Colossal Gardner, ch. 13


Hypercubes. George Gamow and Martin Gardner both put a lot of effort into describing higher dimensions, since the idea of curved (or warped) space is tied to universe systems of 4 dimensions or more. While String Theory never shows up in Gardner’s book, there is one mention of M-Theory in the addendum for the Hyperspheres chapter, which speculates on 10 or 11 dimensions, most of them wrapped up in a really small space. If M-Theory ever pans out, it could relate gravity to the strong and weak forces, so there’s a reason to consider 4 dimensions and up kind of seriously. On the other hand, Gamow and Einstein argued that our universe is closed in such a way that a spaceship traveling in a straight line out away from our sun would eventually return from the opposite direction. Gamow’s discussions indicate that there might be a left-for-right mirror flip along the way, so that the pilot would be able to put a left-handed glove on his right hand (or that he might be upside-down compared to when he left).

Most of the discussions of hypersolids seem to come down to explaining what a tesseract is, and how it would look to 3D observers. If you’re not familiar with the idea, first take a point on a piece of paper. If you extend the point in one direction, you get a line segment – with 2 points, 1 line, 0 squares, 0 cubes and 0 tesseracts. If you take that line segment and extend it in a direction perpendicular to the segment, you get a square – with 4 points, 4 lines, 1 square, 0 cubes and 0 tesseracts. Extend the square in a direction normal to the plane and you get a cube – with 8 points, 12 lines, 6 squares, 1 cube and 0 tesseracts.


(Rotating tesseract from a 3D perspective, from the wiki article.)

The trick is to then extend the cube in a direction normal to the 3D space, which in effect makes 8 cubes that overlap without touching each other (there’s the original cube, 6 cubes extending from the 6 faces of the original cube, and the final cube you get when you stop pushing in the 4th direction) – with 16 points, 32 lines, 24 squares, 8 cubes and one tesseract.

You can calculate the number of faces, etc., for a hypercube of any dimension “n” by using the simple binomial (2x+1)^n. So, for a 4D cube, write out (2x + 1)(2x + 1)(2x + 1)(2x + 1):
(4x^2 + 2x + 2x + 1)*(4x^2 + 2x + 2x + 1)
(4x^2 + 4x + 1)*(4x^2 + 4x + 1)
16x^4 + 16x^3 + 4x^2 + 16x^3 + 16x^2 + 4x + 4x^2 + 4x + 1
16x^4 + 32x^3 + 24x^2 + 8x + 1
(where the x^4 coefficient gives the number of points, and the x^3 coefficient gives the number of squares, etc.)

To go to a 5D hypercube, just multiply the above equation by (2x + 1) again.

The rest of Gardner’s article is just an examination of what the hypercube would look like to us if we use 3D projections or slices. He goes on to give examples of hypercubes in art, including Dali’s Corpus Hypercubus, and Heinlein’s “-And He Built a Crooked House” short story.


(Dali’s Corpus Hypercubus, from the wiki entry.)

Gardner closes by saying that Heinlein’s idea of the house dropping out of 3D space could have a possible analogue in giant stars that undergo gravitational collapse, according to J. A. Wheeler. The density of a quasar could be great enough to curve space to the point where the mass drops out of space-time “releasing energy as it vanishes.” This might explain the enormous amount of energy emanating from quasi-stellar radio sources.

Puzzle:
If the longest line that can fit into a unit square is the diagonal, with a length of sqrt(2), and the largest square that can fit into a unit cube has an area of 9/8 and a side of 3/4 * sqrt(2), then what’s the largest cube that can fit into a tesseract?

 
(All rights belong to their owners. Images used here for review purposes only.)

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