Discrete non-Euclidean geometry for kids via trivial arithmetic

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:

X addition table

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.

nTop addition 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.

Addition as a metric

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.

Addition and subtraction fields

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

nTop sum field and epigraph

nTop difference field and epigraph

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.

Magnetic tile assemblies of the epigraphs

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?

Flattened magnetic tiles

(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:

nTop addition field

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