Summary
A point, with its lack of length, area, and thickness, can be said to be the foundation of geometry.
When you fill that with ink and drag it in one direction, you get a line. Because points don’t have area, lines likewise don’t have width and thickness. The reason why we call lines one dimension is that there is only one degree of freedom: east and west. Here, we can describe the location of a point on the line as P(a).
When you fill the line with ink, again, and drag it in one direction, the trace will be a face, a two-dimensional shape. Because lines don’t have width and thickness, faces don’t have thickness either, but it does have volume. The reason why we call faces two-dimensional is that the degree of freedom isn’t just east and west, but north and south as well. Here, we can describe the location of a point on the line as P(a, b). a denotes the extent of movement east to west, while b denotes the extent of movement north to south.
If you fill a face with ink yet again and drag it perpendicular to the face, you get a 3D object. It’s three-dimensional because, for one, it has a volume, and for another, it can move in not only cardinal directions but up and down as well. A point on the 3D object can be described as P(a, b, c), where a denotes the extent of movement east to west, b denotes the extent of movement north to south, and c denotes the extent of movement up and down.
What you get when you keep on doing this, you get what’s called a hypercube, of which a point on it can be described as P(a, b, c, d). Of course you can’t do that, but if you keep at it, you can get a n-dimensional figure, and a point on it can be described as P(a1, a2, … an). This is an expansion of space of which is only natural in mathematics.
To help you understand, let’s assume a space consisting of a single line. Whenever it moves, it can only move left and right along the line. Whenever you move point P you’re moving on top of a single line, and whatever point there is can be met by moving P. With a single degree of freedom, it is one-dimensional. Points can be sorted by positive and negative in reference to the origin, meaning they can be sorted in 2(21) ways.
Let’s move on to faces. If it were a line, you’d be able to specify the locations of points using the extent to which they moved left and right, but in a face, you can move up and down, left and right. This necessitates a different method of specifying the location of a point, and we call it either Cartesian coordinate system, or rectangular coordinate system. With these, you can have two lines be orthogonal to each other, call the horizontal line x and the vertical line y, and know the location of a point Q using the knowledge of how far they are from the lines. Since points moving on the lines don’t affect the other line, the components are independent to each other. Therefore, a face is two-dimensional, and is divided into 4(22) ‘quadrants’. Axes don’t belong to any quadrants.
Now, let’s move on to the third dimension. In order for three lines to be orthogonal to each other, you have to suppose a line orthogonal to a face made by two lines orthogonal to each other. Here you can describe the location of point R as R(a, b, c), the components of which move independently. The third dimension is divided into 8(23) ‘octants’, and the axes, just like before, don’t belong to any quadrants.
By taking a look at this trend, we can imagine that the fourth dimension would be four lines orthogonal to each other and divided into 16(24) parts, and that the fifth dimension would be five lines orthogonal to each other and divided into 32(25) parts. We can only imagine, however: we can’t, for instance, draw these on a paper.
Because we haven’t been there, we can’t know what happens there. Using the certainty, rigor, and natural extension of mathematics, however, we can take a glimpse of the higher dimensions, all thanks to mathematics.
While the importance of mathematics, which can both reduce and expand, is demonstrated, we still have to learn how to apply this in our day-to-day lives. This is to develop the ability to solve real-life problems through mathematical thinking.
Implication
Mathematics lets us think things that are beyond our ability to think intuitively.
Underlying context – Background behind
There are concepts of which we can’t think of intuitively. Mathematics helps us think those things.
Notions – Key ideas
Dimension: count of numbers required to describe the location
Index – Source(s)
How to Think Like Mathematicians / 피타고라스 생각 수업
05 Going Beyond Vertex, Edge, and Face to nth Dimension | Expansion / 점, 선, 면을 넘어 n차원으로 – 확장
Trajectory – Where I’m headed now
Let’s find out ways to apply mathematics to real life.
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