As a beginning student of elementary (euclidean plane) geometry, so far, I have gotten the impression that there are two major approaches to geometries: naive vs axiomatic. Being a humanities student I have found books taking the latter approach rather inaccessible; I suspect it has a lot to do with my ignorance of linear algebra and real analysis. Books taking the naive approach (e.g. Kiselev's) also present difficulties for a beginner (of geometry as well as of math in general). Consider the following exercise from the aforementioned book:

Exercise 5. Give an example of a surface other than the plane which, like the plane, can be superimposed on itself in a way that takes any one given point to any other given point. (Hint: the required example is not unique.)

Having read everything that preceded this exercise, I only have (i) a vague idea of what a 'surface' is, and (ii) been given no information about surfaces other than the plane. I imagine a plane that has a little bump in the middle; does that count as another kind of surface? I simply don't know how to imagine new surfaces, new spaces, etc. The primary reason I want to study geometry is to learn about such things and how they relate to each other. I feel like the questions I ask, such as:

  • what exactly is a surface?
  • how do you tell whether two surfaces are equal (in some sense) or not?

are very obvious and probably stupid from the perspective of an experienced mathematics student or someone who is mathematical gifted, but having no proper mathematics background and having no mathematical gifts, I could use the guidance of a computer software that would let me explore such objects, to see what they look like, what properties they have, which of their properties are preserved under which transformations, and so on. No book I've looked at has and could answer all my stupid questions, but a controlled environment would let me find out the answers to those silly questions. Thank you if you've read this far; my main question I can express as follows:

Question. Are there computer programs (or books with lots of pictures of elementary concepts) that could help a non-mathematician study elementary plane geometry from scratch?

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    $\begingroup$ For your problem, think spherically. $\endgroup$ – André Nicolas May 13 '14 at 5:13
  • $\begingroup$ @AndréNicolas Thank you. I think I got the idea. $\endgroup$ – Readingtao May 13 '14 at 5:16
  • $\begingroup$ You are welcome. I could have said think cylindrically (infinitely long both ways). Unconnectedly, the free program Geogebra lets you experiment. $\endgroup$ – André Nicolas May 13 '14 at 5:18
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    $\begingroup$ The boundary of a doughnut is hard to visualize? You must have healthy eating habits. $\endgroup$ – André Nicolas May 13 '14 at 5:50
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    $\begingroup$ With respect to complicated objects modern mathematicians prefer to talk about isomorphism, instead of equality. This terminology helps to reduce usage of “up to” mantra. For surfaces it is a tricky question: two surfaces can be isomorphic as Riemannian manifolds, but different (even not equivalent up to Euclidean isometries) as subsets of Euclidean 3-space. Which surfaces are and aren’t equal depends on how to think about a surface. $\endgroup$ – Incnis Mrsi Nov 2 '14 at 17:40

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