Geometry was my best math subject, and it served me well during my CAD jockey years. But one thing I could never wrap my head around was the notion of a 4th dimension. People smarter than me would try explaining it to me at a bar, and while I could comprehend the individual words coming out of their mouths, I could never put the concept together in my head.
Finally, after seeing this video by the folks behind a game called Miegakure, I can start to wrap my head around it:
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Miegakure is an interactive puzzle game that lets you mess around with different shapes while toggling back and forth between the 3D and 4D world.
The fourth dimension in this game is not time, it works just like the first three: it is a mathematical generalization. [The game] plays like a regular 3D platformer, but at the press of a button one of the dimensions is exchanged with the fourth dimension, allowing for four-dimensional movement.
Your ability to move in the fourth dimensions in addition to the usual three allows you to perform miraculous feats like seeing inside closed buildings, walking through walls, stealing objects from closed containers, binding two separate rings without breaking them, etc...
These actual consequences of the mathematical formulation of 4D space have been thought about for more than a century (in the 1884 novella Flatland for example) but it is the first time anyone can actually perform them, thanks to the video game medium.