I am a mathematical beginner. As we all know, the graph of a one-dimensional function is a curve, and the graph of a two-dimensional function is a surface. What is the graph of a three-dimensional function? For example, $f(x,y,z)=x^2+y^3+5xyz$. Someone said that we cannot draw the graph of a three-dimensional function. Could someone explain why? Thank you in advance for your kind help.

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    $\begingroup$ $C=x^2+y^3+5xyz$ level curve sets or contour lines can be drawn on CAS.. like Matlab, Maple, Mathematica $\endgroup$
    – Narasimham
    Feb 12 at 11:22
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    $\begingroup$ Or you can assign a color to each value of $f(x,y,z)$ and find ways to plot that in 3D. Morale: don't believe everything that someone says. $\endgroup$
    – Kurt G.
    Feb 12 at 11:25
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    $\begingroup$ It really depends what you mean by "the graph of a function". If you mean, the set of $(x,y,z)$ that solve $f(x,y,z)=0$, that's a surface (which you can draw in, say, Geogebra). If you want to plot $w=f(x,y,z)$ on a $w$ axis as you vary $x$, $y$ and $z$, then you're asking "why can't we draw in 4D?". It's worth pointing out that we also can't draw in 3D (well, 3D printing aside): when you visualise a surface, you're looking at a projection of that surface in 2D. With 4D, you can do the same thing, but it can be harder to interpret because we're not used to seeing projections of 4D space. $\endgroup$ Feb 12 at 11:37
  • $\begingroup$ Of course we see in three dimensions, if not four. With one function presented to the left-eye and a separate function presented to the right-eye, it is usually possible to view these two separate 2-dimensional plots in "3-D", as the brain is very good at processing these sorts of images. Another dimension is color. Actually color is often split up as Red Green Blue (RGB) or other parameters, perhaps giving another 3 dimensions of possibilities. So, thinking out of the box, one can think about how to best present 3-dimennsional quanties. I even made a program that would plot w/ sound. $\endgroup$ Feb 13 at 11:14

1 Answer 1


I mean the answer is we can visualise them, but it's just not possible to draw them! Obviously , the limitation here is that a three dimensional function actually delves into the fourth dimension, whereas we obviously live in a 3D space and so it's not possible for us to draw such a function.

However, a nice representation of a 3 dimensional function could be to set one of the variables to time, and see how the shape of the graph changes throughout time. (which closely links to the ideas of level sets)

That is not to say that we can't analyse such functions, it is just significantly harder to visualise them.


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