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Climbing the dimensions (part 1)

Climbing the dimensions (part 1)
by Claudio Avi Chami on 10-07-2016 at 7:00 am

 Translated and adapted from an article by Jaime Poniachik
The novel Flatland was written en 1884 by Edwin A. Abbot. This novel describes a fantastic, two-dimensional, flat world. Hence the name of the novel. This world has living beings. They have only two dimensions and they move in a plane which they cannot abandon.

It is not difficult to imagine a bi-dimensional reality. An amoeba, for example, laid on a flat surface is essentially living on a two- dimensions reality. It will explore and get food moving its pseudopods up and down, left and right, but not above or below its “body”.

In the same way that there is a three-dimensional world (ours), and the amoebas two-dimensional world, we can try to imagine how a four-dimensional world would be. Please note that we are talking about a four dimensional spatial world. Time is commonly regarded as an additional dimension, but it is not spatial (and we cannot easily move on it back and forth).

The Mystery of the Yellow Room

In “The Mystery of the Yellow Room”, suspense master Mr. Gaston Leroux (author of “The Phantom of the Opera”), proposes the following enigma: A woman that has been attacked nearly to death, is found in a locked room. A similar case is proposed by Edgar A. Poe in his story, “The Murders in the Rue Morgue”. In both cases, the authors finally give the reader a plausible explanation of how the crimes were executed inside rooms apparently locked from the inside.

A three-dimensional being could execute the perfect crime in a bi-dimensional world. A crime that even Leroux or Poe would not be able to explain.

In a bi-dimensional world, a locked room could be, for example, a rectangle. No creature from this bi-dimensional world would be able to enter the locked room, unless it forced its doors or opened a hole in its walls (the rectangle perimeter).

But a three-dimensional assassin could easily enter the locked room from the third dimension, commit the crime and abandon the locked room, leaving no trails behind.

In the same way, a criminal from a four dimensional reality could easily enter a perfectly locked three-dimensional room without touching its walls, its ceiling or its floor. It almost makes you look behind your back in awe.

A peek into hyperspace
What is the aspect of a four-dimensional object? We will try to imagine an hypercube, a cube of four dimensions.

To try (somehow) to imagine the hypercube we will explore its analog bodies on lesser dimensions.
Let’s start by the square. The square is a figure. It is flat, it has only two dimensions. It can be built using one-dimensional elements. Let’s say that we take a pair of segments. Then we connect their vertices using another pair of equal length segments. Using one-dimensional elements (the segments) we have built a two-dimensional figure, the square. This can be seen in figure 1.

Figure 1 – A square, a 2-D figure built using 1-D segments

We can also imagine that the square “connects” between two 1-D universes, the upper segment and the lower segment.

Now let’s climb to the third dimension. A cube can be built in a similar way as we built the square. This time we draw a square in one plane, and then another square in a parallel plane. If we connect each one of the vertices of the top square with the bottom square, we get a cube, as shown in figure 2.

Figure 2 – A cube, a 3-D body built connecting 2-D squares

Now let’s imagine we have a cube floating in space. We have another cube floating on another three-dimensional space. If we connect all eight vertices from one cube with those of the other cube, we have built an hypercube.

Representing such a beast is a little more complicate than telling how to build it. A possible representation (in two dimensions!) of an hypercube is shown on figure 3. If we were able to see four dimensions, we would see that all its edges are ortogonal by pairs, as they are in the square and in the cube.

Figure 3

– An hypercube (tesseract), a 4-D body built connecting 3-D cubes

Counting the elements

A square has a single face, four vertices and four edges
We climb another dimension to 3-D. There we find that the cube has six faces, eight vertices and twelve edges.
When we climb to the fourth dimension, we find that the tesseract has sixteen vertices.
On the following table we can see elements for each figure or body, starting from the first dimension and climbing into the fourth dimension:

Vertices Edges Faces Bodies
Point 1
Segment 2 1
Square 4 2 1
Cube 8 12 6 1
Tesseract 16 32 24 8

The quantity of vertices for each figure follows a simple rule, it is 2 to the power of the dimension. But the formula for the quantity of edges, faces, etc. are a bit more complicated.

Also read: Climbing the dimensions (part 2)

My blog: FPGA Site

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