I'll begin by saying what the Three Gap Theorem is. I'll show some examples and then explain a simple way you can play with the theorem using only paper and pencil. After that I'll outline the theorem's history, and lastly I'll explain how the graphics were produced

Theorem

Draw a circle and mark a point anywhere on its circumference. Then choose an angle and rotate the point through that angle about the circle's centre. 

Mark this point, then repeat the process, using the same angle, to get a third point. Carry on in this way, always rotating the current point through the same angle to get the next point.

Eventually the total angle travelled will exceed a full turn and the circle will be traversed a second time. As long as the angle chosen was not a simple fraction of a whole turn, the new points will not coincide with existing points but be new points, each splitting the gap between two points that were formerly adjacent into two smaller gaps.   

We can carry on indefinitely and each time we go round the circle again new points will split existing gaps into gaps of smaller size. This will lead to an increasingly complex situation and we naturally expect each successive snapshot to contain more and more different gap sizes

But it doesn't!. The number of distinct gaps is only ever two or three, never more. Moreover if the number of gap sizes is three, the size of the largest gap will be the sum of the other two gap sizes.

This is the Three-Gap Theorem.

Examples

These examples were generated by a Python program which I 'll say more about later. If you zoom the browser in you can see the number of points I stopped at in each and the gap sizes in decreasing order. It's interesting that, for example, from 3 to 4 points increases the number of gap sizes from two to three, but then from 4 to 5 it drops to two again. Another interesting feature is that in every case there appears to be a two-fold symmetry.

History

The behaviour of the gaps was described by the noted mathematician Hugo Steinhaus in 1957 but he only conjectured that the number of distinct gaps never exceeded three. The theorem was first proven in 1958 by Sós, Surányi and Świerczkowski, and since then different proof have been found, although none that seem to me very simple. The area is still the object of active research; for example I believe the two-fold symmetry was provem fairly recently and at least one researcher is studying whether generalizations exist to e.g. three dimensions.

Pencil and paper investigation

You don't need anything complicated to play around with these gaps, just pencil and paper. I drew a circle round a large cup, then marked made two marks on a strip of paper and stepped off the points round the circle., which as easy to do and worked quite well I felt.

You can see three distinct gap lengths. I think Steinhaus must have been playing around in a similar way when he made his discovery, don't you?

Computing

I'd never heard of thie Three Gaps Theorem until a few days back, but when I stumbled across it, I thought it was really surprising and fascinating. So first I knocked up a quick and dirty Python program, which simulated moving round a circle in same size steps as described above. Once I had made a list of the points I calculated the gaps and sure enough there were only two numbers. After I tweaked the numbers a bit I managed to get three numbers sometimes, but never more. So my experiments seemed to corroborate the theorem.

But I wanted something a bit more dramatic to illustrate it and although I once wrote a Python program to draw pie charts doing it all myself by hand requires too much heavy lifting and so I asked 'Google AI' to produce a program. I was very specific about what I wanted and I spelt out things like the colours and what we were trying to simulate. The AI was aware of the theorem so that helped a lot.

I asked for a legend that would list the gap sizes and specified that I wanted the gaps shown by 'pizza slices', i.e. coloured sectors, because I thought would stand out a lot better than just using differently coloured circular arcs

On the first attempt, the AI made a mistake and we got a Python error so I told it what the error message was and it diagnosed what had done wrong and corrected it and now it worked.

However the colours were a bit wishy-washy and I wanted something a bit more dramatic so I managed to get it to change the colours and then I had the program that generated the pictures that I've shown earlier in this post.

I thought the AI did a good job and it wrote code in a couple of seconds that would've taken me probably a couple of hours at a minimum so I thought as a productivity aid it was quite impressive. I thought it is very interesting that it not only corrected its mistake, but explain to me what it had done wrong.

I hope you found this introduction to the Three Gap Theorem interesting, and if you've done the pencil and paper exercise, you may have got some intuition about why the theorem is true.

That's very far from a proof of course, and at the moment I didn't feel that I understood any proof well enough to try and sketch it out in a block post, but I'm working on it. Hopefully, I'll be able to come up with something accessible and convincing!

Problem

Given an arbitrary complex quadrilateral ABCD, draw lines BE and DF from vertices B and D to the midpoints E and F of the opposite side. This divides the quadrilateral into two triangles, shaded blue, and a quadrilateral (shaded pink).

The question asked which is greater: the proportion of the quadrilateral that is shaded blue or the proportion that is shaded pink.

Solution

If a question asks 'Which is greater, X or Y?' the answer is often neither, they are the same, and so it is here. To see this, we add some extra lines.

Consider triangles ABE (blue) and EBD (pink). They share the same height GB and because E is the midpoint of AD they have the equal base lengths. So they must have equal areas.

A similar argument applies to triangles BDF (pink) and FDC (blue) and shows their areas are also equal.

