Thinking
In Four Dimensions
Height, width, and depth are three very important dimensions
that allow us to judge shapes on a three dimensional plane. Through seeing this
we are able to measure the surface area of an object, and/or its distance
between another object. If we had no depth, and could only see the height and
width of things, then we would constantly be bumping into everything! So that third
dimension is a very important factor. Using it I will explain why hypercubes
are not truly four dimensional, and I will show what one will look like on a
geometric plane.
Albert Einstein in his Theory of Relativity said that Measurements of various quantities are relative to the velocities of observers. Velocity is a measurement of change, which
happens over time. If the quantities
were to be our three dimensions of height, width, and length then the fourth
dimension is velocity which we can say is time. What we have now are geometric
shapes named "Hypercubes" which are said to have four dimensions.
There are many Hypercubes that can be made, but a 'Tesseract' is one of the
most simplest of the shapes. A Tesseract is a cube with a cube inside of it
creating a total of eight cubes within itself. In short, it is cube-ception:
People have
put this in a three dimension simulation of where it rotates within itself
which in essence would give it velocity, however on a two dimensional plane the
shape remains stagnant, and a cube could also be rotated on a three dimensional
simulation. In a two dimensional plane, as you see above, it only becomes an
exploitation of depth to create an object within itself. So I believe that it
is not a true four dimensional shape.
A four
dimensional shape will have to be something that does not remain the same in
any plane of conception. So I formed a theory that a four dimensional shape
could be drawn if our perception of it was constantly changing. One way this
could work would be to draw it as a paradox like the "Penrose Stairs"
drawing. So after a long night of critical thinking, simple line drawing, and lots
of erasing, I was able to place a four dimensional shape on a two dimensional
plane:
As you look
at the shape your vision of it teeters between it being a cut out of four
squares connecting to each other, and a cube. It fails to make sense of how it
would exist in three dimensions, but you perception keeps adjusting itself to
try and make sense of it. This constant change in perception of the shape gives
it that fourth dimension of velocity. And through this knowledge we can create
a hyper four dimensional shape:
Which creates a total of eight four dimensional shapes, like
the Tesseract does with making eight three dimensional shapes.
I feel that
modern math has missed conceived what four dimensions really look like. Hypercubes
are only an exploit of surface area in three dimensions. True four dimensional
shapes maintain constant change over time which gives it velocity. And velocity
is relative to the other quantities being measured. This is my understanding of
'outside the box' thinking and I hope this inspires you to think critically
about how dimensions are formed.
Works Cited
Einstein, A. Relativity: The Special and General Theory,
New York: H. Holt and Company (1916 (translation
1920)), Print.
Ernst, Bruno. The
Eye Beguiled: Optical Illusions. Benedikt Taschen. 1992, Print.
T. Gosset: On the
Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan. 1900, Print.
Hanson,
J.R. International Journal of Mathematics
Education in Science & Technology. Harrisonburg:
Taylor & Francis Ltd. 2014,
Print.
Grosse, Harald. Communications in Mathematical
Physics. Munster: Springer Science & Business
Media B.V. 2014, Print.
DeVito, Jason. Differential Geometry & its
Applications. Tennessee: Elsevier Science, 2014. Print
