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.

There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.

Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)
The vertices of the *runcinated 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.

Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)
The vertices of the *biruncinted 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.

Note: (*) Symmetry doubled for A

_{k} graphs with even

*k* due to symmetrically-ringed Coxeter-Dynkin diagram.

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

Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)
The vertices of the *biruncitruncated 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.

Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)

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

Runcicantitruncated heptapeton
Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)
The vertices of the *runcicantitruncated 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.

Biruncicantitruncated heptapeton
Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)
The vertices of the *biruncicantittruncated 6-simplex* can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.

Note: (*) Symmetry doubled for A

_{k} graphs with even

*k* due to symmetrically-ringed Coxeter-Dynkin diagram.

The truncated 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.