Tag Archives: philosophy

Math Moments – Extending Spaces into New Dimensions

So far, we have been exploring a 1-dimensional “space” of points along an infinitesimally thin line. Mathematicians call this a 1-dimentional space. The concept of symmetry implies that the line looks exactly the same in both directions. Therefore, this line must extend on forever in both directions, there is no point on the line where it ends. The line looks exactly the same on both sides of every point.

The concept of point symmetry in 1-dimensional space also means that if you were to reverse the direction of the “ruler” that measures the distance between any two points in this space, you will still measure the same distance between the points. This concept of invariance of direction is one of the most important characteristics of space. Of course, we need a lot more than just one line to describe all the space around us.

In order to expand the dimensions of space beyond this 1-dimensional space, we must assume that there is some point that is not in this space, a point that is not on the line. When we find a point that is not on the line, we can create a new line between any point on the line and the new point. This defines a new 1-dimensional space (a line) for every point on the original line, all of these new lines pass through the new point.

Since there are an infinite number of points on the original line, we see that there are an infinite number of 1-dimensional lines that can be drawn through the new point all in different directions. The space formed by this infinite number of lines is called a 2-dimensional space. It is called 2-dimensional due to the “language” necessary to describe the location of any point in this space. To locate any point, we must choose a reference point, then first (1) we need to choose one of the lines passing through the reference point, and second (2) we use the methods we have already discussed to determine the location of a point along this reference line. Thus, the location of a point thus requires 2 numbers, one (1) to locate the reference line, and the other (2) to determine the location of the point along this line.

The concept of point symmetry in 2-dimensional spaces also comes into play. It means that measuring distances is the same in every direction. This invariance of direction means that if we have a “ruler” that measures a distance along any of the lines in this space, and we change the direction of the ruler, the length of the ruler does not change.

Imagine how the world would look if measuring lengths in one direction were different than lengths in another direction. If the invariance of direction was not true, then when we changed direction, the distance between things would shorten or lengthen, depending on which direction we faced. This is not what we experience in real life.

We can use the same concepts that we used to extend 1-dimensional space to 2-dimensional space to extend into spaces into any number of dimensions. How many distinct points do you need to generate a 3-dimensional space? Think about it. We will discuss how it can be done later.

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Math Moments – The power of symmetry

If we were to consider any two points in the universe (A and C) and draw an imaginary line segment between these two points, there would be only one singular point on that line segment that preserves symmetry (under reflection). This is the midpoint (B) on the line segment which is exactly the same distance from each of the endpoints. You can reflect (flip) either one of the endpoints through the midpoint and it would fall exactly on the other endpoint. Such symmetry applies to all points on the line segment that are the same distance from this midpoint.

If you were to pick the midpoint as the reference point and the positive direction pointing toward one of the endpoints, then from the midpoint, the line segment in the positive direction looks exactly the same as the line segment in the negative direction. You could conceptually switch the positive direction with the negative direction, and nothing would change. This is called reflection symmetry, for obvious reasons.

If you have reflection symmetry, then anything that exists at a certain distance along the positive direction, also exists at that same distance in the negative direction; just like looking at things through a mirror. We use such symmetries in atomic physics all the time to make things easier.

For example, the electric field between two identical electrons, as observed from the midpoint between them, looks exactly the same when you are facing one electron as it does when you are facing the other. From the midpoint, everything is exactly the same in both directions, you cannot tell the difference between the electrons nor any of their physical properties from this perspective. To tell the difference, you would need to “break the symmetry” by introducing something else that is not symmetric to the midpoint.

So, once you have seen what exists on one side of the “mirror” we also know what will exist on the other side. It makes the math (and the diagrams) so much easier. We always should seek out these points of symmetry when we are doing the math, it makes everything so much more…symmetric.

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Math Moments – How many Points are on a Line Segment?

