A pentatope is a simplex embedded in R4. It is the "simplest" 4-dimensional shape (though "simple" is a relative term when discussing 4D geometry...). It has five vertices, ten edges, ten faces, and five tetrahedra. Its dual polychoron is itself and it is also known as the hypertetrahedron or the 5-cell since it has five vertices. To imagine one projection of it, imagine a pentagram with straight outside edges instead of a circle.

Don't hurt yourself trying to imagine it, though. It's impossible for a three-dimensional being to conceive such four-dimensional shapes.

Three-dimensional beings can only perceive four-dimensional objects as animated, three-dimensional cross-sections, just as a two-dimensional being could only perceive a three-dimensional object as animated, two-dimensional cross-sections. A straighforward animation of the pentatope, as well as links to several other pages of four-dimensional objects, can be found here:


Leave me a /msg if this link ever breaks.

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