Mr Jones in the Fourth Dimension
The professor was standing at a blackboard. "As you all know, a four-dimensional point can be represented by a set of four coordinates (x, y, z, w). Now consider a unit tesseract. Its vertices' coordinates are all 0 or 1. It itself is the set of all points such that 0 <= x, y, z, w <= 1. By proceeding in this way..." All this was too much for Mr Jones, who had slept badly the previous night. He couldn't help dozing off.
But he had an extraordinary dream. He met his friend, Mr Thomson, who wanted to play paintball with him. Only everything seemed to look strange to him. There was no paint about. In the end they decided to play chess. "Don't you mean Chesseract?" Mr Jones still didn't know what that was. "Don't worry, I'll teach you the rules." After a long and hard-fought game, Mr Thomson won. And so Mr Jones wanted to see what was clearly a new house. "I see you've got a very big dining room. I can even see sculptures on all six walls!" But there were no sculptures. As Mr Thomson explained, those were paintings. Meanwhile Mr Thomson was idly drawing a cube on a piece of paper. To Mr Jones, the paper seemed solid. Even the cube could rotate like the ones he saw, unlike the drawings that he was familiar with.
And so he visited the basement. What he saw was a strange world. He could see his professor on what seemed to be an insubstantial sheet. Moreover, he could see his brain. He could see inside him, his heart beating. He could see the lecture hall. He could even see himself sleeping. He decided to pick up the professor. "So, by using analogy you can work out the number of faces, vertices, edges, and cells. Now consider... AAAAARGH! What are these five spheres doing right next to me? How can they change size?" Soon his eyes turned to a bucket of paint outside the lecture hall. It was full but it seemed freshly spread. He picked it up and wrapped it around a few pellets. The paintball game would start soon.
Mr Jones took a shot at Mr Thomson. It looked sure to hit. Then Mr Thomson leapt in a direction strangely unfamiliar to him. Only then did Mr Jones realise that this was the fourth dimension. The paintings were three-dimensional, the paper was three-dimensional, and Mr Thomson could draw a real cube as easily as he could draw a square.
Furthermore, his world was in a sheet in the basement. He could pick up his professor just like he could pick up a paper figure. He could see inside him, as he could see a brain drawn on a picture of a man. His fingers intersected his world in a sphere, just as his fingers make circular cross-sections with the surface of water. The paint was insubstantial in this 4-D world, just as a painted picture of a bucket was. But despite knowing all this, he couldn't quite identify a strange buzzing sound. It was only when he woke up that he realised that it was his professor. He had slept through the entire lecture. "And so you can find out the surface volume of a hypersphere. This shows the limits of analogy in 4 dimensions and above. You should now be aware of the reasoning used to find out about the 4th dimension."
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