Romanesco Broccoli: the mascot of awesome mathematical stuff.
You deserve cool things. Mathematical things, in the Adventure Time sense. All too often math is treated like a chore, but it is, and should be, mind-blowingly awesome. It could be the coolest subject you ever study, but textbooks seem to have the special quality of sucking the joy out of everything.
Let’s start with an episode of Adventure Time as an example, specifically the one where Finn gets glasses that make him super smart, and invents a device for blowing bubbles that have any number of dimensions. It’s possibly the most concise explanation of higher-dimension geometry I’ve ever seen. A shadow falling on a flat surface is a projection of an object in two dimensions. A two dimensional bubble casts a one dimensional shadow. A three dimensional bubble casts a two dimensional shadow. A four dimensional bubble casts a three dimensional shadow into our three dimensional space. Another way to think about it is like this: every object of the next-highest order of dimensions has an object of the previous order as its “face.”
A square has a line for each of its four faces that define it in two dimensional space. A cube has a square for each of its six faces. A tesseract or 4D hypercube – a four dimensional object – has a cube for each of its eight three dimensional faces. Watch this video, which shows a tesseract rotating so you can get a good look at it and understand how it’s put together. It may help to realize that none of the cubes pass through each other during the rotation. When they squash and stretch, it’s not a small cube passing through a bigger one, but rather the bigger faces popping out towards you and the smaller ones passing in the distance behind. The lines are all the same size, and the angles are all 90 degrees. It’s perspective, not changing actual shape. Eventually you’ll get it.
Or, read a breezily-written book on mathematics. Here’s Looking at Euclid is one of the best introductions to math that I’ve encountered. Also, this won’t be the last about mathematical mathematics on this blog. Here’s a teaser, in case the broccoli wasn’t a tip-off.