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