My daughter, “X”, who just turned five, has been enjoying adding and subtracting small whole numbers in her head. She is just starting to write, so we made a little table for her to practice. Here she is adding:

When she finished, we went into nTopology and plotted the sum field, reflecting around the origin. Looking at it this way, we can think of the sum as adding the distances to two infinite lines, as defined by the values of our rows and columns. We call this sum field a kind of *norm field*.

On the other hand, we can place a pawn on our zero and count as it walks down and right in the table. Somehow, the table always knows exactly how many steps the pawn takes. We can interpret this board as a *distance field* to the origin.

What happens if we repeat the same process with subtraction? Notice how the sum and difference contour lines and slopes are perpendicular at every point.

In nTopology, we can also construct *epigraphs* of our sum and difference fields:

Let’s fabricate those epigraphs from magnetic tiles, relating positive and negative Gaussian curvature to the convexity and concavity of curves. Positive curvature comes from the sum field, and negative curvature from the subtraction field. Ball-like ellipses and horse-saddle hyperbolas are easily found on the surface geometry couch, especially if it has a cover like ours.

For the next discussion, consider also assembling four squares. We can compare those four squares to the four equilateral triangles on the spherical model and the four diamonds (double equilateral triangles) in the hyperbolic model. What happens when we take the pieces around each center apart and look at them flatted?

(It might help to arrange the pieces next to each other, if your student doesn’t do it first.)

X quickly observed that the excess angle on the hyperbolic side completed the elliptical side, so they were on average flat.

To complement addition, here’s the subtraction image from nTop, in case you want to print it: