A Mathematician's Guided Tour Through Higher Dimensions

September 26, 2021 6:09 AM

The Journey to Define Dimension - "The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it."

also btw...

• The four-dimensional life of mathematician Charles Howard Hinton - "From bigamy to belief in ghosts, Hinton was no ordinary mathematician." (previously)
• How to understand Einstein's equation for general relativity - "Mathematically, it is a monster, but we can understand it in plain English."[1]

posted by kliuless (8 comments total)

Ow! This hurt Hulk's head.
posted by y2karl at 11:41 AM on September 26, 2021

The first article was really interesting. I got lost in the middle, but the visualizations make you think you can almost grasp the concept and fractals are always awesome. There is quite a lot buried here: "However, in 1877 he discovered a one-to-one correspondence ...Intuitively, he proved that lines, squares and cubes all have the same number of infinitesimally small points, despite their different dimensions. "

From the third article:
"Yet despite its success over more than 100 years, almost no one understands what the one equation that governs general relativity is actually about. Here, in plain English, is what it truly means." I know all the words in this article, I read it all and do not understand any of it, but I appreciate the effort. (Haha, autocorrect turned 'appreciate' into 'survived', also accurate.) There seems to be a lot I've forgotten since college physics, most of this doesn't even ring a bell, but fascinating!
posted by lemonade at 12:04 PM on September 26, 2021

Intuitively, he proved that lines, squares and cubes all have the same number of infinitesimally small points, despite their different dimensions.

The proof (at least the one I know) for this is so easy that it’s hard to imagine it ever could have been in doubt; yet it wasn’t clear until it was proven. The thing about standing on the shoulders of giants is, it does let you see further, but sometimes it means you can no longer see the ground.
posted by Joe in Australia at 4:31 PM on September 26, 2021

This is not the behavior we would want for a coordinate system; it would be too disordered to be helpful, like giving buildings in Manhattan unique addresses but assigning them at random.

Oh, like in Prague?

Then, in 1890, Giuseppe Peano discovered that it is possible to wrap a one-dimensional curve so tightly — and continuously — that it fills every point in a two-dimensional square. This was the first space-filling curve. But Peano’s example was also not a good basis for a coordinate system because the curve intersected itself infinitely many times; returning to the Manhattan analogy, it was like giving some buildings multiple addresses.

Oh, like in Prague?
posted by aws17576 at 7:52 PM on September 26, 2021 [7 favorites]

I still say that if you want to learn yourself a good deal of basic physics that the c. 1985 Caltech series Episode 1: Introduction - The Mechanical Universe - YouTube is a great way to start.

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.

Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor.

I watched most of that at 15 before spending the next summer at Caltech cramming in three semesters worth of physics/calculus (that whole three volume green book with the wavy yellow lines on the front). It's a well done sorta like Cosmos or NOVA public television general audience targeted series.

Off to see if the High Dimensions article gets to hedgehogs and infinitely big but still nowhere to go cold dark hell of being trapped as a unit N-sphere inside an N-cube as N approaches infinity.

(graar, it's supposed to be #mathmonday)
posted by zengargoyle at 9:03 PM on September 26, 2021 [5 favorites]

Zengargoyle I think I would have remembered the hedgehogs, but your beloved unit n-sphere/n-cube definitely are there. Thanks for the recommendation.
posted by lemonade at 11:37 PM on September 26, 2021

Any n-dimensional spherical cows? The teseract is mind twisting enough, a 4D sphere (or torus, omg) is just... just.
posted by sammyo at 3:45 AM on September 27, 2021

As someone who regularly attempts to teach the principles of general relativity to students who will never use it to solve problems, I'm impressed by the last article on Einstein's field equations. So much so that I'm probably going to make it assigned reading.

I'd be very curious to hear from anyone who has strong opinions about it as an introduction for people who may not have thought much about the topic before.

(The rest of the post was also neat!)
posted by eotvos at 7:47 AM on September 29, 2021