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= Strategies A brief outline with some thoughts to flesh out.... == Strategies for polytopes * Start with lower-dimensional analogs (in the tradition of Flatland) * Projecting into a lower-dimensional space * Unfolding * Cross-sections over time as higher-dimensional figure passes through lower-dimensional space For 4D objects, begin with the tesseract (or 4D hypercube, the 4D analog of a cube) and the pentachoron (or 4-simplex, the 4D analog of a tetrahedron). == Strategies for curved figures Again, begin with the simplest figure: a 4D hypersphere (a.k.a. glome). Somewhat unintuitively to non-mathematicians, this is also termed a 3-sphere, because any point on its "surface" can be specified with three coordinates. (The familiar 3D sphere, such as a globe of the Earth, is called a 2-sphere for analogous reasons; you can specify any point on its surface with only lines of latitude and longitude.) * Slicing (gluing successive cross-sections together) * Suspension (connect every point on a 3D sphere to two points in higher space, like a hammock; this yields a topological glome, though not a geometric one. * "Cone" gluing (similar to the above; take two 3D spheres and connect each one to a point in 4-space (a different point for each). Paste the two spheres together, point-for-point, around their surfaces.) * Hopf fibration

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