Fourth Dimension Part 1: Beyond the three axes of space

 

 

hcube-slice-edge

Ever imagined what the world would look like if viewed from a higher dimension? What if, our universe is just a small toy kept in the showcase of some higher dimensional being? These questions might seem a page taken out from a science-fiction novel, but in fact, it lies on the thin line separating sci-fi and science.

Before delving further into the 4th dimension, let us revise some facts:

In 2nd dimension there exists four infinite regions each known as “quadrant”. Quadrants separate 2-d space into four distinct regions. Similarly, in 3rd, we have eight such regions bounded by either positive or negative side of each of three axes x, y and z – “octants”. By this logic, the Nth dimension has 2n regions.

The 4th dimension therefore, consists of 16 infinite regions known as “sedecants”. (sedeca meaning 16). Throughout this article, we shall use the letter ‘w’ to indicate the fourth dimensional axis.

 

  • TESSERACT /HYPER-CUBE/4-D CUBE

The most common example of a four-dimensional object is a “tesseract”, or hyper-cube, first used in 1888 by Charles Howard Hinton in his book “A New Era of Thought”

So what is a hyper-cube?

A hyper-cube is a 4-d geometric shape formed by stacking infinite number of cubes along the w- axis.

As difficult as it might sound at first, we are going to discuss just that in depth, very soon. But, before we begin, let us clarify some basics.

 

  • NATURE OF DIMENSION

What is a dimension?

It is a measurable extent and a definition of an aspect. Mathematically, we can work with hundreds of dimension without working up a sweat, but problem arises when we try to represent them visually. Dimensions are of two types:-

  • Spatial or space dimension
  • Temporal or time dimension

 

In our reality, the three spatial dimensions and the temporal dimension comprises “space-time” continuum, as explained by Einstein in his general theory of relativity. Time is therefore, considered the fourth direction and if searched properly, ample evidence supporting this theory can be found on books, magazines and the internet.

But time and space are not interchangeable. They are not equal to one another. One cannot convert 500 seconds into an equivalent measurement in length. Hence, when we say fourth, we are particularly referring to the temporal dimension.In this article, we are going to focus strictly on the 4th spatial dimension also known as Minkowski’s fourth dimension.

Not much speculation is needed while performing purely mathematical operations in the form of size-4 matrices. But when approached from a purely intuitive state, not only is the 4th dimension mind-boggling, but also begs the question whether at all our mind were designed to handle such complexity in the first place. Our relation with the fourth dimension is no different than “flatland”-ers with the third.

Even to begin comprehending the nature of 4th spatial dimension, we must use first use analogy of flatland objects mentioned in the following article, previously published.

Click here to read the previous article on 2nd dimension.

  • UNDERSTANDING THE 4th DIMENSION THROUGH ANALOGY

The above article in essence explores the 2-dimensional world in details, as to how objects such as squares, circles and triangles would interact with each other in flatland. We shall understand the 4-d world through an analogous progression, i.e. define the existing dimensions in terms of its lower dimensions.

Analogy 1: Infinite stacking up of one dimension to get a higher dimension

The most typical example of a 4th dimensional object is a tesseract, as discussed earlier. To understand a hypercube, we must use an analogy and define a square and a cube in terms of a lower dimensions.

What is a rectangle?

 A rectangle can be defined in number of ways but is essentially an infinite number of line segments stacked together in a direction perpendicular to the direction of a line segment (say x-axis in this case).

If we were to strictly limit ourselves to one dimension(i.e. the x-axis), and ignore the possibility of any other perpendicular dimensions, then a square is nothing more than a line.

This is what happens in flatland. But when we increase the scope, i.e. increase our dimensional space by one, we see the true nature of 2-d space. A cube is similarly, an infinite number of squares stacked upon another in a direction perpendicular to both the existing dimensions.

A hyper-cube, therefore, by analogy, is an infinite number of cube placed together in a direction perpendicular to the existing three dimensions. Due to the fact that we cannot define the fourth dimension, it is imaginary and simply does not exist in the 3-d space. A “true” cube has no length in the fourth dimension

tesseract
Sides of tesseract/hyper-cube parallel to four axes of space

Analogy 2

Another way of understanding higher dimensions is through how it is actually plotted on paper. In this analogy, we will consider the figures hollow and will be concerned with only the terminal figures.

When we say, ”terminal”, we are referring to the bounding shapes which is a dimension smaller than the solid. For e.g.:

The terminals of a square are the 4 bounding line segments. Similarly, a cube has the 6 faces as its terminals. By this logic, a tesseract must have 8 terminal cubes (8 because twice the number of dimension it possesses, which is 4 in this case). But there’s a constraint to this system that can be easily missed in a hurry – the four lines holding the two cubes lies strictly in the w axis and are perpendicular to all the three axes x, y and z. This means the fourth dimensional segments (marked in yellow in the diagram)cannot be represented as components of the three axes.

As accurate as these theories might be, it is difficult to imagine a direction other than the existing three, and hence we are forced to take a different approach which will be discussed in the following article.

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Click the following link to read the next article in the Fourth dimension series:

Fourth Dimension Part 2: Understanding 4-d space through strings

 

 

 

 

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