# 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?

• Your son sounds really cool. Oct 23 '15 at 14:52
• Your son has some potential of a mathematician within him Oct 23 '15 at 15:12
• When I was 16 I noticed that if you look at a grid paper through a magnifying glass, lines which are parallel on the paper are now distorted and while they remain parallel if they sit close enough to the focal point, they will bend and ultimately meet at the edge of the glass. Oct 23 '15 at 15:47
• One step: what the teacher drew is not "a straight line," in the sense that a straight line is an ideal. The teacher drew something that, within physical abilities, was a good representation of a straight line. The fact that it is a representation of a straight line is useful for explaining why, by changing the underling mechanics of the paper, it can represent something else, like a circle. (or a line embedded in a non-Euclidean manifold!) Oct 23 '15 at 18:23
• I would encourage him to experiment for himself, rather than starting a "formal" approach too soon. Get him a beach ball and some erasable dry markers! A good question would be to explore the sum of the angles of triangles on a sphere. You might also want to throw Mobius strips and Klein bottles into the melting pot and see where they lead his imagination. Oct 23 '15 at 22:46

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.

• I've read the Flatland, and of course, I enjoyed it a lot. but I don't know why it didn't come to my mind when facing with his question. I guess, at the back of mind, I was looking for something more ready made and directly related to his question without doing any effort in my side. Silly me Oct 23 '15 at 16:47
• I don't remember flatland being non-euclidean so much as multi dimensional. I could be mistaken remembering. In any event the thought process of examine the idea of space is the same and complimentary even if flatland is strictly euclidean as I remember it to be. A modern "sequel" is "sphereland" by burger. Oct 23 '15 at 17:21
• Sounds like you can just give Flatland to your son, and he will make the effort of bringing it to bear. Oct 23 '15 at 17:45
• @fleablood You might be right. I've not read Flatland for a while and I probably have conflated it with its treatment in the second book I cite, which does consider non-Euclidean cases using the analogies of Flatland. Oct 23 '15 at 17:56
• There's also Flatterland, although that might be to advanced. $\;$
– user57159
Oct 23 '15 at 22:00

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.)

• I should've mentioned that I have a relatively sound background in mathematics. Thus, the last bit of your answer was the most useful part for me. Having said that, many thanks for all parts of the answer that made the question self-standing. Oct 23 '15 at 17:29
• Okay, sorry for being repetitive. I was trying to identify the stripped down tenets of non-eucludean geometry if I had to give the "big picture" to someone in 5 minutes. the mc escher pictures really help. You have a smart kid there. Oct 23 '15 at 17:56
• In spherical geometry there are no parallel lines. All lines eventually meet cross and grow apart. This doesn't sound right. Imagine a globe of the Earth, one of the most familiar spheres we know. Longitude lines appear parallel locally but eventually curve together and meet at the poles, but latitude lines appear parallel locally... and actually are parallel, never crossing each other. They're even called that in official geographic language, such as "the 45th parallel." Oct 23 '15 at 20:34
• I think only great circles count as lines. Other than the equator latitude lines are lesser circles and don't, I think, count as lines as they will have have notable curvature in relation to the surface of the sphere which great circles wouldn't. I think. I could be wrong. Oct 23 '15 at 21:28
• You can drop the "I think"s. "Lines" in non-euclidean settings have to minimize curve length between all curves connecting two points. Great circles do that. Other circles on the sphere (whose centers are not the center of the sphere) do not. (And before someone criticizes me for saying "minimize", I am assuming a positive definite metric. No need to complicate matters here.) Excellent answer, by the way. Oct 23 '15 at 23:36