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3. Everything / Maths, Science & Technology / Mathematics

Tesseracts

When one thinks of dimensions, the first thought may be of alternate realities or parallel universes. A more mathematically-orientated person may think of the dimensions of length, width and depth. A physicist would then go on to cite the fourth dimension, believed to be time.

However, what many may not think of is the fourth spatial dimension, in which tesseracts (also known as hypercubes) exist.

Unfolding the Tesseract

A tesseract is, in essence, a four-dimensional cube. It is very similar to a 'regular' three-dimensional cube, and in fact, many of the explanations in this entry will be comparisons of a cube and a tesseract.

Anyone who has taken high-school geometry would recognise the unfolded version of a cube. It looks like a lower case 't', divided into boxes. In a similar vein, a drawing of a cube looks like a square with two parallelograms attached to the sides. But wait! How can that be a cube? Cubes are made of squares, not parallelograms. The answer is simple - perspective! This is a representation of a cube as a person would see it in two dimensions.

Now that we have cubes figured out, we can move on to tesseracts. A tesseract can also be unfolded from four dimensions into three. It is made of eight cubes, which, unfolded, look like a tower of four cubes, with four more cubes attached to the second-storey cube. To fold in a tesseract in the fourth dimension, fold the top of the top cube into the bottom of the bottom one, and the four cubes inwards, bringing the second storey up, down, and sideways into the other cubes.

Of course, all this folding is done in the fourth dimension, where it is fully possible to do this. It is comparable to folding two-dimensional squares in the map of a cube through the third dimension so that they connect with each other.

Tesseracts can also be made in the third dimension. This is similar to drawing a 2D picture of a 3D cube. To make a model of a tesseract, construct two wire-frame cubes, one smaller than the other. Toothpicks and marshmallows work well. Place the smaller cube in the centre of the larger one, then connect the corners of the cubes with toothpicks. That's a tesseract.

Do you see the eight cubes? No? All you see is a small cube surrounded by flat-topped pyramids within a larger cube.

Why?

Perspective! The eight cubes can be found in the small cube, the six pyramids, and the outside of the figure. Think of it this way; in the same way that the actual shape of a cube is distorted when drawn in two dimensions, so is the actual shape of a tesseract distorted when reproduced in a three-dimensional model.

Another concept that may be difficult to understand is the outside of the figure representing another cube. Another comparison between two and three dimensions may be necessary. A two-dimensional being would have a hard time, when considering a cube, understanding how you could be on the underside of a plane, as you would be when standing halfway around a cube. In a tesseract, the six sides of the 'outside' cube are folded around the rest of the model, so although you stand 'outside' in three dimensions, in a four-dimensional object you are just inside another cube.

Moving in a Tesseract...

...or Slim, our two-dimensional friend.

The easiest way to think of moving in a tesseract is to first consider a cube in the eyes of a two-dimensional being. Let's call our two-dimensional being 'Slim' just for ease of use. Slim has known nothing but forward, backward, and side to side. He cannot move up or down or even look up or down. Thus, when Slim suddenly encounters a cube - a three-dimensional object - he has no idea what is going on!

Because Slim is two-dimensional, and moves as such, he must move on the surface of the cube, not within the three-dimensional space inside the cube. Let's say Slim begins on the top of said cube. He soon reaches the edge, but because he perceives things only in two dimensions, he does not see the 90° turn leading downwards, or, if he continues in the same direction, the next three 90° turns that take him around the cube and back to where he started.

Naturally, this would be quite frightening to this unassuming two-dimensional creature, so he moves in the other direction, to the right, hoping to escape this strange thing he has stumbled across. This, naturally, also leads him back where he started.

One might also think of a one-dimensional creature, who came upon a square of two dimensions, and proceeded to make its way all the way around the edge of said square, returning to where it started.

So what about a three-dimensional creature moving within a four-dimensional tesseract? Let's create another character; we'll call him...

Fred, a Three-dimensional Creature

Fred is walking along one day, when, similar to Slim, he suddenly falls out of his dimension and into a tesseract. This tesseract is a house, a very plain house, in fact, it's just a plain-looking cube. Fred walks in.

But wait! There are doors on all four walls, plus stairs going up and down! Those couldn't be there, this is just a plain cube. So Fred adventures through one door and finds himself in another room, with exits on all six sides once again. Continuing forward, Fred finds himself in yet another room, and continuing again, he finds himself back where he started. But why is this?

Think of our model of a tesseract. Let's assume Fred begins in the centre cube. He walks through a door and finds himself in one of the pyramids. Walking in the same direction, he is inside another cube, represented in our model by the outside of the figure. Continuing in the same direction, he goes to the other side of the model and enters the pyramid opposite the one he was previously in. He then continues back into the centre cube.

So there you have it. Tesseracts are a very difficult concept, which requires you to literally 'think outside the box'.

Visual and Other Aids

Some science fiction authors have been inspired by the concept of tesseracts, such as:

  • Robert A Heinlein in And He Built a Crooked House

  • Lewis Padgett in All Mimsy were the Borogroves

  • Carl Sagan in Space

The h2g2 three-dimensional diagram of a tesseract:

       
            ______________________       
      _,.-""|`.                ,'|""-.._
     +----------------------------------+
     |`-.   |    `.        ,'    |   ,-'|
     |   `-.|      +------+      |,-'   |
     |      `-._ ,'|      |`. _,-'      |
     |      |   +------------+   |      |
     |      |   |  |      |  |   |      | 
     |      |   |  |______|  |   |      |
     |      |   |-J        F-|   |      | 
     |      |,-'| F        J |`-.|      |
     |      +---|J----------L|---+      |
     |     /    |F          J|    \     |
     |    /    ,'------------'.    \    |
     |   /   ,'                `.   \   | 
     |  /  ,'                    `.  \  |
     | / ,'                        `. \ |
     |/.'                            `.\|  
     +----------------------------------+

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Entry Data
Entry ID: A510986 (Edited)

Written and Researched by:
Flaming Moose
Adrenochrome
HenryS

Edited by:
You can call me TC - Trillian's Child


Date: 26   March   2001


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