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I'm interested in graphing equations with no real solutions. I know in the real plane no such pair $(x,y)$ will satisfy the equation, and I can list out a few solution pairs. However this does me no good as graphing a complex number by itself is $2-D$ and is $x$ and $y$ are both complex it would require some sort of $4-D$ graph. So what to do?

Ahmed S. Attaalla
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    You can't draw four dimensions on paper, however you can imagine four dimensions in your mind. Just think of a line and each point on the line is a three space (a slice). You understand the graph by holding many slices in your mind at once. Or you can think of a plane each of whose points correspond to a plane. That's what you need to do here. For each $y$ think of the solutions $x^2=1-y^2$ in $\Bbb C$ and try to vary $x$ and see if there's a pattern. – Gregory Grant Dec 25 '15 at 01:58
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    well.. $ x = i \sin \theta $ and $ y = i \sin \theta $ are solutions for $\theta \in \mathbb{R}$, thus you'd need to draw the solutions in $\mathbb{R}^4$ – Jeb Dec 25 '15 at 01:59
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    It's worthy of note that this makes a circle on the plane where $x$ and $y$ are imaginary, since one may take solutions to be of the form $x=iu$ and $y=iv$ where $u^2+v^2=1$. Of course, one may realize that there are complex solutions to $u^2+v^2=1$ - it's not easy to imagine the graph even when the real picture is clear. – Milo Brandt Dec 25 '15 at 02:08
  • Theoretically you can graph something four dimensional by treating the fourth dimension as time and displaying an animation of three dimensional objects. I've seen this and honestly it doesn't really illuminate much for me, but I'm not a great example because I'm strongly left eye dominant and don't perceive depth in the usual way. You could give it a shot. – Matt Samuel Dec 25 '15 at 02:49
  • Thanks everyone, appreciate you taking the time to comment/ answer on Christmas Eve. Wish you a happy weekend. – Ahmed S. Attaalla Dec 25 '15 at 03:15

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Here is an idea: plot the solutions in $(\text{Re}(x), \text{Im}(x), \vert y \vert)$ space, and use a hue determined by $\text{arg}(y)$, also varying the brightness to convey the orientation of the surface. Alternatively, use the $(\vert y \vert, \text{arg}(y))$ cylinder, let $\arg(x)$ again correspond to hue, and let $\vert x \vert$ correspond to brightness or saturation. There are other ways of doing this too of course, the idea is that you can pack one, two, or even three dimensions (think RGB) into the color.

Dan Brumleve
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  • And how would I do this? – Ahmed S. Attaalla Dec 25 '15 at 03:13
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    What software are you using? Or else do you have a bunch of different colored markers? – Dan Brumleve Dec 25 '15 at 03:15
  • I tried looking up some programs, couldn't find any. I don't know nothing about computers/programming if that's where you're getting too. – Ahmed S. Attaalla Dec 25 '15 at 03:19
  • Well you can do it on paper, get a bunch of different colored markers and sort them into a color wheel, assign each color an angle, and go to work. You can use a piecewise linear approximation and mark each point with a blob or short line segment, then blend the boundaries later to make an appealing plot. – Dan Brumleve Dec 29 '15 at 07:45