Non-Euclidean Geometry for Children

I should've asked this question two years ago when my son (at that time, 9 years old) came to me and said: "Dad, today in school our teacher drew a line on a paper and said this is a straight line, it goes from both directions and doesn't meet itself". I answered, "Okay, what is wrong with that". He said, "but I think it meets itself". I told, "how come?" Then, he drew a line on a piece of paper, made a cylinder with the paper and showed me how. At that time, I knew there is an element of truth in his idea, but being afraid of destroying the originality of his thinking I didn't add anything and instead promise him to think of his idea. I didn't keep my promise! But last night, he (now eleven) came back to me with the same idea thinking of what happens on a sphere. Again, I promised him that we discussed it later on (that is supposed to be today). The question is how can I help him to develop his ideas. The question for you is: Are there any sources available?

A classic introduction to some aspects of non-Euclidean geometry is Edwin Abbott's Flatland. It would be good for your son now because it doesn't require any technical apparatus.

(Flatland was used as inspiration for this scene in Carl Sagan's Cosmos. Sagan mentions Abbott by name.)

Another book is Geometry, Relativity and the Fourth Dimension. If you are relatively competent in mathematics yourself this book will give you lots of fodder: the explanation of the five Euclidean axioms, what happens when we throw out the parallel axiom, ... The diagrams in the first couple of chapters alone should give you things to talk to your son about. I have used it with teenage children of friends.

Actually, there is a perfectly nice non-euclidian geometry that does not require any extra imagination: spherical geometry. Just do the usual math and discover tons of cool stuff such as existence of a triangle with three right angles.

Smart kid. don't discourage him! It's okay to say, he's discovered some aspects of non-eucludean geometry that isn't generally taught and you, yourself, might not know all the answers but he should google or try the local library.

Euclidean geometry assumes space is basically "straight" and the shortest distance is straight line. Curving lines on a cylinder or a sphere aren't "really" lines because they aren't straight. You can have a cylinder or a sphere as objects but lines are presumed to be flat and unbendable. 3 d space is similarly "flat" and straight.

Non-eucludean geometry allows that space can be "curved". There are two types of non-eucludean geometries: spherical and hyberbolic. Spherical presumes space "bulges" and hyperbolic presumes space "pinches inward".

Spherical has a perfect model in Euclidean space, the sphere. Imagine space bends like sphere but in an an imperceptible and unmeasurable way. Heading out in a straight line will ultimately return on yourself and two lines, even if they start out going apart will ultimately bend toward each other.

Hyperbolic, sometimes called saddleseat, doesn't have a good model but the pictures by mc escher of the circles with receding horizons are pretty close. Lines go on forever and don't meet themselves, but they bend away. Any two lines grow apart.

The two important ideas are parallel lines. In Euclidean geometry, a pair of parallel lines always remain the same distance apart. There are only one pair of parallel limes a certain distance apart. (This is a casual and very imprecise paraphrasing of euclid's 5th postulate. The actual wording is precise and clear, but very awkward.) In spherical geometry there are no parallel lines. All lines eventually meet cross and grow apart. Two lines heading in the same direction with the same relative interior angle (which in Euclidean space would be parallel) eventually meet. In hyperbolic geometry there a multiple parallel lines at a distance.

I should rephrase that. One form of euclid's 5 postulate is that given a line, a point not on the line, there will be exactly one line parallel to the first line through that point. This pair of parallel lines will remain "straight" and equal distance apart and have interior angles of 180 degrees. In spherical geometry there are no parallel lines through the point. There is one line with an interior angle of 180 but it eventually gets closer. In hyperbolic there are many parallel lines through the point (all intersecting each other) many at 180 degrees but not all parallel lines need to be at 180. These lines may get closer and then go apart or they may just go apart. But they don't stay the same distance.

.....

So, references. Um, local library "recreational mathematics". Collections of short essays by Martin Gardner or Isaac asimov. Also "sphereland" by burger as a non-euclidean companion piece to "flatland" by abbott. (Your son will love flatland, although it is strictly euclidean.)