To conceptualize a mathematical point, it’s necessary to first imagine the third, second, and first dimensions. We must imagine the first three dimensions in order to achieve the nothing that is a point. Grasp the real and bring it down to something infinitely small and abstract.
Anything with mass, or more specifically volume, lies in the third dimension. The third is the most tangible. Everything you feel is in the third dimension. It is the easiest to picture, but the hardest to draw (without being impossible to capture like the fourth dimension). Any real object in this world lies in the third dimension. The third dimension holds polyhedrons, and spheres, and platonic solids, and any morphus object without a name. It also carries certain massless things like planes which hold the second dimension and vectors which hold the first.
One of the most basic equations for something that lies in the third dimension would be the plane equation: z=a(x)+b(y)+c. This equation is some eternally linear square that only crosses the z axis on an exponentially thin string. This plane looks like a piece of paper, if that piece of paper stretched itself into infinity and made itself untouchable.
Nothing makes me feel as small as infinity.
Diego Velázquez captures the concept of infinity better than I ever could. Velázquez composed his masterpiece Las Meninas so that the scene stretches beyond the viewer. A priest and a young dwarf look beyond the frame of the painting into something somewhere behind the viewer. They look past the audience into something of deep importance. The mirror near the center of the painting gives a hint as to what lies beyond the viewer. The mirror contains the image of the king and queen, looking down at their daughter who can be seen in the painting’s center. The mirror carries the implication that the king and queen stand behind the viewer, and with that comes the infinity of the third dimension.
I can’t look at Las Meninas for too long. When looking at it, I feel stuck in a world that spreads behind me and in front of me. The queen's chamberlain, Don José Nieto Velázquez, stands in a door leading to a room unseen in the painting, as if taunting me with the idea that the infinity behind me is echoed in an infinity ahead of me. This painting makes me feel like the ghost of the future looking into the past.
A plane carries this implication as well. When looking at a mathematical diagram of a plane, to properly imagine what we are seeing we must look beyond ourselves. What we are looking at is more than just the two-dimensional drawing, and more than anything that could physically exist in this universe.
The second dimension is a little stickier. In essence, the second dimension is the third dimension stripped of its mass. In the second dimension, a pyramid becomes a triangle, a cone becomes either a triangle or a circle, a sphere becomes a circle, and a cylinder becomes either a circle or a rectangle. Although the plane we just created lies in the third dimension, it is a two-dimensional object.
The plane has no mass, no weight to it. Everything lying on the plane is deeply and viscerally two-dimensional.
In the second dimension, anything with weight or volume is drained of its substance until it can lie flat on a page. If the third dimension lives in objects the second dimension lives in their shadows. A mathematician would say something three-dimensional can be represented by an “x,y,z” axis while a two-dimensional object only needs two axises “x,y”.
Another way to imagine the second dimension is to look in a mirror. The image on a mirror carries no weight and has no volume. Mirrors also contain something deeply important to the second dimension, movement and time. A point on the second dimension can move in any way its function allows. In this way the image on a mirror can still contain any movement constrained only by physics. If you’ve ever looked in a mirror, and I assume you have, the reflection staring back at you is a weightless, massless, two-dimensional version of yourself.
The idea of a mirror reflecting some other version of its viewer is cross cultural. An old Roman legend touches on this idea. In Ancient Rome there was this belief that mirrors would temporarily capture the soul of the person looking into it. They also believed that if you break a mirror while looking at it, the mirror will then capture a small piece of your soul. In the Jewish faith, when someone dies, their family members will cover all the mirrors in the house for fear the soul of the deceased loved one may become trapped. In medieval Japan, mirrors were used to project the spirits of their Gods. In more recent times, there is the idea of Bloody Mary, a spirit trapped in the world of mirrors.
The soul mirror archetype is reflected in the idea of some weightless soul. This archetype implies some connection between the two-dimensional reflection of a person might have some correlation to a soul. Although humanity has never lived in the second dimension, a soul mirror archetype still exists which assigns deep significance to some two-dimensional force. What the second dimension lacks in volume, it makes up for in assigned significance.
Let’s move now to the first dimension. The first dimension is a single axis, a line. It is the second dimension without shape. The shadow of a shadow. A triangle represented in the first dimension is either one point, two points, or a line, depending on where the triangle is cut. A circle is either two points or one. A rectangle becomes either two points, a line, or one point.
There is a trick for seeing an object’s representation in the first dimension. First I would take a cheese wire and wrap it around my fingers so that they become the color of wine, and the wire is taut enough to cut through anything. When an object sliced into two parts with that cheese wire, the residue left on it would then resemble that object’s shadow in the first dimension. Points still live on the first dimension but they are now restricted to one line with points that slide back and forth representing where and when the cutting must happen.
In a one-dimensional world, the only concepts that exist are time and direction. There is only a constant sliding between left, and right. There is still some concept of direction, but only forward and backward. A map’s compass cut in half.
A line, in its most visceral form, is the connection between two points.
Light and color are often described as a spectrum, and in that sense, they exist in the first dimension. Every color lies somewhere between the bluest point and the reddest point on this spectrum. By definition, there must be some ultimate blue point, more blue than an emperor butterfly, a peacock in the moonlight, or the blue man group. Bluer than that Eiffel 65 song, or the feeling of ice on skin just before it gets painful. On this line there exists some infinitely red point as well. This point is redder than the skin of strawberries mashed into an indistinguishable pulp. Redder than the rolling stone cover. Redder than the anger of a broken teenager as he makes dents in his solo cup and speeds past a stop sign. At either end of this spectrum, there exists a point of infinite blueness and redness, impossible to capture but not impossible to imagine. Every shade of purple and red and blue lie between these two.
Results of the Myers Briggs test are shown on four lines. A series of lines to represent what personality traits we identify ourselves as. The first dimension is used to provide some rough snapshot of personality, of who someone is. Every human being can be represented by four points lying on the Myers Briggs lines.
The second dimension captures the soul, the first dimension captures its essence.
In America, political ideologies are first-dimensional: some entity is either left, right or in the middle. We use the first dimension to describe the politics of entire countries and continents. A single line describes the ideals of a generalized county, or state, or the country itself.
A point is a line without length. When we strip the third dimension of its volume, the second dimension of its shape, and the first dimension of its length, what do we have left? What lies in this supposed dimension of zero? In some sense, we have nothing left. But, there is still the idea of where that line was. There is the location of where we stripped the third dimension down to nothing. What we have left is the imprint of something with length or mass, but represented by something without. A point is a way for us to express eternal emptiness.
The third dimension stripped to nothingness is a point.
To make up for this nothingness, the point creates everything. A line is made up of several points stacked on top of each other. More specifically the first dimension is infinitely many points crowded together in single file. The second dimension is made up of stacking lines forever to give them shape. It is comprised of infinitely many points made of infinitely many lines. It is an even greater infinity of points than the infinity of points that make up a line. The third dimension is made up of infinitely many planes made of infinitely many lines made of infinitely many points. When we strip away all of those infinities, we are left with a point that can duplicate itself infinitely again to recreate what we have just stripped.
TeresaJean (TJ) Herleth graduated from NAU in the spring of 2021 with a double major in English and Mathematics. She was a manager for this magazine.
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