Half measure polytope

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In geometry, half measure polytopes are a class of n-polytopes constructed from an n-measure polytope where half of the vertices are removed (or alternately truncated). The 2n facets become 2n (n-1)-half measures polytopes and new (n-1)-simplex facets are formed in place of the removed vertices.

They have been named with a demi- prefix to the measure polytopes: demicube, demitesseract, etc.

This alternate truncation operation is also called a snub when applied to omnitruncated polytopes.

They were discovered by Thorold Gosset, although only the demipenteract (and lower regular forms) were included in his regular and semiregular polytope list. He called it a 5-ic semi-regular. Higher forms were excluded because defined semiregular to mean only regular polytope facets could be used.

The entire family forms uniform polytopes.

Infinite sequence: (Measure form --> Half measure form)

2-polytope: square --> digon, (degenerate into edge element)
3-polytope: cube --> regular tetrahedron (demicube)
4-polytope: tesseract --> regular 16-cell (demitesseract) (8 and 8 tetrahedra)
5-polytope: penteract --> uniform demipenteract (10 16-cells and 16 5-cells)
6-polytope: hexeract --> uniform demihexeract (12 demipenteract and 32 5-simplices)
7-polytope: octeract --> uniform demiocteract (14 demihexeract and 64 6-simplices)
...