Looking at the source of your post it seems plausible that you maybe wanted to enter linebreaks at some places. This can be achieved by typing two spaces or <br>. For example 9 vertices<br/>36 edges <br/>84 triangles<br/>. For more details see Markdown help.

Hi Richard - thanks for posting your answer - Since you have now posted the answer, let me ask you an obvious follow-up question. I am sure you can answer it as easily as you answered my question about the two 84's. We know that: 1) the 240 vertices of the 4_21 are the 240 roots of the group E8 <br> 2) the 240 roots of E8 contain the 72 roots of E6 (and therefore, E6 is a subgroup of E8) <br> 3) the 72 roots of E6 are the 72 vertices of the 1_22. So, with respect to the {84,72,84} of the 4_21, where are the 72 vertices of the 1_22? (see next comment also)

I think Wendy has already said that the 72 of the 1_22 are NOT the 72 in the {82,72,84}. So, how exactly is the 1_22 positioned with respect to the 2 mutually inverted copies of a birectified enneazetton (each 84 vertices) and one expanded enneazetton (72 vertices

I don't see how this is an answer to the question, "Should MO perhaps give numerological abductive questions a little more benefit of the doubt?"

@Gerry Myerson: in fact it rather was an answer to his contained question: "Are the two 84's in the {84,72,84} decomposition of E8 's root-system non-coincidentally related to the two 84's in row 8 of OEIS A135278?"

@DavidHalitsky: the "72" in "{84,72,84}" is the vertex set of x3o3o3o3o3o3o3x, i.e. the expanded 8-dimensional simplex, whereas the 72 roots of E6 are the vertex set of the 6-dimensional Gosset polytope 1_22. That is, the latter is clearly subdimensional, and thus those numbers, even so equal, have nothing in common geometrically, I fear. (Might be wrong: if you're lucky, there might be a corresponding projection, which I don't see know.)

@Dr.RichardKlitzing yes I understand that. That is what Wendy said also. Then where is the 1_22 ? It must be SOME 72 vertices of the 4_21, correct? If so, how many of the 72 1_22 vertices are in the two 84's, and how many in the 72?

@GerryMyerson - in case you're wondering, I did not ask Richard to post his answer. He did so of his own accord, much to my own surprise. But I am grateful to him for sharing it, wherever he chose to do so.

The largest shared set between A8 (ie the 72), and E6, is A5 (30 vertices). The remaining 42 vertices are spread equally between the two sets of 84.

@wendy.krieger - Thank you !!!! You will soon see how very important and useful that information is. Also, we will be able to see this clearly when you complete your matrix multiplication mapping from the usual {128,112} E8 coordinates to the most natural coordinatization of the {84,72,84} E8 decomposition.