How can we imagine the 4th dimension?

The fourth dimension - very simple

“Can you imagine the fourth dimension?” My friend Peter recently asked me. We meet once a month, talk about this and that, while drinking a beer or two. I was almost embarrassed to have to answer him: “Introduce? No! ”We mathematicians can calculate in four-dimensional spaces without any problems. "But to be honest, the four-dimensional space is beyond my imagination." Peter was disappointed: "I would have expected that from you mathematicians."

What do you say on it? Good advice was expensive, so we looked into our beer glasses, took a sip, and put the glasses back down. Then an idea occurred to me: “Maybe I can explain something to you after all. You know what a cube is. There is also a four-dimensional cube, a cube in four-dimensional space. And I can tell you everything about this four-dimensional cube. "

"That sounds better," muttered Peter and asked me: "Then let's go with your 4D cube!" I reached for a beer mat and said: "First we take a step back from the three-dimensional cube and look at a 2D one. Cube. ”“ I beg your pardon? ”Peter's astonishment was real. "A two-dimensional cube is nothing more than a square." He understood that immediately: "Sure, all right angles and four corners."

"Wonderful. Let's count the corners first. The square has 4, the cube 8, and the 4D cube so ”- I paused a little to give it a chance -“ exactly: 16. ”“ And how many edges? ”Peter had become brave. “Let's first look at how many edges go through a corner. With the square there are 2, with the cube 3, and with the four-dimensional cube ", here I raised my voice suggestively," there are 4. "

“What else is there?” “Good question. With a square there are edges, with the cube there are squares, and with our 4D cube - 3D cube! ”“ But how many? ”Asked Peter. "If you really want to know: How many 3D cubes are there at one corner of the 4D cube?"

“There are 4 edges - and every three of them form a cube. And since you can choose 3 out of 4 edges in exactly 4 ways, there are 4 3D cubes at each corner. "

“That means a total of…?” “Well, 16 corners times 4. That gives 64. But every 3D cube has 8 corners and can be viewed from every corner. So there are a total of exactly 64 divided by 8 equal to 8 normal cubes in the 4D cube. ”Peter took a long sip. "But I still can't imagine the 4D cube." How would you build a normal 3D cube? ”Peter answered spontaneously:“ Cut out the sides of the cube and glue them together. ”“ Exactly. Could the side surfaces be cut out so coherently that only a few edges have to be glued? "

After a second Peter remembered: "Four squares in a row from top to bottom and then one to the right and left."

“Great!” I said, “that's called a network of the cube. And that's exactly how it works with the 4D cube. "" Why - exactly like that? "

“With an adjustment, of course. The mesh is in three-dimensional space, not in a plane, and consists of normal 3D cubes. Four on top of each other, and then another one on the right, one on the left, one in front and one in the back. "

“And I should be able to imagine that?” “Why not, it's only in three-dimensional space!” I blasphemed. "Salvador Dalí painted a picture with exactly this construction: a crucifixion scene, where the cross is the network of a 4D cube."

“As usual with Dalí. Blasphemous, but brilliant ”, Peter kept the last word as always.

August 17, 2004