# Hexicated 7-simplexes

In seven-dimensional geometry, a **hexicated 7-simplex** is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.

There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.

The simple **hexicated 7-simplex** is also called an **expanded 7-simplex**, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the *hexipentisteriruncicantitruncated 7-simplex* is more simply called a *omnitruncated 7-simplex* with all of the nodes ringed.

In seven-dimensional geometry, a **hexicated 7-simplex** is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.

Its 56 vertices represent the root vectors of the simple Lie group A_{7}.

The vertices of the *hexicated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .

A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:

The vertices of the *hexitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .

The vertices of the *hexicantellated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .

The vertices of the *hexiruncinated 7-simplex* can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .

The vertices of the *hexicantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .

The vertices of the *hexiruncitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .

In seven-dimensional geometry, a **hexiruncicantellated 7-simplex** is a uniform 7-polytope.

The vertices of the *hexiruncicantellated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .

The vertices of the *hexisteritruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .

The vertices of the *hexistericantellated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .

The vertices of the *hexipentitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .

The vertices of the *hexiruncicantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .

The vertices of the *hexistericantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .

The vertices of the *hexisteriruncitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

The vertices of the *hexisteriruncitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

The vertices of the *hexipenticantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .

The vertices of the *hexisteriruncicantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .

The vertices of the *hexipentiruncicantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .

The vertices of the *hexipentistericantitruncated 7-simplex* can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .

The **omnitruncated 7-simplex** is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A_{7} symmetry of the regular 7-simplex. It can also be called the *hexipentisteriruncicantitruncated 7-simplex* which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.

Like all uniform omnitruncated n-simplices, the **omnitruncated 7-simplex** can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .