Rectified hecatonicosihexapentacosiheptacontahexaexon

The rectified hecatonicosihexapentacosiheptacontahexaexon, or ranq, also called the rectified 321 polytope, is a convex uniform polyexon. It has 56 icosiheptaheptacontadipeta, 126 rectified hexacontatetrapeta, and 576 rectified heptapeta. 2 icosiheptaheptacontadipeta, 10 rectified hexacontatetrapeta and 16 rectified heptapeta join at each demipenteractic prismatic vertex. As the name suggests, it is the rectification of the hecatonicosihexapentacosiheptacontahexaexon.

The rectified hecatonicosihexapentacosiheptacontahexaexon contains the vertices and edges of a small cellated demihexeractic prism, a tetrahedral-truncated tetrahedral duoprismatic prism, a triangular-small rhombated hexateric duoprism, a hexagonal-small cellidodecateric duoprism, a small rhombated octaexon, and small bicellihexadecateron.

Vertex coordinates
The vertices of a rectified hecatonicosihexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,±1\right)$$ and all permutations of first 6 coordinates
 * $$\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±\frac12\right)$$ and all even sign changes and all permutations of the first 6 coordinates
 * $$\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,0\right)$$ and all permutations of first 6 coordinates
 * $$\left(±\sqrt2,\,0,\,0,\,0,\,0,\,0,\,0\right)$$ and all permutations of first 6 coordinates

Representations
A rectified hecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:


 * o3o3o3o *c3o3x3o (full symmetry)
 * xoxoo3oxooo3ooooo3oooxo3ooxox *c3ooooo&#xt (E6 axial, icosiheptaheptacontadipeton-first)
 * ox(oo)xo3oo(xo)oo3oo(oo)oo *b3oo(oo)oo3xo(oo)ox3ox(ou)xo&#xt (D6 axial, rectified hexacontatetrapeton-first)
 * xoo3oxo3xoo3ooo3oox3oxo3oox&#zx (A7 symmetry, hull of 2 small rhombated octaexa and 1)