If you have fun with the concept of being able to zoom in to a single Point forever, always getting closer and closer but never arriving, you are one step closer to understanding the concepts of advanced math and calculus.
You saw that with any two points you could always find a point that is exactly in the middle. You can then take this midpoint as a new end point and find another point that is in the middle of it. Since points have no size, you can keep on cutting these line segments in half forever.


No matter how close two points are together, you can still find a point in the middle. Thus, there are an infinite number of points in any line segment, no matter how short it is. This is an example of the concept of a countably infinite set of points. It is possible to write out list the location of these points as their distance from the reference point: 1/2, 1/4, 1/8, 1/16, … and so on forever.


You could just as well cut the segments into thirds, with the location list of points: 1/3, 1/9, 1/27, … and so on forever. Notice that this list of points does not have any of the same points as the one in the last paragraph. So, it appears like you could go on forever filling in all the gaps by dividing the segments up evenly and still never fill in all the gaps. So how many points can fit into the line segment between any two endpoints, no matter how short? Can we ever find a way to fill in all the gaps between points? To fill in all the points between any two points, we would need what is called an uncountably infinite number of points. This is a concept that mathematicians have not yet resolved.

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Math Moments – The Number Line

Let’s imagine that you have a universe with only two points in it. You now draw a line segment between the two points and choose one of them to be the reference point. The location of any point along the line segment is now determined by the distance it is from the reference point. Now make a third point on the line segment that is exactly half-way between the two points. This new point is located at exactly half of the distance between the two end points.


Can you see the symmetry that is formed by these three points? The distance from the center point to either of the two end points is exactly the same. If you now were to choose the center point to be the reference point, then both end points would be the same distance away, but in opposite directions.


You would now need more than just the distance measurement alone to determine the location of points along the line. With three points, now direction becomes important. Again, using the axiom of choice and can choose which direction from the center point is positive, and which direction is negative.


We can then, finally, determine the location of every point on the line by specifying both a distance and a direction (positive or negative) from the center point. This arrangement of determining the locations of points on a line is called a “number line”. We call the distance and direction of each point on the line the Point’s coordinates. A Point’s coordinates uniquely determine its location on the line. Finally, we have a way of determining the location of a Point.


The center point is a special point of symmetry between the two endpoints. We can “rotate” the direction (exchange negative for positive directions) and the location (distance and direction) of the two identical endpoints would be exactly the same. This is called “rotational symmetry”. It means you can switch reference directions and the list of all the point coordinates will be the same. Remember that since you can’t tell the difference between points, the order of the list of points is not important.


Please believe me, taking the time to understand these concepts makes math so much more fun and easier later on. Learning these concepts is like learning the basic vocabulary of the math language.

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Math Moments – 1-Dimensional Space

How can the exact location of the Point be described? This is a conceptual problem. It is also where some very advanced concepts of math (found in calculus and number theory) come to play. After some thought, it becomes obvious that if there was just one point in the universe, all by itself, it does not have an absolute location until you look at where it is relative to another point in the universe.


If there were only two points in the universe you could imagine a line segment drawn between the two points, and then there is some distance between the two points, you could pick one of the points as the reference point. Then you could describe the location of the second point as being a certain distance from the reference point. You only need one (1) distance measurement from the reference point to describe the location of the Point. This is why such a line is called 1-dimensional space.


You have now discovered quite a few interesting concepts, including the concept of the number “2”. Two identical objects in space that are distinguished by the distance between them. You have also discovered the concept of the number “1”, the concept of distance, and you are also starting to realize why Einstein’s theory is called the theory of relativity. Distances and times are only measurable relative to a reference point, and you can pick any of the points to be the reference point.


The idea that you can pick any of the points to be the reference point is called the axiom of choice. An axiom is something that is assumed to be true even though it has no definite proof. The axiom of choice forms the conceptual foundation for all mathematics.


Another concept that comes from using the distance from a reference point to describe location is the idea of symmetry. You can exchange the locations of two points, and you have not changed anything. It is impossible to tell the two points apart, and both points still have the same locations relative to every other point. This is called exchange symmetry. Symmetries makes math and physics much easier.

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