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.