Torus
Topology
If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.
Flat torus
A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:
Flat torus
Other tori in S3 having this partitioning property include the square tori of the form Q ⋅ T, where Q is a rotation of 4-dimensional space R4, or in other words Q is a member of the Lie group SO(4).
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