Walker

Ann and Bob - a paradox

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I just stumbled across this paradox which was only discovered (or invented?) quite recently. It's called the Brandenburger-Keisler Paradox and I'm still trying to get my head round it.

Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong.

Does Ann believe that Bob’s assumption is wrong?

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Richard Walker

Fritillaries

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There beautiful flowers are snakes-head fritillaries, in flower a couple of weeks early than usual.


They are widely cultivated, but also grow wild in southern and central Britain, although it is not clear whether it is native or a garden escape. See Wkipedia for more on this. 

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Richard Walker

A Square Year

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Next year will be a square year. 2025 = 45 x 45, and this is the only square year in this century, because 44 x 44 = 1936 and 46 x 46 = 2116.

By the way, here is a neat trick for squaring numbers that end in a 5. Suppose the number is X5, where X is some series of digits that form a number in their own right. Work out X times X+1, then pop 25 on the end.

Take 135 as an example. 13x14 = 182, so we get 18225, which is indeed 135 squared.

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Richard Walker

A Lakeland Home

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Richard Walker

New blog post

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Richard Walker

Word of the day

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Muscatorium: a pope’s ceremonial fan.

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Richard Walker

Banana Skin Joke

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I slipped on a banana skin. It didn’t suit me so I took it off again.

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Richard Walker

Musical Joke (sorta)

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What was Chopin‘s favourite pasta? Spaghetti Polonaise!

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Richard Walker

Ullswater 6 March

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Richard Walker

One Liner

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I saw a sign saying “Table top sale”. I thought “Strange, why don’t they sell whole tables?” 

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Richard Walker

Dad Joke

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What do you call a group of policeman standing in the middle of a field?

Copse.

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Richard Walker

What Fairy Tale?

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Richard Walker

Nalbinding

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Edited by Richard Walker, Sunday, 3 Mar 2024, 02:27

You've probably met the Zen kōan of "clapping with one hand". Well, nalbinding is knitting with one needle. It is far older than knitting and examples from 6500 BCE have been found in a Judean cave. Wikipedia is full of information about the craft [1].

Nalbinding is also known from the Viking era, and the tradition is strong today in Scandinavia. Here's an attractive example fro the late 19th century. [2]


The word means "needle binding", and the common root of "nal" and "needle" may perhaps be related to ancient Greek νήθειν = "spin". This connection is appealing but I couldn't find any strong evidence.

[1] https://en.wikipedia.org/wiki/N%C3%A5lebinding

[2] https://commons.wikimedia.org/wiki/File:1_par_vantar_n%C3%A5lbundna_av_vitt_ullgarn_med_dubbelvikt_krage_-_Nordiska_museet_-_NM.0005802B.jpg

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Richard Walker

Did you know?

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Edited by Richard Walker, Saturday, 24 Feb 2024, 00:54

The city of Melbourne Australia was, for a short time in the early 19th century, known as Batmania? Named after someone called… Batman.

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Richard Walker

My Dwelling

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I live in a very modest house. It doesn’t like talking about itself.

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