Sunday 8 April 2018

A guide to the fourth dimension

The idea of this article is to imagine what it would be like to be in four dimensions. Of course, since we live in only three dimensions, it is impossible to visualize a four-dimensional space, but we can show how everything would work in the fourth dimension by using mathematics.

To start with, we need to know what a transition to a higher dimension is like. The start is zero dimensions. In a zero-dimensional world, it is impossible to move, and there would be no space around, and no amounts of anything. There can only be something, or nothing. The shape of a 0-dimensional world is a simple dot.

To go to the first dimension, imagine taking two zero-dimensional worlds(dots), and connecting them. This world is simply a line. In the first dimension, nothing can move past anything else, and any object is basically a line, with varying length. The ends of each line are zero-dimensional points.

To make a two-dimensional world, imagine taking two one-dimensional lines, and connecting them at every point. The result is a flat sheet. An infinite world of two dimensions is called a plane in mathematics. It would be the same as in three dimensions, except that one of the directions is missing, so it would be like moving pieces of paper around on a flat table.

Stepping up to three dimensions can be done by taking two parallel planes and connecting them at each point. This results in a three dimensional space. This is the dimension that we are the most familiar with. Many more things are possible in three dimensions than in two. The second dimension to us would be so thin that it could not influence the third dimension. There are still two-dimensional things in our world, though, like shadows, surfaces and images.

The best way to imagine the fourth dimension would be to recognize that the third dimension is to the fourth dimension as the second is to the third. A four dimensional space would be the space in between two volumes that are separated from each other in only the fourth dimension, which is at right angles to each of our three dimensions. Of course, we know of no evidence that a four dimensional world exists, but it is possible in mathematics.

The simplest way to imagine these transitions through the dimensions is to imagine a hypercube. A hypercube is an object in any dimension, where its lengths in each dimension are the same, and it takes up the maximum amount of space for a given side length. For example, a 1-dimensional hypercube is a line. Stepping up to two dimensions makes it into a square, and in the third dimension it becomes a cube. The 1-dimensional hypercube has two 0-dimensional endpoints. The 2-dimensional hypercube has four 1-dimensional sides and four 0-dimensional corners. The 3-dimensional hypercube is a bit more complicated, with six 2-dimensional faces, twelve 1-dimensional edges and eight 0-dimensional corners. As the hypercube goes up through each dimension, it gains a different property. The next step is where it starts to get weird.
An animation of a hypercube

A 4-dimensional hypercube is called a tesseract or 8-cell. The tesseract is to the cube as the cube is to the square. Because a square has four sides and a cube has six, a tesseract has eight sides. These 'sides' are all three-dimensional cubes and are called cells. The two additional sides are extended in the fourth dimension as follows: if you go straight ahead all of the way around a cube or a square, you will go around three other sides before you get back to the starting side, and this is the same with a tesseract. Following the normal pattern of hypercubes, each cell of a tesseract has six square sides. Going through any one of those sides takes you onto another side. At each of the 32 edges, three cells meet, and four cells meet at each of the 16 corners. In normal three-dimensional space, four cubes can meet at an edge, and eight can meet at a corner, so at those areas the tesseract would be especially warped from three dimensions.
The Dali cross

One way to make this all simpler is by unfolding the tesseract, just like how you can unfold a cube into two-dimensional space. In a cube, the resulting shape looks like a cross made of six square panels. In a tesseract, the shape would look like a four-sided cross made of eight cubes, four cubes tall with four additional cubes sticking out in all four directions. This is called a 'net' of a tesseract, which is known as the Dali cross.

The common way to display a tesseract(see below) is to have a cube with a smaller one inside it, linked to the bigger one by twelve walls, one for each edge of the inside cube. The inner cube is the cell that is facing away from us in the fourth dimension, and the space in between the inner and outer cubes is divided by the walls into six semi-trapezoidal shapes. These shapes are six of the other cells. The last one, which is facing toward us in the fourth dimension, is the bigger cube and has all of the other cells shown inside it.

This is the classical projection of a tesseract onto three dimensions. It is three-dimensional, but it has all of the same edges, corners and faces of the actual tesseract. The only thing that is changed is the shape. It is what the shadow of a tesseract would look like if it was rotated in the right way.

There are many other shapes in the fourth dimension besides a tesseract, which include 6 shapes that are closely related to the five three-dimensional platonic solids. There is also a shape related to the circle or sphere, which is called the 3-sphere or glome. It consists of all of the points at a certain distance x(in four dimensions) to a certain point. The cross-section of a 3-sphere is a sphere, just like how the cross-section of a tesseract can be a cube.
The shadow of a rotating 8-cell

Visualizing the fourth dimension is impossible, because we live in only three dimensions, however, there are three different ways we can make a small understanding of the fourth dimension. The first is by using three dimensional graphs that capture some of the elements of the fourth dimension. These can be like shadows or cross-sections of four-dimensional objects. They give some of the information about these objects, but never all of it. The second is by keeping in mind that four dimensions is to three as three is to two, so we can imagine how three dimensional objects would relate to a two-dimensional world.

The third is by using our own fourth dimension: time. Even though we can influence the future and it is impossible to look into it, and we can only see three dimensions, time can still be compared to a fourth. We can represent a four dimensional object by using time as one of the axes. Maybe we do live a four dimensional world.

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