1. The pentachoron is a four dimensional pyramid with a tetrahedral base. 2. Let us consider a four dimensional system of coordinates. 3. Therefore, there are two commuting subgroups of the set of four dimensional rotations. 4. I suppose 50 to a hundred years from now common usage will usually reflect four dimensional thinking. 5. Consider gravity defined on a compact region of spacetime, with the topology of a four dimensional ball. 6. Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as gravitational instantons in non-compact ).7. This set of expanding and contracting spheres represents a hypersphere in four dimensional space ( a 3-sphere ). 8. Arbitrary four dimensional rotations have 6 degrees of freedom, each matrix represents 3 of those 6 degrees of freedom. 9. One of the more complex spin-offs was Dr . Mervin Verbit's twenty-four dimensional measure. 10. Given a four dimensional vector, and pretending that it is a quaternion, we can rotate the vector like this:
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