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
No comments:
Post a Comment