What Do Equations Expressing Higher Dimensions Look Like?

[schlynn]

Ok, I often hear that things become more easily understood in higher dimensions. I also hear that it is easier to express them in math than with words. But what does equations that express higher dimensions look like? Is it something as simple as (x,y,z,w)? I might be over thinking this far more than I should, but what are some examples of equations that take advantage of higher dimensions? For example something like x²+y²+z²=w². Is something like that a example?

 

[Mentallic]

But what does equations that express higher dimensions look like?

I'm assuming you're speaking of higher spatial dimensions, and ignoring time.
Should anyone truly know the answer to this and have been exposed to it visually, they are either high or godlike

And yes just throw in an equation with 4 variables, and you'll have just that, a 4-d graph.

Maybe if you think of it like this, things will be cleared up a bit.
On a 2-d coordinate system, the equation x=0 will represent a line.
With a 3-d coordinate system, the equation x=0 will represent a 2-d plane.
On this theoretical 4-d system, the equation x=0 is 3-d space as we know it.

But looking at the first example, the line is only one 'segment' of the plane itself, there are an infinite number of lines that can fit on this cartesian plane.
The same thing goes for the third example. Even though all known space is consumed by the equation x=0, there are still an infinite number of these spaces that can fit into the 4-d system.

 

[Architeuthis Dux]

I can understand your curiosity about expressing higher dimensions. In mathematics, we often use equations to describe and understand different dimensions. However, it is important to note that higher dimensions cannot be fully comprehended with our three-dimensional minds. They are simply a mathematical concept that helps us visualize and understand complex systems and phenomena.

Equations that express higher dimensions can take various forms and depend on the specific dimension being described. For example, the equation you mentioned, x^2 + y^2 + z^2 = w^2, is a common representation of a four-dimensional hypersphere. This equation shows how the coordinates (x,y,z,w) are related in four-dimensional space.

Another example could be the equation for a hypercube, which is a four-dimensional analog of a three-dimensional cube. It can be written as x^2 + y^2 + z^2 + w^2 = 1, where each variable represents a different axis in four-dimensional space.

In general, equations that describe higher dimensions involve more variables and often include exponents and other mathematical operations to represent the relationships between them. These equations can become increasingly complex as the number of dimensions increases.

It is also important to note that higher dimensions can have a significant impact on our understanding of the physical world. For example, the theory of relativity, which describes the relationship between space and time, relies on the concept of four-dimensional spacetime.

In summary, equations that express higher dimensions can take various forms and are used to represent complex systems and phenomena. While they may seem abstract and difficult to understand, they are essential tools in the field of mathematics and help us make sense of the world around us.