In both cases the blue and pink areas are equal and therefore the overall proportions shaded pink and blue respectively must be the same.

In quadrilateral ABCD, E and F are midpoints of the sides they lie on. Which colour covers the greater area, the red or the green?

(From 7.1 in Charming Proofs, by Alsina and Nelsen, 2010)

You've probably heard of the famous Golden Ratio, . It satisfies the equation , which means that if we have a rectangle we can cut off a square and be left with a smaller rectangle whose sides are in the same ratio as the original rectangle. 

This works because the ratio is equal to , by virtue of the equation given above.

The golden ratio has many interesting and important mathematical properties and also crops up in art, architecture, music and many other places in science and culture. The number of words written about it must be in the hundreds of millions and the subject too vast even to survey in this blog post. I just want to look at two methods of constructing with ruler and compasses. The first is from Euclid ca. 300 BCE. Here is his construction.

He starts with a square, draws a line from the midpoint of its base to an opposite corner, then draws a circular arc to intersect the line formed by extending the base. The ratio between the length of the extended base and that of the original square is then , as can be shown using Pythagoras.

The second construction was discovered by the 20^th century designer and mathematical enthusiast George Odom. Here it is, very simple, and has the additional elegance of including an equilateral triangle, one of my favourite polygons.

Inscribe an equilateral triangle in a circle. Draw a line through the midpoints and of two of its sides, to meet the circle at . Then . Isn't that neat? Odom must have been very please to discover it. It was later published in the American Mathematical Monthly as problem E3007.

I will post my solution on Friday. In the meantime if you want to have a go please do put solutions whole or partial in the Comments. It would be interesting to see what ideas and thoughts people have.

It's well known that Pumpkin Seed oil is dichromatic - a thin layer of it is a greenish-yellow but a thicker layer is a dark red.

Yesterday I noticed that another oil seems to have a similar property. There was a tiny drop of Rape-seed left in the bottom of the bottle and I noticed it appeared green, although in bulk the oil was not green at all but a golden yellow. Here it is - definitely green.

Here is what a bottle of this oil looks like when full - definitely yellow.

What's going on, why is the colour different? I think the explanation, very roughly, is that the light passing through the oil is made up of different colour components. The light is partially absorbed by the oil but some colours are absorbed more than others. If the light has to travel a significant way through the oil the blue and some of the green get preferentially absorbed and the remaining components are perceived as yellow. But if we only have a thin layer most of the blue and green survives and now the colour we perceive is green.

I first learned about the properties of Pumpkin Seed oil in 2021 and wrote a blog post about it then/ I bought some Pumpkin Seed oil back then and tried it some experiments. Here's a photograph I took. I hope you can see that at left where the oil is deeper it has a dark red hue, whereas at right where is only a thin oil layer the colour is green/

A few days back I posted a geometry problem, Question 1182471 from Math Stack Exchange, and then have a proof in a follow-up post. My proof was correct, I believe, and a decent effort , I thought.

But I must still have been thinking about it subconsciously, because yesterday I suddenly saw a much shorter and simpler solution, one that is also more insightful because it throws up an interesting link to optics, to electronics, and to the infamous Crossed Ladders Problem †

Here's the problem amd the (new) proof.

Now when I got to the equation at the end I realised I had seen it before! In optics it comes up in the Lens Formula, in electrical circuits it is the rule for the resistance formed by parallel resistors, and in the crossings ladders puzzle it is the height of the point where two ladders cross, expressed in terms of the heights the ladders reach up the walls of an alley.

The equation is also related to the Harmonic Mean, which has many uses, and to things like Egyptian fractions.

† Look it up. A problem that sees easy at first but isn't, it leads to 4^th degree equations.

A friend has a rowan tree, also called the mountain ash, in her garden, which set me wondering (a) if I could get hold of some rowanberry jelly, just to try; and (b) where name rowan comes from.

The answer to (a) is no, unless I make myself or pay an exorbitant delivery charge. The answer to (b) is that it's probably named for the bright red colour of its berries. The immediate origin seems to be Scandinavian (the Swedish is rönn) but there is a suggestion that it may go back ro the Proto-Indo-European root *ruidh-, 'red'.

This is the PIE colour word we are most confident of, because words that can be traced back to it, with meanings related to 'redness', are found in so many branches of the Indo-European language family. Some examples are English ruddy, Sanskrit rudhira, Polish rudy, Welsh rhudd, Lithuanian raudona, Latin rufus, Greek erythros.

Other English cognates include red, of course; rust, russet, rouge, roan and ruby. A particularly interesting one is ruddock - Britain's favourite bird, known today as the robin, but called in Old English rudduc, 'little red'. Here's a later example I found in the Middle English Compendium (sote = 'sweet' and the two thrushes are the song and the missel.)

How sote this seson is..The thrustelis & the thrusshis..The ruddok & the Goldfynch. 

Picture credit: George Chernilevsky, Vinnytsia, Ukraine, Rowan Berries, on Wikimedia, Licensed under Creative Commons.

