In seven-dimensional geometry, a "'runcinated 7-orthoplex "'is a convex uniform 7-polytope with 3rd order truncations ( runcination ) of the regular 7-orthoplex.
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In six-dimensional geometry, a "'runcinated 6-simplex "'is a convex uniform 6-polytope constructed as a runcination ( 3rd order truncations ) of the regular 6-simplex.
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The runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra ( vertex figures ), cubes ( face prisms ), and triangular prisms ( edge figures ).
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The Cartesian coordinates of the vertices of the " runcinated 8-simplex " can be most simply positioned in 8-space as permutations of ( 0, 0, 0, 0, 0, 1, 1, 1, 2 ).
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If two polytopes are duals of each other ( such as the tesseract and 16-cell, or the 120-cell and 600-cell ), then " bitruncating ", " runcinating " or " omnitruncating " either produces the same figure as the same operation to the other.
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