lets construct a 4D cube, it has 2^4=16 corners and 2*4=8 3D faces and 24 2 dimensional faces. the 8 3D faces contribute 4 edges and the total edges is 8*4=32. so can you know these properties by deduction or visualization somehow? maybe there is an Euler characteristic?
anyway I guess people are discussing fiber bundles and cotangent bundles here but I wanted to ask this