In 8-dimensional geometry, the **2**_{41} is a uniform 8-polytope, constructed within the symmetry of the E_{8} group.

Its Coxeter symbol is **2**_{41}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.

The **rectified 2**_{41} is constructed by points at the mid-edges of the **2**_{41}. The **birectified 2**_{41} is constructed by points at the triangle face centers of the **2**_{41}, and is the same as the rectified 1_{42}.

These polytopes are part of a family of 255 (2^{8} − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

The **2**_{41} is composed of 17,520 facets (240 2_{31} polytopes and 17,280 7-simplices), 144,960 *6-faces* (6,720 2_{21} polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 2_{11} and 483,840 5-simplices), 1,209,600 *4-faces* (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.

This polytope is a facet in the uniform tessellation, 2_{51} with Coxeter-Dynkin diagram:

E. L. Elte named it V_{2160} (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
It is named **2**_{41} by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
**Diacositetracont-myriaheptachiliadiacosioctaconta-zetton** (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)

The 2160 vertices can be defined as follows:

16 permutations of (±4,0,0,0,0,0,0,0)
1120 permutations of (±2,±2,±2,±2,0,0,0,0)
1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1)

*with an even number of minus-signs*
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram: .

Removing the node on the short branch leaves the 7-simplex: . There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 2_{31}, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4_{21} polytope.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 1_{41}, .

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

The **rectified 2**_{41} is a rectification of the 2_{41} polytope, with vertices positioned at the mid-edges of the 2_{41}.

Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (acronym robay) (Jonathan Bowers)
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E_{8} Coxeter group.

The facet information can be extracted from its Coxeter-Dynkin diagram: .

Removing the node on the short branch leaves the rectified 7-simplex: .

Removing the node on the end of the 4-length branch leaves the rectified 2_{31}, .

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1_{41}.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.