Climbing the dimensions (part 2)

In the first part of this article we tried to present a way to capture the essence of the tesseract. We did that by “climbing” the dimensions from the point (no dimensions), through the segment (1-D), square (2-D), cube (3-D) and finally tesseract (4-D).

In the following figures we present other attempts at visualizing what we can barely imagine, a 4-D solid.

Figure 2-1 – Another representation of the tesseract

Figure 2-1 represents the tesseract by using segments which are all of the same length. Figure 2-2 shows another representation. In this one, two cubes (one blue, one pink) are shown with their connection in the fourth dimension. The image can be explored to see that all eight cubes forming the tesseract are there. The cubes are deformed because of the perspective used.

Figure 2-2 – Yet another representation of the tesseract

Unfolding the tesseract

Let’s go back to our 3-D world for a minute. A cube, as we know, has six square sides. Six squares, connected in certain ways, can be folded to form a cube. Most of us have done something like this in elementary school. There are several ways in which six squares can be connected so to form a cube when folded. One of them is shown in figure 2-3.

Figure 2-3 – An unfolded cube

An unfolded 3-D cube is a 2-D image. Hence, an unfolded 4-D tesseract is a 3-D body. One of the possible ways to unfold the tesseract is shown on figure 2-4. If we could fold the tesseract (in the fourth dimension), the faces marked with the same letters in the figure would be in contact.

Figure 2-4 – An unfolded tesseract

The unfolded tesseract appears in a famous picture from Dali, the crucifixion of Christ.

According Wikipedia: “Just as the concept of God exists in a space that is incomprehensible to humans, the hypercube exists in four spatial dimensions, which is equally inaccesible to the mind”.

Figure 2-5 – Tesseract in art – Dali’s “Corpus hypercubus”

To end this two part series, I invite you to watch a video of world famous Carl Sagan, speaking of some hypothetical interactions of 3-D beings with 2-D beings:

Also read: Climbing the dimensions (part 1)

My blog: FPGA Site