Normally, complex numbers are plotted on a 2d plane where real numbers are the x coordinate and imaginary numbers are the y coordinate. I'd like to plot complex numbers in such a way that I can represent it in one dimension.

Sure, I could plot the complex number(s) in a 2d-plane, take cross-sections of it, and line them up, but that isn't very "mathematical" and it loses a lot of beauty that many images on the complex plane have (hint: fractals).

I was wondering if there were any methods of plotting numbers on the complex plane. Maybe using imaginary numbers to determine spacing between real numbers, or using some other algorithm to combine the real and imaginary numbers and listing that. I'd like to find something that isn't just listing the real and imaginary values together on a number line, but something that actually preserves the beauty of arrangements in the complex plane in a one-dimensional environment. This question could also be more generally taken as "how to plot 2d coordinates in one-dimension" if that solves the problem.


I ended up using a hilbert curve to travel through the points. This wasn't the "mathematical" approach I'd originally hoped for, but it worked for my purposes.

  • 1
    $\begingroup$ Maybe you are looking for a "nice" isomorphism between $\mathbb R^2$ and $\mathbb R$, which probably does not exist (as a constructible one, with the preserving properties you want). See this and this. $\endgroup$
    – Pedro
    Oct 19, 2017 at 23:52
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    $\begingroup$ Perhaps a space-filling curve would be useful, as nearby points tend to remain nearby. en.wikipedia.org/wiki/Z-order_curve for a simple example. $\endgroup$
    – Claude
    Oct 20, 2017 at 0:22
  • $\begingroup$ @Claude At first glance, that looks really good! One thing I noticed is that it looks astonishingly similar to the 3-dimensional version of the hilbert curve. Maybe that's how it was generated in the first place? Either way, that looks good. $\endgroup$
    – Byte11
    Oct 20, 2017 at 0:42

2 Answers 2


Plotting complex numbers on a number line(or indeed any set on a line) gives us a total ordering, and vice versa. Now, to have this plot maintain nice properties translates into nice properties of this total ordering. In particular, we of course want to be able to shift on this number line. In addition, many of the fractal pictures come about because of the properties of complex multiplication, so we will want to preserve some information about complex multiplication as well.

To see that this cannot happen, see Total ordering on complex numbers

  • $\begingroup$ hmmm...ok. I can't say that I understand it very well because I don't know a lot of the jargon and symbols that are being used in the linked answer, but I'll learn that and see what all that means. Do you know of any ways that I could maybe approximate this without necessarily keeping all of its properties. $\endgroup$
    – Byte11
    Oct 20, 2017 at 0:26

You could represent the complex numbers using arrows on a line. The location of the start of the arrow is the real part, and the length/direction of the arrow is the imaginary part. For example, this picture shows how two numbers -1+3i and 5-2i look in this representation. (A real number is just a dot in its usual place on the line.) complex numbers as arrows

It doesn’t preserve most of the spatial properties of complex numbers like rotation or length, but it still might produce an interesting result.

I don’t know if anyone else has used this method before, so I don’t know if it has a name.


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