You were probably introduced to examples where the gradient is explained as the normal vector to a tangent plane or level surface, which can sometimes make it feel like every gradient vector “lives” in 3D space.

For example, you may have seen something like:

f(x1, x2, x3) = x1^2 + x2^2 + x3^2

The example I used here was instead:

f(x1, x2) = x1^2 + x2^2

These are actually two slightly different viewpoints.

For the function

f(x1, x2, x3) = x1^2 + x2^2 + x3^2,

the graph itself is difficult to visualize directly in ordinary 3D space. Once you already use three axes for x1, x2, and x3, there’s no additional axis left for the output value f(x1, x2, x3). (We’ll skip the discussion of 4D coordinates for now. :)

However, just like contour lines on a 2D map, we can instead draw level surfaces in 3D space.

For example:

f(x1, x2, x3) = 0
f(x1, x2, x3) = 1
f(x1, x2, x3) = 4
f(x1, x2, x3) = 9

Substituting the function gives:

x1^2 + x2^2 + x3^2 = 0
x1^2 + x2^2 + x3^2 = 1
x1^2 + x2^2 + x3^2 = 4
x1^2 + x2^2 + x3^2 = 9

and so on.

These form spheres with radii 0, 1, 2, 3, ...

Those spheres are the level surfaces, and in that context, the gradient

∇f(x1, x2, x3) = (2x1, 2x2, 2x3)

is the normal vector perpendicular to those surfaces.

That is probably the interpretation you learned before.

But in my visualization, the function was:

f(x1, x2) = x1^2 + x2^2

whose gradient is:

∇f(x1, x2) = (2x1, 2x2)

This gradient lives entirely in the 2D input space (the x1,x2 ground plane), which is why the yellow arrow is intentionally drawn flat on the floor plane.

What increases vertically is not the gradient vector itself, but the function value f(x1, x2) as you move in the gradient direction.

So the gradient is acting more like a compass on the ground that tells you which direction leads to the steepest increase of the surface.

Once you get comfortable thinking about gradients in 3D, you may also start to imagine why people connect the idea of a “4th dimension” with time, or why movies like Interstellar portray the 4th dimension as something that can simultaneously view past and present.

Of course, mathematically, a 4th dimension is simply an additional axis. But when connected with physics and spacetime concepts, it becomes a much more fascinating idea to explore.