Intuition Fails in High Dimensions

title: Intuition Fails in High Dimensions

Intuition Fails in High Dimensions


A 2D plane with X and Y axis. On it you mark points (1,0) and (0,1). And through them you draw a straight line. Even without looking at the image below, one can get an idea about how the graph would look like.

X-Y plane with your imaginary line

Now let’s imagine a 3D plane with X, Y and Z axis. Through this 3D structure, a plane passes that intersects the Xaxis at (2, 0, 0), Y axis at (0, 3, 0) and Z Axis at (0, 0, 6). Such a plane is tough to imagine in our heads, but if we try we would end up with something that looks like this.

X-Y-Z with our plane

With that, we get to step into the next higher dimesion. Planes in dimensions higher than 3 are refered to as hyperplanes. But first let us address just where does the fourth axis even point to. Let’s call it W axis. And similar to previous cases of new Axis creation, this W axis would be perpendicular to pre-existing axes (X, Y and Z). Just like Z was perpendicular to X and Y axes.

X-Y-Z-W Axes

It is important to understand that the W-axis as depicted here is perpendicular to all of the other coordinate axes. We may be tempted to try to point in the direction of W, but this is impossible because we are confined to 3-dimensional space.

Because we live in a world that is 3-dimensional, its difficult for us to comprehend the world that has dimensions higher than 3. This is the reason for our intuition and imagination to be of limited help in higher dimensions.

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