Why is it called hyper-plane? "In geometry a hyperplane is a subspace of one dimension less than its ambient space." However, the Greek prefix hyper- means "'over', usually implying excess or exaggeration". So why do we call a hyperplane a hyperplane, while it actually has _less_ dimensions than the original space? Shouldn't it be called hypoplane rather?

When we're working in a general dimension $n$, the general case we're thinking of is that $n>3$. In that case a "hyperplane" has _larger_ dimension than the 2-dimensional planes we're used to from solid geometry.

The "plane" is the important part of the word; "hyper" just reminds us that "this is not necessarily your ordinary kind of plane from 3D geometry".