Happy 2025, y’all!

As you may know, I love playing around with numbers. Not necessarily doing long, involved mathematical proofs — I was never great at that sort of thing — but more just poking around with series of numbers like Fibonacci’s sequence or fun things like primes and pentagonal numbers and the like.

As it happens, 2025 is a pretty cool number. It has a lot of factors (numbers that when multiplied together give you the original number) and partitions (the same as factors but for addition). It’s also a perfect square! 45 x 45 = 2025. 

But breaking that down a little more generates some nifty stuff. 45 is equal to 9 x 5, so 45² = 9 x 5 x 9 x 5, which can be rearranged to 9 x 9 x 5 x 5, which in turn means 2025 also equals 9² x 5² (or 81 × 25), making it the product of two squares.

It’s also the sum of three squares: 40² + 20² + 5².

Here’s where it gets cool. 45 is the sum of the first nine integers (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9), so 2025 is equal to the square of the sum of the first nine integers:

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)²

But it’s also equal to the sum of the first nine integers cubed:

2025 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ 

WHAT.

Work it out if you don’t believe me! But it’s weird to me that you can add a series of consecutive integers starting at 1, square that sum, and it’s equal to adding up the cubes of those individual integers too (I first learned this just the other day in a random TikTok video). But it turns out this is a well-known idea called Nicomachus's theorem. The proof is not too hard if you know a bit of high school math, but too long to reproduce here. That link explains it. It also deals with triangular numbers, which I also enjoy playing with. Basically, I have fun seeing patterns in numbers, and it’s amazing how often that kind of stuff turns up.

What does this mean for the coming year of 2025? Nothing, really, unless you’re into numerology, which you shouldn’t be because it’s pseudoscience. But the math is still fun.

And I hope you have a great 2025… no matter how you add it up.

Ultra-compact dwarf galaxies are a bit of an enigma. They’re tiny, only a few hundred light-years across, but very dense, packed with many millions of stars in them. They resemble globular clusters or tidally stripped dwarf galaxies, but it’s not clear how they form, or if there’s just one process that makes all of them. Maybe there are a lot of pathways to their formation, and we don’t know them all yet.

New research has found an interesting way to make them [link to journal paper].

Clusters of galaxies are collections of hundreds or thousands of galaxies living in close proximity. There’s gas in between the galaxies, and as the galaxies plow through that gas the pressure can strip away the gas internal to the galaxy (as I’ve written many times, it’s like opening up a window to let the wind air out your car after your dog or other travel companion makes an unfortunate smell).

This creates long, thin tendrils of gas that extend away from the galaxy downwind, and in fact these kinds of galaxies are seen and are called jellyfish galaxies. The tendrils of gas extend into intergalactic space. But what happens to them?

Using a very sophisticated simulator, astronomers modeled a cluster in a computer using known physics, let the galaxies move around, and then identified jellyfish galaxies within. Zeroing in on one that had the right characteristics, they found that in that case, the gas blown out formed stars in clumps, and at a high rate. These clumps formed tiny, dense galaxies on their own that look a lot like ultra-compact dwarfs (or UCDs). Even better, they found that the stars in the original galaxy don’t get pushed into the clump by the wind, which means any stars seen in the UCD were born there. It’s a brand new teeny galaxy! 

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