I have recently irritated a number of long-time contributors to this site by:

i) posting multiple (but not duplicate) questions involving the roots of the algebraic-group $E_8$

2) posting this relatively large number of questions over a relatively short period of time.

3) posting these questions without providing sufficient context that would explain my reason for posting them.

So, I want to first apologize for (1-3).

But I also want to very sincerely say "thank-you" to this site for allowing me to post these questions, for the following reason.

The reason I posted these questions was to try and get insight into a very specific problem confronting the research team of which I am a member. This problem was very simple, and can be stated as follows:

4) My team has a set of 240 elements which can be split into two groups of (144,96) in a principled way, i.e. this split is "principled" because it arises as a natural and inexorable consequence of certain fundamental biomolecular "first principles" on which all molecular biologists agree

5) My team also has several reasons to believe that these 240 elements do, in fact, instantiate the roots of $E_8$.

6) but our problem arose because we could not see how our intuitively obvious and principled split of the 240 into (144,96) could be reconciled with the usual split of the 240 into (128,112), i.e. the split according to the usual coordinatization of the roots of $E_8$ as given in the "Construction" section here:


And all of the "odd" questions which I recently posted about the roots of $E_8$ were in fact posted in an attempt to gain some insight into how this reconciliation could be accomplished.

And this attempt succeeded, but NOT for the reason one might expect, i.e. I did not gain insight as a result of any answers to the questions I posted.

Instead, the very process of posting these questions forced me to sharpen my thinking about our (144,96) split, and as a result, I was able yesterday to finally see that there is a natural and principled way to reconcile this split with the usual (128,112) split.

And this is why I want to express my deepest gratitude to this site. If this site did not exist as a place where I could sharpen my thinking about our (144,96) split by posting questions about $E_8$ to multiple knowledgeable communities, I truly believe that the reconciliation of our (144,96) split with the standard (128,112) split would have taken a lot longer. And in fact it might well be that I never would have seen how to accomplish this reconciliation.

So, thank you so much for tolerating my presence here over the past few weeks.

I owe this site an intellectual debt which I will never be able to repay.

And if you want to know why I feel that way, it is because my team can now:

i) CERTAINLY show that biomolecular codon space (the space of genes) instantiates two opposed instances of 4$_2$$_1$, while biomolecular amino acid space (the space of proteins encoded by genes) instantiates two opposed instances of 1$_2$$_2$

ii) PROBABLY show that biomolecular codon space instantiates two opposed copies of $E_8$ (AS WELL AS their associated Coxeter groups), while biomolecular amino acid space also instantiates two opposed copies of $E_6$ (AS WELL AS their associated Coxeter groups).

And since we can now make those two claims with some degree of confidence that they might be correct, we can also make a third claim which goes to the "heart of the matter"

iii) the particular way in which amino acids are encoded by codons (and proteins by genes) can PROBABLY be shown to follow as a natural consequence of the particular way in which the 72 roots of $E_6$ occur as a subset of the 240 roots of $E_8$.

