Geometry of nth dimension

[mathworker]

In this pdf it was written ,

" we probably have very little in our background that gives us geometric insight to the nature of \(\displaystyle R^5\). However, the algebra for a line in \(\displaystyle R^5\) is very simple, and the geometry of a line is just like the geometry of \(\displaystyle R^1\)"

I would be grateful if you help me look in how geometry of line will be in \(\displaystyle R^5\) or what is the geometry in dimensions greater than 3

 

[mathbalarka]

Re: geometry of nth dimension

what is the geometry in dimensions greater than 3

It's similar to how you define geometry on $\mathbb{R}^3$, i.e., by defining each point as real 3-tuples and usual operations on them. For n-space geometry, define a real n-tuple instead with the same usual operation acting analogously over them.

The geometry of a line over $\mathbb{R}^5$ is similar to the geometry of a line in $\mathbb{R}^1$ in the sense that the lines in the former can be projected into any lower order spaces without the loss of any dimensions. That is, line is a one-dimensional object.

 

[mathworker]

I am rather asking for a way to visualize the geometry of \(\displaystyle R^n\) for \(\displaystyle n>3\) ,I hope I am making some sense

 

[mathbalarka]

I am rather asking for a way to visualize the geometry of \(\displaystyle R^n\) for \(\displaystyle n>3\) ,I hope I am making some sense

That is something hard, then, as we are in the 3-space. The best one can do is to draw out Schlegel diagrams or perspective projections of objects of 4-space or higher.

 

[Deveno]

When one first encounters lines, it is often in the form of a function in the plane, such as:

$y = mx + b$.

If one is not given the slope $m$, but rather two points $(x_1,y_1),(x_2,y_2)$ that the line passes through, one uses a formula such as:

$y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

With a bit of algebraic "arm-wrestling" we can see that this is just the line:

$y = \dfrac{y_2 - y_1}{x_2 - x_1}x$

"shifted" so it passes through the point $(x_1,y_1)$.

The quantity $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ just depends on the 4 (constant) numbers $x_1,y_1,x_2,y_2$.

We can re-write the line $y = \dfrac{y_2 - y_1}{x_2 - x_1}x = mx$ as the set of all points in the plane of the form:

$\{(t,mt): t \in \Bbb R\}$

or, equivalently as: $t(1,m)$, or even as:

$t(x_2-x_1,y_2-y_1)$

If we call the points $(x_1,y_1) = P,\ (x_2,y_2) = Q$, then one way to describe the line going through $P$ and $Q$ is:

$t(Q - P) + P$, which goes through the point $P$ when $t = 0$, and goes through the point $Q$ when $t = 1$.

This last form generalizes nicely to $n$ dimensions (where each point has $n$ coordinates), we can also write a line as:

$L = \{t(Q-P) + P: t \in \Bbb R, P,Q \in \Bbb R^n\}$

The "one-dimensional-ness" of the line $L$ is reflected in the fact that we have one parameter ($t$) even though it is moving through two points in "$n$-space".

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As far as "visualizing" the geometry of $n$-dimensional space, it is rather hard to do "concretely": the best we can do is look at various 1,2 and 3-dimensional "shadows" of such a space. This is rather like imagining that the "top,front and side" views of a 3D object are actually all the same thing which can't be fully portrayed on a flat (2D) piece of paper. But mathematically, an $n$-dimendsional object behaves much the same as the lower dimensional objects we CAN visualize, it just has "more dimensions". One can have a "quasi-visualization" of 4-dimensional objects using 3-dimensional animation (remembering that although we are seeing an object "evolve" over time, what we are actually looking at is a "static" object from different 3D vantage points...analogous to representing a static sphere to a 2-dimensional being as a point that grows to a circle of the same radius of the sphere, and then shrinks back to a point again).

Because of the difficulty of "seeing" things in higher dimensions, we do something else entirely: we focus on the mathematical properties, which turn out to be much more tractable.

Although at first, this appears to be conceding defeat of some sort...it is actually liberating: we no longer have to think of higher-dimensional objects as "spatial" in nature, but as just depending on $n$ linearly independent variables. This lets us use the mathematics to describe things we might not have thought of as "spatial", such as: game outcomes, colors, economic trends, or collections of functions. Geometry becomes freed from its "earth-bound" origins, and becomes a tool for investigating MANY types of things.