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Maps and navigation have played an important role throughout human history. Advances in these areas allowed for the expansion of trade networks and empires. Mathematically there are a number of interesting things. A number of different projects are possible with this material. Find what most interests you. 
  • Map projections. There are many ways to represent a sphere on a flat sheet of paper. All  methods create some kind of distortion. The familiar Mercator has the amazing property of preserving angles. In mathemaitcs transformations that preserve angles are called conformal and of of widespread use throughout mathematics. The first steps in this project would be to investigate (and maybe invent your own) different map projections and what properties they have. 
  • Navigation. Hows does a sextant work? How does GPS work? Nice geometry behind these things. Other devices?
  • Imagine you are a flat 2 dimension creature on the surface of the earth. How can you figure out the shape of the earth. Plane, cylinder , sphere torus? There are numerous answers, some of which involve the kind of geometry you know about (for example, angles in a triangle and area of a circle).  The corresponding question about the shape of the universe is difficult and is an important topic is astronomy. The math behind this is maybe too advanced, but some things about curvature you can understand. You could still write about this at a more science level.