A point is the most fundamental concept and mathematical object in geometry. Just because it is fundamental, doesn’t mean it exists without depth. A point represents a location. It can be 1-dimensional, 2-dimensional or 3-dimensional. In fact, a point can have however many dimensions it needs.
Though not necessarily simple, a point is an object that exists at some location in space (Euclidean n-space) defined by its coordinates in each of the dimensions (n-dimensions in n-space) it exists. We often describe the coordinate of a point as an n-tuple with n coordinates. When “n” is refereed to, it can be any number. A tuple is just an ordered list of elements that can be zero up to n.
Points form the basis of all other geometric objects, such as a line, plane, space or hyperspace. Each of these geometric objects are a higher dimensional object of the previous element in the list. Within each of these objects there are an infinite number of points that define the given object.
Example for your mind
Think of a point like an address. You live in the Milky Way Galaxy. You live on planet Earth. You live in a given country with a specific region, province or state. You live in a particular city, town or village. You likely have a street and an address. Each layer in the onion defines another aspect for your location with more and more precision.
A point could be large, say 1, to generalize a location. Or a point can be very small indicating precision or significance, say 1.000000000000001. Think of the planet Earth as a point in space. From the perspective and relative size of the of the Milky Way galaxy, Earth is very small, but it still has a point location at a macro level. Where you are currently sitting reading this represents an even more precise location, both on the plant and within the galaxy perspectives. All the way down to the smallest atom at the center of the planet. Everything has a location and it is unique.
What is relevant depends
Traditionally in high school level mathematics, we work in 1 or 2-dimensional space. These are likely referred to as a number line and coordinate plane respectively. We also tend to associate some letter to each dimension, most likely x and y corresponding to each axis.
In college mathematics courses, you may expand into 3-dimensional space or even higher depending on your major. In 3-D space, the third character associated is typically z.
Points in 1-Dimensional Space
When we work in 1-dimensional space, we are limited to one axis or a number line. Typically this is the x-axis. It could go from from -4 to +4, 0 to 4, -4 to 0 or -n to n, where n represents any number. Shift the numbers left or shift the numbers right. It makes no difference. The only constraint is it only exists in one dimension.
Points in 2-Dimensional Space
In 2-dimensional space, we gain an additional axis, the y-axis. The added dimension allows us to give more details about a points location. It also opens up additional concepts in geometry, algebra and calculus.
Points in 3-Dimensional Space
In 3-dimensional space, we gain an additional axis, the z-axis. This axis provides a feature of depth, rather than thinking of a point in a flat 2-D space. 3-dimensional space, also opens up additional concepts in geometry and calculus.
Points in Hyper-dimensional Space
When get past the traditional 3-dimension space, considered physical space, we enter into hyper-dimension space. When working with data, you often handle information that is hyper-dimensional, by ascribing aesthetic features, such as color, shape, size, etc, with each attribute representing some piece of additional information or dimension.
Think about a point in a physical 3-D space. You can add time, which denotes where a point exists in space at a particular moment. A point could be assigned some piece of information to characterize it. The gravitational value could be ascribed to a point. You could define it’s electromagnetic behavior and more. Although these are just examples of various elements of a point coordinate, they demonstrate that a point can be defined more and more as more information is added in higher dimensions.
For More Information
- Point: http://mathworld.wolfram.com/Point.html
- Euclidean Space: http://mathworld.wolfram.com/EuclideanSpace.html
