Explanation
In this lesson, we learn how to understand \(3D, 4D,\) and higher dimensions in real life from a mathematical point of view.
In mathematics, a dimension can be thought of as one independent piece of information.
So, the number of dimensions depends on how many independent inputs we use.
For example:
\(3D\) usually means we use three coordinates:
$$(x, y, z)$$
These tell us the position of a point in space.
But we can add more information, such as time, temperature, color intensity, wind speed, or pressure.
Each extra independent input can create a new dimension.
\(3D:\)
A \(3D\) point has three pieces of information:
$$(x, y, z)$$
For example, a point on a face or a building can be described by its position:
right / left
forward / backward
up / down
So, \(3D\) means:
location in space
\(4D:\)
To make a \(3D\) point into \(4D\), we can add time.
For example:
$$(x, y, z, \text{time})$$
This tells us where the point is and when we are looking at it.
So, \(4D\) can mean:
$$3D \text{ space } + \text{time}$$
\(5D:\)
To make it \(5D\), we can add another input, such as temperature.
For example:
$$(x, y, z, \text{temperature, time})$$
Now we know the location, the temperature at that location, and the time.
\(6D:\)
To make it \(6D\), we can add another input, such as color intensity.
For example:
$$(x, y, z, \text{temperature, color intensity, time})$$
Now the point has six pieces of information.
The key idea is:
In mathematics, the number of dimensions equals the number of independent inputs.
Formula / Rule
A simple rule is:
$$\text{Number of dimensions } = \text{number of independent inputs}$$
Examples:
$$3D \longrightarrow (x, y, z)$$
$$4D \longrightarrow (x, y, z, \text{time})$$
$$5D \longrightarrow (x, y, z, \text{temperature, time})$$
$$6D \longrightarrow (x, y, z, \text{temperature, color intensity, time})$$
We usually put time at the end, so it is easier to read the location first and the time last.
Example
Suppose we choose a point on a face.
First, we describe its \(3D\) location:
$$(0, 4, 2)$$
This means:
\(x = 0 \longrightarrow \) no movement right or left
\(y = 4 \longrightarrow \) moved forward
\(z = 2 \longrightarrow \) moved up
So, this is a \(3D\) point.
Now add time:
$$(0, 4, 2, 1)$$
This is \(4D\) because we added time.
Now add temperature:
$$(0, 4, 2, 37, 1)$$
This is 5D because we added temperature.
Now add color intensity:
$$(0, 4, 2, 37, 0.5, 1)$$
This is \(6D\) because we added color intensity.
So, each time we add one independent input, the number of dimensions increases by one.
Video Explanation
Practice
Continue Learning
- Length, Area, and Volume: What’s the Difference?
- Difference between \(1D, 2D,\) and \(3D\)
- \(3D, 4D,\) and Beyond in Real Life (Current lesson)