Hyperspace: Up, Out, and Away Looking at the world through 4-D glasses
Tesseract: Cube to the Fourth Hyperspace—space beyond three dimensions—is one of the most mind-expanding ideas in modern science. For neophytes, the classic introduction to higher dimensional spaces is Edwin A. Abbott's 1884 novel Flatland, the tale of a square who lives in a two-dimensional world. Mr. A. Square can't imagine a third direction perpendicular to his world—up but not north—until he dreams of Lineland, a one-dimensional world. Thinking about the jump from line to square, he makes the leap from square to cube.
As a 3-D creature, you may have trouble picturing a fourth direction perpendicular to our three—out but not north or up. But, like Mr. A. Square, you can make the leap from cube to hypercube by first thinking about the jump from square to cube. Try filling in the blanks in the paragraphs below.
1. To build a three-dimensional cube, make two copies of a two-dimensional square. Lay one of the squares flat on the ground and raise the other in the third dimension the distance of one side length. This doubles the four corners of the original square for a total of eight corners. Connect each corner of the bottom square to the corresponding corner of the top square, for a total of 12 edges. The completed cube has six square faces: one each top and bottom and four around the sides. If the edge length is one foot, then the farthest distance from one corner to another is ___ feet.
2. To build a four-dimensional hypercube, called a tesseract, make two copies of a three-dimensional cube. Lay one three-dimensional cube flat on the ground and raise the other in the fourth dimension the distance of one side length. This doubles the ___ corners of the original cube for a total of ___ corners. Connect each corner of the bottom cube to the corresponding corner of the top cube, for a total of ___ edges. The completed hypercube has ___ cubical hyperfaces: one each top and bottom and ___ around the sides. If the edge length is one foot, then the farthest distance from one corner to another is ___ feet.
4-D TICKTACKTOE The Flatlanders' world has two perpendicular axes: north-south and east-west. We three-spacers have one more axis—up-down—perpendicular to the first two. In four-space, there is yet another axis, perpendicular to the first three. William Sleator, in his novel The Boy Who Reversed Himself (1986), describes a visit to four-dimensional space by using the words ana and kata for the fourth axis equivalent of up and down.
To help you imagine ana and kata, here is a 4-D version of ticktacktoe. Nine 3x3 boards are arranged in three columns of three boards each. Each column of boards represents the three floors of a 3x3 cube. Each row also represents three floors of a 3x3 cube but stacked ana and kata instead of up and down. A ticktacktoe consists of three markers in a straight line. The line may be either all within one board or extend in a straight line horizontally, vertically, or diagonally across three boards. For example, the orange, green, and purple squares below show three potential ticktacktoes that start with the red square.
Two players take turns placing X's and O's in the squares. Winning by getting three in a row is easy: The first player can always win by playing first in the middle square. So the goal of our game will be to get as few ticktacktoes as possible. The game ends when all the cells are filled with X's and O's.
1. How many ticktacktoes does each player currently have?
2. In how many different places could each player make one move and complete a ticktacktoe?
3. How many potential ticktacktoes include the red square? How many potential ticktacktoes are in a 3x3x3 hypercubical board? Hint: First answer the question for a standard 3x3 and a 3x3x3 board.
4. Can a game end in a draw?
A number of 20th-century artists portrayed higher dimensional spaces in their paintings. In Salvador Dali's Corpus Hypercubus, the traditional cross made of six squares, which can be folded into a cube as shown above, is replaced by a higher-dimensional cross made of eight cubes, which can be folded into a hypercube as shown below.
The cubical "faces" of the hypercube do not fold along hinged edges, as they would in three-space. Instead, they stay joined along entire square faces. To us three-spacers, some of the cubes appear squashed, distorted, or even turned inside out.
1. [Easy] Which of the strips below cannot be folded into a cube? Hint: Hold the red square still and fold the others around it.
2. [Not too difficult] How many ways are there to unfold a cube?
3. [Difficult] Which of these shapes cannot be folded into a hypercube? Hint: Hold the red cubes still and fold the others in front.
4. [For geniuses only] How many ways are there to unfold a hypercube?
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