Sheep can be trained to recognize human faces, even from photographs. In a 2017 study at Cambridge University, researchers trained sheep to recognize photographs of four celebrities (Fiona Bruce, Jake Gyllenhaal, Barack Obama, and Emma Watson). They learned to distinguish a celebrity’s face from another face 8 out of 10 times, and their performance dropped only about 15% when they were shown photographs taken at an angle. When an unfamiliar photograph of a handler was inserted randomly in place of a celebrity, they chose that photo 7 out of 10 times.

“During this final task the researchers observed an interesting behaviour. Upon seeing a photographic image of the handler for the first time — in other words, the sheep had never seen an image of this person before — the sheep did a ‘double take’. The sheep checked first the (unfamiliar) face, then the handler’s image, and then unfamiliar face again before making a decision to choose the familiar face, of the handler.”

Sheep are long-lived and have relatively large brains, so it’s hoped that studying them will shed light on illnesses such as Huntington’s disease.

Also: Pigeons can distinguish Monet from Picasso, and rats can distinguish spoken Japanese from spoken Dutch. “A previous study by Porter and Neuringer (1984), who reported discrimination by pigeons between music and Bach and Stravinsky, and the present study suggest that pigeons have abilities that enable them to identify both musical and visual artists.”

Many people know that you can form a pentagon by tying a strip of paper in a simple overhand knot.

Stephen Bleecker Luce’s seamanship manual of 1863 tells how to fold a blade of grass into an octagon (below):

It is first doubled short over itself, then 1 under 2, — leaving a space, then 2 over 1, and down through the centre of the triangle; next 1 over 2, and down through the centre, coming out on the opposite side, and so on until an octagonal figure is formed.

That’s pretty terse, but I think I’ve almost managed to do it tonight. Keep your eye on the drawing of the finished piece, and don’t form and flatten the finished shape until you’ve done all the weaving.

(Via The Ashley Book of Knots.)

The sum or difference of any pair of the numbers {150568, 420968, 434657} is a square:

420968 + 150568 = 756²
420968 – 150568 = 520²
434657 + 420968 = 925²
434657 – 420968 = 117²
434657 + 150568 = 765²
434657 – 150568 = 533²

In the summer of 1997 mathematician Lionel Levine discovered a sequence of numbers: 1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, … . It’s made up of the final term in each row of this array:

1 1
1 2
1 1 2
1 1 2 3
1 1 1 2 2 3 4
1 1 1 1 2 2 2 3 3 4 4 5 6 7
1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 9 10 10 11 12 13 14
…

The array is built from a simple rule. Start with 1 1 and regard each line as a recipe for building the next one: Read it from right to left and think of it as an inventory of digits. The first line, 1 1, would be read “one 1 and one 2,” so that gives us 1 2 for the second line. The second line (again, reading from right to left) would be read “two 1s and one 2,” giving 1 1 2 for the third line.

And so on. That’s it. It’s excruciatingly simple, “yet it seems likely that the 20th term say is impossible to compute,” wrote N.J.A. Sloane that November. At that time only 15 terms were known, the last being 508009471379488821444261986503540. Two further terms have since been found, bringing us up to 5347426383812697233786139576220450142250373277499130252554080838158299886992660750432. And the 20th is still out of sight.

See the link below (page 15) for more.

(Neil J.A. Sloane, “My Favorite Integer Sequences,” in C. Ding, T. Helleseth, and H. Niederreiter, Sequences and Their Applications: Proceedings of SETA ’98, 2012, 103-130.)

In 2004, Canadian musician Andrew Huang wrote a song that encodes the first 101 digits of π.

Also: A “piku” is a haiku whose word lengths reflect the digits of π:

How I love a verse
Contrived to unhusk dryly
One image nutshell

A set of points has diameter 1 if no two points in the set are more than 1 unit apart. An example is an equilateral triangle whose side has length 1. What’s the smallest shape that can cover any such set? A circle of diameter 1 won’t cover our triangle; part of the triangle projects beyond the circle:

Of course a larger circle would work, but what’s the smallest shape will always do the job? Surprisingly, no one knows. When French mathematician Henri Lebesgue posed the problem to Gyula Pál in 1914, Pál suggested a modified hexagon (in black):

Here Pál’s shape manages to surround a circle (blue), a Reuleaux triangle (red), and a square (green), each of diameter 1, and in fact it will accommodate any such set. Its own area is 0.84529946. Will a smaller shape do the job? Well, yes, but the gains get increasingly fine: In 1936 Roland Sprague whittled Pál’s shape down to 0.844137708436, and in 1992 H.C. Hansen reduced it further to 0.844137708398. At this point observers Victor Klee and Stanley Wagon wrote, “[I]t does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.” But in 2015 John Baez reached 0.8441153 with an exquisite adjustment to two regions in Hansen’s shape; the smaller of these would span only a few atoms if the shape were drawn on paper.

Is that the end of the story? No: Last October Philip Gibbs claimed a further reduction to 0.8440935944, and the search goes on. In 2005 Peter Brass and Mehrbod Sharifi showed that the universal cover must have an area of at least 0.832, so there’s room, at least in theory, for still further improvements.

(Thanks, Jacob.)