Examples of non-Euclidean spaces

First, we created shapes that closed in upon themselves -- in one case, by insisting that 5 equilateral triangles meet at every vertex rather than 6 and in another by starting with a flat disk and adjusting it so that the diameter stayed the same but the circumference was smaller.
Both of these shapes had less of something (triangles, circumference) than would be predicted if we force the object to lie flat.
These are, as you read, examples of ways to model what we calll elliptic space.
The Greek root word of elliptic means coming up short, or deficient.
Next, we created shapes that rippled out in all directions. Unlike the first pair, both of these could be forced to lie flat in any one place, but that would force it ripple more dramatically elsewhere, sometimes even folding in on top of itself. In one case, we created this shape by forcing 7 equilateral triangles to meet at every vertex rather than the 6 neessary to lie flat; in the other, we again started with a flat disk but this time adjusted it so that while the diameter stayed the same, the circumference was larger.
Both of these shapes had more of something (triangles, circumference) than would be predicted if we force the object to lie flat everywhere.
These are examples of ways to model hyperbolic space.
The Greek root word of hyperbolic means excessive.