# Pentellated 6-simplexes

In six-dimensional geometry, a **pentellated 6-simplex** is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple **pentellated 6-simplex** is also called an **expanded 6-simplex**, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the *pentisteriruncicantitruncated 6-simplex*, is called an *omnitruncated 6-simplex* with all of the nodes ringed.

The vertices of the *pentellated 6-simplex* can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on facets of the pentellated 7-orthoplex.

A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of:

Its 42 vertices represent the root vectors of the simple Lie group A_{6}. It is the vertex figure of the 6-simplex honeycomb.

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

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

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

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

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

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

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

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

The **omnitruncated 6-simplex** has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-simplex.

The omnitruncated 6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through the origin and the seven vertices of the 6-simplex.

Like all uniform omnitruncated n-simplices, the **omnitruncated 6-simplex** can tessellate space by itself, in this case 6-dimensional space with three facets around each hypercell. It has Coxeter-Dynkin diagram of .

The pentellated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A_{6} Coxeter plane orthographic projections.