When bad news gets me down, I often get insomnia. I wake up in the middle of the night, start thinking about how we're all doomed, and can't easily stop. To break out of these doom loops, I do elaborate visualization exercises. They don't really put me to sleep, they just calm me down. Then later I can fall asleep.
Here's what I've been doing this week. I visualize this shape made of two interpenetrating tetrahedra, called the 'stella octangula' or 'stellated octahedron'. Notice that these two tetrahedra are 'dual': each corner of the yellow one is above the center of a triangle in the red one, and vice versa.
Then I imagine the yellow tetrahedron moving 'up' into the 4th dimension while the red one moves 'down'. At some point the distance between each corner of the yellow tetrahedron to the 3 nearest corners of the red one equals the distance between any 2 corners of the yellow tetrahedron. Then I've got a 4d shape called the 'cross-polytope'. All its faces are regular tetrahedra.
There are easier ways to think about the cross-polytope, which is one of the six 4-dimensional regular polytopes. So the real challenge is to visualize how this way of getting it leads to the same result.
My go-to way to think about the cross-polytope is to imagine the 4 coordinate axes in 4-dimensional space and put two dots on each axis, one unit away from the origin in each direction:
These are the vertices of a cross-polytope. It's the 4d analogue of an octahedron. Just as the octahedron has equilateral triangles as faces, this guy has regular tetrahedra as faces. Can you see those tetrahedra in this picture, and count them? Don't worry — if you're too busy now, you can do it when you're lying in bed at 3 am thinking about global warming and the decline of democracy. Start by visualizing this picture:
But the hard part is to visualize the cross-polytope this way and then rotate it in your mind so you see it as made of two tetrahedra, one red and one yellow, and an edge connecting each vertex of the red one to the 3 nearest vertices of the yellow one. That's been keeping me busy all week.
By the way, everything I just said has a 3d version! The 3d analogue of the cross-polytope is a regular octahedron. The corners of a regular octahedron are
But here's a flattened picture of an octahedron:
See the two interpenetrating equilateral triangles? If you move one up, and move one down, they can become two opposite faces of a regular octahedron.
Actually this sort of trick works in any dimension. Take two regular \(n\)-simplexes, dual to each other and interpenetrating; then move one 'up' into the \((n+1)\)st dimension, and the other 'down'. At some point their vertices will be the vertices of an \((n+1\))-dimensional cross-polytope. In 3 dimensions this is easy for me to visualize, while in 4 dimensions I can just barely visualize it... though it's getting easier every night.
August 16, 2025
Astronomers have found a truly huge black hole! It's in the massive galaxy in the center here, called the Cosmic Horseshoe. The blue ring is light from a galaxy behind the Cosmic Horseshoe, severely bent by gravity.
This black hole is 36 billion times the mass of the Sun. It's not just 'supermassive': any black hole over 10 billion times the Sun's mass is considered 'ultramassive'. Not many have been found.
To me the coolest part is that the Cosmic Horseshoe has swallowed all the other galaxies in its group: it's part of something called a 'fossil group', which I hadn't heard about until today.
Our galaxy, the Milky Way, is part of a group too: the Local Group. Many smaller galaxies have already fallen into ours. Eventually the Milky Way and Andromeda may collide and form a single bigger galaxy. This will be an elliptical galaxy — too disorganized to have spiral arms. So it's not surprising that if you wait long enough, galaxy groups form a single big elliptical galaxy which eventually eats the rest.
Some ancient galaxy groups have already done this, and they're called 'fossil groups'. The Cosmic Horseshoe is the big bully in this particular fossil group: it's 100 times heavier than the Milky Way. It's surrounded by a halo of very hot gas, 10 million Kelvin, emitting lots of X rays. But most of its mass can't be explained by stars, gas and dust, so we say 90% is dark matter. All this is completely typical of a fossil group, except the Cosmic Horseshoe is bigger than average.
Fossil groups show us what the future will be like. Big galaxies will eat the rest, and big black holes at the center of these galaxies will eventually eat most of the matter. Dark matter — whatever that is — takes longer to fall in. But there's plenty of time.
Here's the new paper about this ultramassive black hole. Luckily, it's free to read:
Below is a picture from this paper. The blue blob in a box at left is called the 'counter-image' of the galaxy that's making the Einstein ring. When light from behind a massive body is bent by gravity, it often forms a ring, called an 'Einstein ring', together with a counter-image.
I've always been fond of cryptozoology — the study of probably mythological animals like the Loch Ness monster — but I'm happiest when these creatures turn out to be real, or at least based on real animals. So I'm surprised I'd never heard about the Cat Sith and the Kellas cat.
The 'Cat Sith' was a creature from Celtic mythology. It looked like large black cat with a white spot on its chest. But when it thought no humans were watching, it would walk on its hind legs. They were fierce, and would attack at the slightest provocation. On Samhain — the original name for Halloween — they would roam the land at night. Wise families would leave a bowl of milk out for them.
But as Christianity took hold, the Cat Sith's identity began to change. Instead of being a fairy cat, the Cat Sith became a witch who could shapeshift into a cat. She could do this 8 times — but if she did it a 9th time, she'd be stuck in that form forever. This may be where the myth that a cat has 9 lives comes from.
By the 20th century, few believed in the Cat Sith... until they found one! Now it's stuffed in the Zoology Museum in Aberdeen. See the picture.
It's a large black cat with a white spot on its chest. Others have been found. They're called 'Kellas cats". They're a hybrid of the domestic cat and the Scottish wildcat, a critically endangered population of the European wildcat Felis silvestris.
Scottish wildcats have been present in Britain since the early Holocene, when the British Isles were connected to Europe by Doggerland. By 1915, they were found only northwestern Scotland. But now their range has increased, and they're being bred in captivity and released!
This is a great intro to the myth and reality of the Cat Sith:
Newcastle writes:
In 1985, Ronnie Douglas, a gamekeeper in Kellas, Moray, was stunned to find a large, black cat with a white chest patch in one of his snares. About a year later, a live one was caught by the Tomorrows World team. Soon, a total of seven additional specimens were collected by alien big cat researcher (in this case, "alien" as in " not supposed to be from around here,” not as in, "extraterrestrials made a pit stop here so their pets could go to the bathroom”) Di Francis, who gave them all to the National Museum of Scotland. There, studies revealed that some of the "Cat Siths” were actually a cross between a domestic cat and a Scottish wildcat. They were then named the Kellas cat by cryptozoologist Karl Shuker after the village where the first one had been found."
Also check out the Wikipedia articles on the Cat Sith, the Kellas cat and Scottish wildcat:
The Scottish wildcat is not black. In fact it looks a lot like a big tortoise-shell cat — see below. So are Kellas cats just Scottish wildcats that have interbred with black domestic cats?
The Scottish Wildcat Conservation Action Plan has launched a captive breeding program. In 2023, captive-bred wildcats were released into the Cairngorms, and by spring 2024 at least two of the cats released the previous year gave birth to kittens.