According to this link:


<https://en.wikipedia.org/wiki/Stericated_6-simplexes>


the stericated 5-simplex "scal" has 105 vertices defined as permutations of (0,0,1,1,1,1,2).


In the course of my team's research into the structure of a particular biomolecular energetic space, we have discovered that there are exactly 21 different ways in which (0,0,1,1,1,1,2) can be assigned to the seven points of the Fano plane so that the sum of the three integers assigned to each line is even. (Of these 21, there are seven sets such that no assignment in any set can be obtained from any assignment in the other six sets.)


Hence, the vertices of "scal" can be used to orgnaizze verticces of the pentallated 6-simplex "staf", since it is clear from this projection of "staf":


<https://en.wikipedia.org/wiki/Pentellated_6-simplexes#Pentellated_6-simplex>


that "heptagonal Fano planes" can be inscribed in "staf" in many various ways.


Has this result (or any portion thereof) been previously reported? If so, can you provide a link or citation?


Also, since the 42 vertices of "staf" occur as a subset of the roots of the algebraic group E8, does anyone know of any discuession of staf and the Fano plane within the context of the sub-algebras of E8?


Thanks as always for considering these questions.



The Fano plane, stericated 6-simplex, and pentallated 6-simplex
As already was outlined in one of the answers to <https://math.stackexchange.com/questions/2070413/for-which-dimensions-does-it-exist-a-regular-n-polytope-such-that-the-distance-o/3492945?noredirect=1#3492945>, it is possible to understand the expanded simplex of any dimension as an axial stack of 3 vertex layers of the mere simplex atop the expanded simplex atop the dual mere simplex (each of one dimension less), i.e.
$$x3o3o3o...o3x=\text{hull}(x3o3o...o3o\ ||\ x3o3o...o3x\ ||\ o3o3o...o3x)$$
Thus esp. each of the following is nothing but the midsection of the following:


* $x3x$ = 2D hexagon, with $6$ vertices
* $x3o3x$ = 3D cuboctahedron ("[co](https://bendwavy.org/klitzing/incmats/co.htm)"), with $2\*3+6=12$ vertices
* $x3o3o3x$ = 4D small prismated decachoron ("[spid](https://bendwavy.org/klitzing/incmats/spid.htm)", aka: runcinated pentachoron), with $2\*4+12=20$ vertices
* $x3o3o3o3x$ = 5D small cellated dodecateron ("[scad](https://bendwavy.org/klitzing/incmats/scad.htm)", aka: stericated hexateron), with $2\*5+20=30$ vertices
* $x3o3o3o3o3x$ = 6D small terated tetradecapeton ("[staf](https://bendwavy.org/klitzing/incmats/staf.htm)", aka: pentellated heptapeton), with $2\*6+30=42$ vertices
* $x3o3o3o3o3o3x$ = 7D small petated hexadecaexon ("[suph](https://bendwavy.org/klitzing/incmats/suph.htm)", aka: hexicated octaexon), with $2\*7+42=56$ vertices
* $x3o3o3o3o3o3o3x$ = 8D small exiated bi-enneazetton ("[soxeb](https://bendwavy.org/klitzing/incmats/soxeb.htm)", aka: heptellated enneazetton), with $2\*8+56=72$ vertices
* etc.


(As was outlined in that very cited answer as well, all these expanded simplices have the common interesting feature of having circumradius = edge size.)


"[scal](https://bendwavy.org/klitzing/incmats/scal.htm)" OTOH is a further Wythoffian member of the 6D simplex family. It is the small cellated heptapeton, aka: stericated heptapeton with Dynkin symbol $x3o3o3o3x3o$ and 105 vertices. It features for facets: $7\ $ $x3o3o3o3x\ .$ ([scad](https://bendwavy.org/klitzing/incmats/scad.htm)s) + $35\ $ $x3o3o\ .\ x3o$ ([tratet](https://bendwavy.org/klitzing/incmats/tratet.htm)s) + $35\ $ $x3\ .\ o3x3o$ ([troct](https://bendwavy.org/klitzing/incmats/troct.htm)s) + $21\ $ $x\ .\ o3o3x3o$ ([rappip](https://bendwavy.org/klitzing/incmats/rappip.htm)s) + $7\ $ $.\ o3o3o3x3o$ ([rix](https://bendwavy.org/klitzing/incmats/rix.htm)es).


Thus indeed scal has scads for facets. And each scad (on its own) can be seen as an axial projection of staf.


It should be noted furthermore that [scal](https://bendwavy.org/klitzing/incmats/scal.htm) has 105 vertices. [Scad](https://bendwavy.org/klitzing/incmats/scad.htm) OTOH has 30 vertices, and there are 7 [scad](https://bendwavy.org/klitzing/incmats/scad.htm) facets in [scal](https://bendwavy.org/klitzing/incmats/scal.htm), so there will be exactly 2 [scad](https://bendwavy.org/klitzing/incmats/scad.htm)s incident at each vertex of [scal](https://bendwavy.org/klitzing/incmats/scal.htm).


--- rk


