C.S. Lewis

C.S. Lewis

Saturday, May 16, 2015

0th to 4th Dimension: Mathematics and Philosophy


Mathematical Introduction

If you stick with this I promise there’s a point at the end!

 0-Dimension:


In geometry, the zero-dimension is a point. A point is an infinitesimally small marker of a specific location with no accompanying volume, height, width, or length. It has no external or internal properties aside from its current location. Several zero-dimensional points may be set in series to create a larger object, such as a line or square, but the object is itself, in its most basic form, a zero-dimensional construct.

1-Dimension:


One-dimensional objects are created from an infinite number of zero-dimension points strung together. The resultant is a line segment. This product, as its title’s numeral designation indicates, has but one property: length. Several one-dimensional line segments may be joined to create a greater whole categorized within a higher dimension, but that object will always be able to be broken down into its one and zeroth-dimensional constituent parts.

 

2-Dimension:


Two-dimensional objects are created from a collection of line segments connected such that they create a flat plane. As such, these may also be broken down into an infinite number of zeroth-dimension constituents. As indicated by the title, two-dimensional objects hold two properties, length and width. Several two dimensional objects can be combined to create a higher-dimensional object that can always be broken down into its smaller contributing parts.

3-Dimension:


Three-dimensional objects are made from several two-dimensional planes connected to create a greater whole that has flat planes extending in degrees off of three basic x-y-z planes. More simply, a 3D object is like a collection of flat planes such that it may be picked up rather than being a plane with no thickness. This dimension has three properties, length, width, and height. Although rarely discussed in high school and many collegiate courses, three-dimensional objects may also be combined to create a new product that may be categorized under a higher dimension.


 

SPECIAL CASE: Circle/Sphere with the same dimension (radius)


Circles and spheres are interesting subsets within two and three-dimensional special sets. This is due to the fact that, for these shapes, all of the dimensions are the same. For example, for circles, the length and width dimensions are replaced by a constant radius coming out of a single origin point. For spheres, the length, width, and height dimensions are replaced by a constant radius.

Beyond a circle and spherical ability to combine multiple dimension types into one, spheres may be used to generalize shapes within all dimensions. This is done through application of something called the “N-Sphere.” An n-sphere is a way to describe dimensions using a generic sphere. For example, a 0-sphere is the set of points on the edges of a 1-dimensional line segment. A 1-sphere is the outer shape, or circumference, of a circular, 2-dimensional flat plane. A 2-sphere is the outer shell of a 3-dimensional sphere. As can be interpolated from the above information, any n-sphere can be described as the surface or shell of a (n+1)-dimensional shape. A 4-sphere can also be called a “quaternionic projective line.” In a nutshell, quaternions are a set of numbers that combine real and complex numbers into sets denoted by “H.” A quaternionic projective line, or 4-sphere, is a smooth, topological shell of the complex quaternion set in question.
EG:         H = a*1 + b*i + c*j + d*k where i, j, and k = √1


This is a 4-sphere depiction of a ring of quaternions. For more information on how this was generated through a sphere-stacking study, click here.
For easier notation, any n-sphere with n<3 is called a “hypersphere,” a 3-sphere can be called a “glome,” and a “unit n-sphere” is simply a sphere with a radius of unity (1) and denoted Sn.

Why do we care about n-spheres? As shown above by the hypersphere, glome, and 4-sphere examples, n-spheres are an interesting example of how circles and spheres are not only a unique geometric phenomenon due to their measurements but also can be used to summarize complex, high-dimension objects and ideas into more understandable, workable forms. This also allows for a much simpler way to calculate the volume and surface areas of higher dimension objects, as is shown in the figure below.


 

4-Dimension:

It is interesting to note that, to this point, all of the dimensions have been strictly lengths. The fourth dimension combines the concept of solid three dimensional space with the 4th dimension, generally considered to be time. This allows for transient (changing with time) analysis. For example, it is through the fourth dimension that changes in position, volume, length, width, etc. can be studied with respect to time. One example of this is the Continuity Navier-Stokes Equation that relates changes in the velocities in x, y, and z defined as u, v, and w with respect to the x, y, and z distance coordinates and changes in time in order to find the properties of continuous motion of a fluid.

Navier Stokes Continuity:               dρ/dt + (d(ρu))/dx + (d(ρv))/dy + (d(ρw))/dz = 0

Philosophical Application:


What is science without art? What is music without wavelengths, amplitude, and frequency? You can’t have one or the other and still be a balanced person, it’s best to learn how to see the science in philosophy and the philosophy in science.

               “What can you possibly pull from a bunch of points, lines, planes, and squiggles?” you might think. While I could sit here and make a billion jokes about thinking outside the three-dimensional polyhedron or some such nonsense, I could also describe a phenomenon that appears quite often in our society today.

A lot of people are content with living day-to-day, in the moment, and are focused on being “happy” with themselves. They simply exist where they are, not moving in a significant manner forward or looking far into the future. This is reminiscent of the zeroth dimension. People like this are perfectly alright with simply existing. For example, they never really put a lot of effort into the mental, physical, or spiritual growth necessary for a well-rounded, healthy person.

There may be somewhat limited expansions and variations on this theme in which people work to some degree, perhaps just getting by in school and work but never truly pushing themselves to the max. These people are reminiscent of the first, second, and/or third dimensions in which there is freedom to move, but still no motion within the time. This limitation shows how people today are not willing to make plans for the future and put in the hours and effort to make their dreams a reality.

People living with a “fourth-dimension” mindset have an understanding that, while life consists of the day-to day living and singular moments, those are simply the smallest components of a greater whole. Much like an atom is the most basic building block of the universe, a moment is the most basic building block of a lifetime. A forward-thinking person will take each moment as an opportunity to advance their future. They will spend their life with the mindset that, were they to die right at that moment, they would not be caught at rest, being lazy, or wasting time. While a person living in a lower-dimension lifestyle might desire a better life and may even work a little to that extent, a person living a fourth-dimension lifestyle will work till their very last breath to ensure the completion of their goals. Much as the mathematic fourth dimension has motion in time, the corresponding people-group will also work over time to make their dreams a reality.

Each day presents unique opportunities that can alter the course of your entire life. A person might choose to be stagnant and live simply for the enjoyments of that day alone, or they can choose to be cognizant and willing to work for those opportunities. Because really. What’s cooler? An “infinitesimally small point with no distinguishable dimension” or a set of many variables moving, adapting, and changing with time? One sounds a lot more fun than the other.


“Neither a wise man nor a brave man lies down on the tracks of history to wait for the train of the future to run over him.” ~Dwight D. Eisenhower