A Function Of 3 Values Which Never Gives The Same Output For Different Numbers

Question

I am currently looking at graphs which describe the movement of an object through spacetime. The graphs would look quite similar to Feynman Diagrams which describe the movement of some subatomic particle through spacetime. However, as you can see on a graph (or at least the kind which I am using), there are only 2 axes - often labelled as $\ x$ and $\ y$ - that represent a 2D plane. However - the spacial axis, $\ x$, can only represent 1 dimension. Therefore I would have to find a way to compress 3 spacial dimensions into 1. I will call this function $\ R$ and it will take the form $\ R(x,y,z)$ where $\ x,y$ and $\ z$ represent the 3 spacial dimensions of our universe.

But what about the movement of an object through space? I will represent this as a matrix where the first column is a vector of the "starting position" of the object, and the second column is a vector of the ending position. With this known, let our object start at position $\ \begin{bmatrix} 1 \\ 4 \\ 3\end{bmatrix}$ and move to position $\ \begin{bmatrix} 2 \\ 5 \\ 7 \end{bmatrix}$ over some time, $\ t$. We can represent this motion in the format: $$\ \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 3 & 7 \end{bmatrix}$$ This matrix represents the movement of an object through space, but not through time. We can set $\ t $ to some value, lets say "1". So now we have the movement of an object through space and a measure of the time it takes. But how would we compress this matrix into two numbers that are inherently 1 dimensional. We can do this with our $\ R$ function, but what is the $\ R$ function?

I tried using Eulers Number, $\ e$ to find this, and I tried the magnitude of both vectors added together and then put into a $\ \sqrt{}$ function, but I can't figure out how I would have a function which, for every input $\ x, y$ and $\ z$, would always give a different output for each input. Similar to: $$\ R(1,1,1) \neq R(1,1,1.000001) \neq R(1,1,1.000002)... $$

Does anyone know of a function which behaves like this?

What exactly do you mean by "I tried using Euler's number to find this" — Ben Grossmann, Jul 26 '20 at 09:42
Because $\ e$ is irrational, I figured that raising $\ e$ to the power of each number and then adding the numbers together would yield different results for each number, but after testing this, I was wrong. — sciencepiofficial, Jul 26 '20 at 09:46
It sounds like you are looking for either something like the maps described here or a space-filling curve — Ben Grossmann, Jul 26 '20 at 09:49
@BenGrossmann after checking the question which you linked, I can clarify that it is indeed what I am looking for. — sciencepiofficial, Jul 26 '20 at 09:52

score 1 · Answer 1 · answered Jul 26 '20 at 09:42

There is no continuous injective map $\Bbb R^3\to \Bbb R$. In fact, this already fails for $\Bbb R^2\to \Bbb R$. Assume $f\colon \Bbb R^2\to \Bbb R$ is continuous and injective. Pick three points $p_1,p_2,p_3\in \Bbb R^3$ (or $\Bbb R^2$ or any $\Bbb R^n$ with $n>1$) where the indices are chosen so that $f(p_1)<f(p_2)<f(p_3)$. Now in $\Bbb R^2$, chose any path from $p_1$ to $p_2$ that avoids $p_2$. Then the image under $f$ of this path must begin at $f(p_1)$ and continuously move to $f(p_3)$ - without ever passing $f(p_2)$. This is impossible.

A Function Of 3 Values Which Never Gives The Same Output For Different Numbers

Does anyone know of a function which behaves like this?

1 Answers1