Court Order

From Chapter 12 of Ken Follett’s novel The Pillars of the Earth:

‘My stepfather, the builder, taught me how to perform certain operations in geometry: how to divide a line exactly in half, how to draw a right angle, and how to draw one square inside another so that the smaller is half the area of the larger.’

‘What is the purpose of such skills?’ Josef interrupted.

‘Those operations are essential in planning buildings,’ Jack replied pleasantly, pretending not to notice Josef’s tone. ‘Take a look at this courtyard. The area of the covered arcades around the edges is exactly the same as the open area in the middle. Most small courtyards are built like that, including the cloisters of monasteries. It’s because these proportions are most pleasing. If the middle is bigger, it looks like a marketplace, and if it’s smaller, it just looks as if there’s a hole in the roof. But to get it exactly right, the builder has to be able to draw the open part in the middle so that it’s precisely half the area of the whole thing.’

How is this done? Inscribe a diamond within a square and then rotate it 45 degrees:

court order

Langley’s Adventitious Angles
Image: Wikimedia Commons

Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:

ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.

(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)

A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.

Even Up

rolling die puzzle

Suppose we cover a chessboard with 32 dominoes so that each domino covers two squares. What is the likelihood that there will be an even number of dominoes in each of the two orientations (horizontal and vertical)?

In fact this will always be the case. Consider the 32 squares in the odd-numbered horizontal rows. Each horizontal domino on the board covers either two of these squares or none of them. And each vertical domino covers exactly one of these squares. So the horizontal dominoes cover an even number of these squares (call it n), and the number of squares remaining in this group (32 – n) must also be even. This latter number is also equal to the number of vertical dominoes, so both quantities are even.

(By Vyacheslav Proizvolov.)


The young specialist in English Lit … lectured me severely on the fact that in every century people have thought they understood the Universe at last, and in every century they were proved to be wrong. It follows that the one thing we can say about our modern ‘knowledge’ is that it is wrong.

… My answer to him was, ‘… when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.’

— Isaac Asimov, The Relativity of Wrong, 1989

(J.R. Deller Jr. wrote, “Education is the process of telling smaller and smaller lies.”)


“It was hard for me to believe. I would look down and say, ‘This is the moon, this is the moon,’ and I would look up and say, ‘That’s the Earth, that’s the Earth,’ in my head. So it was science fiction to us even as we were doing it.” — Alan Bean, Apollo 12

The Cat Gap

The first “true cat,” Proailurus, or “Leman’s Dawn Cat,” appeared about 30 million years ago. But from 25 to 18.5 million years ago, strangely few catlike fossils are found in North America. Biologist Luke Hunter writes:

Following the appearance of the dawn cat, there is little in the fossil record for 10 million years to suggest that cats would prosper. In fact, although Proailurus persisted for at least 14 million years, there are so few felid fossils towards the end of the dawn cat’s reign that paleontologists refer to this as the ‘cat gap’. The turning point for cats came about with the appearance of a new genus of felids, Pseudaelurus.

The gap may be due to changes in climate and habitat, the rise of competing doglike species, an unsustainable “hypercarnivorous” dietary specialization, or some other factor. Modern cats descended from Pseudaelurus.

Kitchen Aid

Three male offspring, aged 9–14 years, of one of the authors were observed to experience visual problems profound enough to imply functional blindness. The visual deficit was evident on almost every occasion when any one of the children of this physician went to the refrigerator and opened the door. The acute visual problem encountered was noted to be part of a consistent behaviour pattern, wherein a few seconds after the fridge door was opened a cry would be heard from the affected child of ‘Mum, where’s the milk?’

— Andrew J. Macnab and Mary Bennett, “Refrigerator Blindness: Selective Loss of Visual Acuity in Association With a Common Foraging Behaviour,” Canadian Medical Association Journal, Dec. 6, 2005

Benedetti’s Puzzle

This is interesting: In 1585, Italian mathematician Giovanni Battista Benedetti devised a piece of music in which a precise application of the tuning mathematics causes the pitch to creep upward.

Avoiding this phenomenon requires an adjustment — a compromise to the dream of mathematically pure music.

Berkson’s Paradox
Image: Wikimedia Commons

Suppose that there’s no correlation between talent and attractiveness in the general population (left). A person who studies only celebrities might infer that the two traits are negatively correlated — that attractive people tend to lack talent and talented people tend to lack attractiveness (right). But this is deceiving: People who are neither attractive nor talented don’t typically become celebrities, and that large group of people aren’t represented in the sample. Celebrities tend to have one trait or the other but (unsurprisingly) rarely both.

The phenomenon was studied by Mayo Clinic statistician Joseph Berkson; this example is by CMG Lee.

Something New
Image: Wikimedia Commons

In June 2006, Iowa paralegal Jane Wiggins looked out the window of her Cedar Rapids office and saw a cloud unlike any she’d seen before. “It looked like Armageddon,” she told the Associated Press. “The shadows of the clouds, the lights and the darks, and the greenish-yellow backdrop. They seemed to change.”

Wiggins sent a photo to the Cloud Appreciation Society, a weather-watching group founded by Gavin Pretor-Pinney, author of The Cloudspotter’s Guide. Other sightings were registered around the world (this one appeared over Tallinn, Estonia), and eventually Pretor-Pinney nominated it as an entirely new type.

The 2017 edition of the World Meteorological Organisation’s International Cloud Atlas included asperitas in a supplementary feature. The name is Latin for “roughen” or “agitate” — “not necessarily gentle or steady, but quite violent-looking, turbulent, almost twisted in its appearance,” Pretor-Pinney said.

It’s not new, really — such clouds have always been up there — but it’s the first formation added to the atlas since 1951. “We like to believe that just about everything that can be seen has been,” Society executive director Paul Hardaker said. “But you do get caught once in a while with the odd, new, interesting thing.”