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    $\begingroup$ You're welcome, but I don't think this is what meta is for. Looking on your profile, it appears you haven't accepted any of the answers (to the few questions that have any). Usually, "thanking" is carried out by accepting and upvoting answers, see What should I do when someone answers my question? $\endgroup$ Dec 3, 2017 at 14:38
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    $\begingroup$ @SimplyBeautifulArt - thanks for taking the time to respond, and my apologies again for my breach of protocol regarding accepting/upvoting answers. "amWhy" has suggested to me that I not post for a while, but when I do return with specific questions arising from my team's on-going investigation into a possible biomolecular instantiation of $E-8$, I will be sure to accept/upvote answers when appropriate. In my defense, I do want to point out that I did explicitly say "thank-you" to everyone who posted any kind of comment to any of my questjons. $\endgroup$ Dec 3, 2017 at 14:48
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    $\begingroup$ I don't think users should upvote or accept answers just because they want to thank the answer-ers. Vote should be used for indicating the quality of the posts. And as an answer-ers, I don't like my answers got accepted not because it answers the questions. @simplybeautifulart $\endgroup$
    – user99914
    Dec 3, 2017 at 17:39
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    $\begingroup$ @JohnMa I assumed that you'd like to thank high-quality answerers and not the lower-quality ones, and that they also sincerely attempt to answer the question. $\endgroup$ Dec 3, 2017 at 17:40
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    $\begingroup$ IMO, the way is to thank the answer-ers in the comment and vote/accept according to the quality. It is quite misleading to say ""thanking" is carried out by accepting and upvoting answers" @SimplyBeautifulArt $\endgroup$
    – user99914
    Dec 3, 2017 at 17:46
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    $\begingroup$ David: Just to be clear, I suggested that you "slow down" and pace yourself with questions asked here, so you can fully digest and think about comments/answers. Do comment a post, if there is something you don't understand (either to address an answerer or commentator of earlier questions). Take notes for yourself about what your thoughts on the answers and comments of a question, any insight you've gained; and try to limit yourself to one or two questions a day (if the topic of the questions are related). $\endgroup$
    – amWhy
    Dec 3, 2017 at 17:47
  • $\begingroup$ @JohnMa I suppose that's also appropriate. $\endgroup$ Dec 3, 2017 at 17:50
  • $\begingroup$ @amWhy - thanks as always for taking the time to stop by and clarify. I really mean it when I say that I very much appreciate your intellectual "caritas". I use the Latin root "caritas" here because if you replace "God" with "order in the universe" in the lead paragraph here: en.wikipedia.org/wiki/Charity_(virtue), you will understand the opinion which I have already formed of you as a person. $\endgroup$ Dec 3, 2017 at 17:51
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    $\begingroup$ @DavidHalitsky : I think what you need is a mathematician/people to work with instead of asking directly here. If you have someone else formulate your question into a mathematical one, IMO some of your questions will be well-received in MSE. $\endgroup$
    – user99914
    Dec 3, 2017 at 18:20
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    $\begingroup$ @JohnMa - thanks for your suggestion. Please believe me that I have tried to find such a person here in the Northeast US, but the problem is that: i) senior people are too busy pursuing their own interests; ii) more junior people are either busy trying to get tenure, or to get from Asst to Assoc,etc; iii) grad students are too busy writing theses. Also, the person would have to be willing to learn some biomolecular fundamentals. If you know of ANYONE I could approach, please email me off-line at [email protected]. And thank you very much again for you comment - very kind of you. $\endgroup$ Dec 3, 2017 at 18:32
  • $\begingroup$ Just wanted to mention that Jyrki Lahtonen has just provided a piece of important and useful information in this thread. math.stackexchange.com/questions/2548030/… This information is so useful and important that it can even be considered an "answer" to the question, so I wish that Jyrki had posted it as an answer instead of a comment ! @JyrkiLahtonen $\endgroup$ Dec 3, 2017 at 22:50
  • $\begingroup$ @amWhy - I sincerely hope you'll find this question considerably more focussed and "well-formed" than past questions of mine . . . if not, please take a moment to tell me what's wrong with this one (I mean that - not being "snarky" . . ) $\endgroup$ Dec 5, 2017 at 2:44
  • $\begingroup$ the very process of posting these questions forced me to sharpen my thinking: you should read blog.codinghorror.com/rubber-duck-problem-solving , written by the co-founder of Stack Overflow. $\endgroup$
    – user117972
    Dec 5, 2017 at 5:43
  • $\begingroup$ @drewbenn - certainly relevant, I agree - thanks very much, $\endgroup$ Dec 5, 2017 at 16:09
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    $\begingroup$ Too melodramatic. $\endgroup$
    – shalop
    Dec 10, 2017 at 14:48

2 Answers 2


I can only speak for myself, but I think in the context of a person legitimately aiming to better understand an area of mathematics with the goal of furthering mankind's understanding of Biochemistry there's no such thing as a bad question...or too many. Some people on this site are a little obsessive with regards to conforming with the rules, and will jump on even the most trivial of violations. Please don't let that discourage you from further inquiries.

Personally, I find your enthusiasm to be refreshing, and I congratulate you and your team on your discovery.

P.S. I myself am fixing to enter medical school. I look forward to reading more about this in my spare time. It never ceases to amaze me, some of the areas in Academia that math can pop up. I remember being shocked when I came across the use of group representations in understanding molecular geometry. Mathematics is truly everywhere.

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    $\begingroup$ thanks so much for taking the time to stop by, and best of luck in your new course of study. Most importantly, thank you for simply being a person who has the requisite sense of wonder at the fit between human math and human cognition of the universe. Too many have lost this wonder because it tends to make one want to do math that does NOT lead to marketable new products, and hence is actually a liability for young people who want lives as well as vocations ... I hope that HSM Coxeter is smiling down on the two of us right now (I mean that VERY seriously . . .) $\endgroup$ Dec 6, 2017 at 20:37
  • $\begingroup$ @Shalop - should I delete it now, or just let auto-delete clean it up eventually? Or, if a mod wants to delete it now, that's OK with me. