A math equation with finite terms that plots every real (x,y) point

Question

Is it possible to create an equation that plots every point?

You can stack plotted equations by moving everything to one side such that it equals 0 and multiplying by another equation that is in the same set-up.

E.g. $$(y-(x^2))\cdot(y-(x^2)-1)\cdot(y-(x^2)-2)\cdot(y-(x^2)-3)=0$$

This stacks 4 quadratic equations. You could stack infinite equations of the form y-x^2+c=0 where c adjusts by infinitesimals from -∞ to ∞, however this equation would have infinite terms.

Is there an equation in finite terms that will plot every real (X,Y) value?

Answer 1

Well, you could try $$x+y=x+y$$ I suppose.

Edit: I'd like to comment on the infinitely-many-quadratics idea in the question. It's quite appealing: the solutions to the equation form an infinite family of parabolas, each shifted vertically from the one before, one for each bracketed expression. It's tempting to try to fill the plane by putting them infinitely close together. But there's a problem.

The left hand side of of the equation lists the quadratics—meaning there are countably many of them, and hence countably many corresponding parabolas.

For a given value of $x$, we need a parabola for each possible $y$ value. That is, one for every real number. But the real numbers are uncountable, so we haven't got enough parabolas.

Answer 2

What about something like $$x=\frac{\{r\}}{1-\{r\}}\cos\bigg(\frac{\{r\}}{1-\{r\}}+2\pi\phi[r]\bigg)$$$$y=\frac{\{r\}}{1-\{r\}}\sin\bigg(\frac{\{r\}}{1-\{r\}}+2\pi\phi[r]\bigg)$$with $r$ from 0 to ∞, & $$\phi={\sqrt{5}±1\over2}$$ (or any other irrational number for that matter), so that you'd be plotting an infinitude of infinite spirals, with phases evenly distributed around 'the clock'.

Actually, I think there might well be shortcomings with this - countability -wise. But something kind of in that vein, perhaps.

Actually, I've just noticed that someone else has addressed this countability problem!

I am beginning to wonder whether it might perhaps be fundamentally impossible ... unless there's some way to circumvent the problem with $$\operatorname{range-of-values-in}\{r\}\cdot\cdot\cdot \operatorname{uncountable}$$&$$\operatorname{range-of-values-in}[r]\cdot\cdot\cdot \operatorname{countable} .$$ If no-one furnish me with a convincing argument to the contrary, I'm rolling with the idea that there's some ultraslick way of doing it! ... like that construction for a distribution of points on a sphere by which Möser's classical limit on the number of closest neighbours points on a sphere (distributed in a certain way) can have was circumvented ... something of that grade of slickity.