Here's a nice problem I found on math stack exchange (question 1182471).

ACE is a straight line and triangles ABC and CDE are equilateral. Prove that CP = CQ.

If you have a solution do put in the comments. I imagine there are a number of different solutions. I'll post mine this Monday coming.

Packing problems ask questions like 'What is the optimal way to pack copies of shape A in shape B?' For example we might be trying to pack equilateral triangles in a square, and the best known arrangement is this. 

We can think of the problem in two equivalent ways

• Choose a size for the square and make the triangles as big as possible
• Choose a size for the triangles and make the square as small as possible

The example above, which I drew in GeoGebra, has been known since 1996 and was discovered by Erich Friedman. It's pleasingly symmetric and you might expect symmetry is the norm. But it absolutely isn't. As increases every new number has its own idiosyncratic pattern, typically somewhat chaotic but with patches of local orderliness. For instance here is the best known arrangement for .

It was discover this month (May 2026) by Emerson Connelly and I found it on the legendary site Erich's Packing Center, maintained by Erich Friedman for the last 30 years. Erich's site has pages for dozens of different combinations. (Pentagons in Dominoes is one I rather like.)

I've known about this site for ages but when I revisited it yesterday I was astonished to find we have just entered a sort of Golden Age of Packing Problems. Of the 45 arrangements of equilateral triangles in squares found on the site, stretching back to 1996, 27 (60%) have been discovered in the last two months, April and May 2026! That's about one every other day.

And it's not just triangles in squares. Across the many different combinations of shapes hosted on Erich's site 46 have seen updated records in 2026, nearly all in that same two months. 

What accounts for this huge upsurge? I asked Copilot and it trawled the internet and proposed a combination of several factors

1. Faster and more sophisticated search algorithms
2. Dramatically greater computing power
3. Widespread parallel experimentation
4. Increasing use of AI/heuristics
5. Rapid online collaboration
6. A problem structure that rewards brute-force discovery

In some ways the problem resembles the search for bigger and bigger prime numbers. Packing problems have practical importance in manufacturing and transport (and prime numbers have practical importance in cryptography) but the search for new records and the application of such extraordinary human ingenuity and massive computing power is not motivated from practical considerations at all. I guess it comes to the spark of human curiosity and the attraction of a challenge.

And perhaps these records are a little like athletic ones; there is scope for a potentially endless series of small improvements and we always strive to reach them.

In 1999 Zoltán Füredi and John E. Wetzel, two covering problems meisters, found a triangle with a remarkable property.^[1]

It can cover^[2] each and every triangle of perimeter 2. It is the smallest region (not just the smallest triangle) that can do this and it is unique. I made a drawing of it using GeoGebra and fitted some sample triangles with perimeter 2 inside it

The length of is , , the length of is and the perimeter of is about .

[1] The smallest convex cover for triangles of perimeter two, Geometriae Dedicata, 2000

[2] To be precise, it can cover a congruent copy of any such triangle.

Can you match each Old English name to the right Dinosaur?

Remember Thorn þ is a 'th' sound, as in 'thin'.

See Comment for information on the Od English versions.

On forums I sign myself like this

Rich 🙂

... but the platform button often suggests a different emoji

🤑 

I was puzzled by this when I first saw it but eventually realised 'rich' is the link; the emoji is called 'money-mouth face', a symbol of wealth (and greed). 

'But rich is a different word', I thought to myself, 'and nothing to do with my name.'

Except... I was wrong. It has, it's fundamentally the same word, but has reached us by two different routes, and thus ended up as two words with different meanings. 

The name Richard means something like 'strong ruler', from Germanic words ric, 'ruler' and hard, 'strong'.  We see the ric element in other names, such as Eric, 'ever ruler' and Wulfric, 'ruler of wolves'. It also survives in the sense of a domain, as in bishopric.

The ultimate origin is the Proto Indo-European (PIE) stem *reg-, which had the sense of being direct and then from that of imposing order, ruling, reigning over, leading, and so on. 

The same PIE stem *reg- gave the Celtic languages rix, 'king', and modern Gaelic still has ri or righ (pronounced 'ree'). The ancient Gauls who Julius Caesar conquered spoke a Celtic language, Gaulish, and their version was rix. This was an early borrowing into Germanic and came down to us as rich.

There may have been influence from French riche (itself a borrowing from Frankish) but from early Middle English on its meaning was widened to magnificence (think 'richly dressed') and nowadays the predominant meaning is wealthy of course. 

And now we come to the freedom fighter. Although familiar with the history I literally had no idea of the etymological connection until I started writing this post

The most famous of the Gauls was Vercingetorix, something like 'great king over fighters', who led a revolt against the Roman rule imposed by Julius Caesar but in the end was forced to surrender. In the modern era he has become a symbol of French spirit and resistance to foreign invasion. Here's an iconic statue to him. See the Comments for more details.