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