Early morning surface thoughts

[magid]

Lying in bed this morning, when I should've been up and about already (of course), I started thinking about area, and how the function for the area of a rectangle (for instance) would look in three-space. Related to half a parabola, but falling away from a ridge spine in the middle. Not an easy surface to describe. Oh, and only in the first octant, technically, though like the parabola modeling area of a square, it's easy enough to reflect over the other axis-planes to get a whole surface, rather than a quarter. The surface for the area of a triangle is related, of course, being half the value at any given point. And that leads me to thinking of surfaces that model volume, of a rectangular prism, for instance, for simplicity. But for that I'd need four-space, which my brain doesn't grok visually. I can think about using matrices to represent the space I'd need, but as a whole function, it's not there. If I hold one variable constant, I can revert to the area of a rectangle surface, and try to think of stacking those, but that doesn't form a true picture in my mind. Oh, interesting, it would be the same problem with the area of a trapezoid, since there's another variable in there. So it's not just solid figures that require four(plus)-space. I just tend to default to the simplest shapes and assume the same for other plane figures, other space figures.