Play around with these geometries in your brain. Then see what happens for real when you do this with high dimensional vectors, like word vectors (Mikolov).

• match the shaddow with its shape
• match the equation with its manifold
• match list of 4d vertices with their 3d projection image
• squash a point cloud or 3-D manifold into 2-D to create a desired shape by adding 3 fiducial points (triangle) and dragging them down to the surface
• afine transformation matrix recognition