Can You Explain Non-Euclidean Geometry to a Kid?

Geometry can be a bit tricky, especially when you start thinking about shapes on paper that isn’t flat. If the paper is round like a ball, then the shapes on this paper are using spherical geometry. If the paper looks like a horse’s saddle, then the shapes on this paper are part of hyperbolic geometry. Both of these are non-Euclidean geometry, which is different from the Euclidean geometry you typically learn about on flat paper.

Lines and Geometry

When you’re learning math, we define a “line” in a special way. If I pick any two points and connect them with the shortest path possible, that’s a line segment. If I then extend the line segment in both directions forever, that’s a line. Lines are always along the shortest distance between two points on that line.

Take a ruler and pencil and draw a straight line on a piece of paper. Now measure exactly 1 unit (inches, cm, etc.) away from the line on the same side of the line in a few different places. Draw a second straight line that goes over all the measurements you made.

Imagine that this line is always exactly one unit away from the first line wherever you measure it. Imagine your flat paper is huge and you can extend these lines as far as you like and they are always one unit apart. Now draw another straight line that crosses both these lines.

Parallel Lines

Wherever it crosses one of the lines, there are four angles around that point. Notice that at both the crossing points or intersections, the angles are the same as at the other intersection.

This is what “parallel” means. It just means two straight lines are always the same distance apart, and a line crossing both will produce the same angles around the intersection.

If we are doing this on a flat surface, this will always work, unless you aren’t careful enough with your measurements or making the lines straight, of course.

Geometry on a Sphere

Now, let’s talk about a sphere, like the Earth. On a sphere, a straight line is not a straight line as we know it. Instead, it’s a “great circle”. A great circle is a circle that has the same center as the sphere. For example, the equator and lines of longitude are great circles.

The meridian at 0 degrees is a great circle. So are all lines of longitude. But notice that all lines of longitude cross at the poles. The equator is also a great circle, and lines of longitude cross it at right angles.

This is very different from what we saw on the flat piece of paper. On a flat paper, parallel lines make the same angles with an intersecting line and never meet. But on a sphere, there are no lines that do not cross. All great circles intersect each other in two places.

Euclidean vs. Non-Euclidean Geometry

When the mathematician Euclid laid out his five postulates, which are definitions and descriptions of things that are obvious and just accepted for geometry, he was thinking about the flat paper. But most of his postulates work on a sphere, too. There is one exception: the one about parallel lines.

So, when we are doing geometry on a flat piece of paper, we often call that Euclidean geometry. But when doing geometry on a sphere or a different curved surface where not all of Euclid’s postulates work, we call it non-Euclidean geometry.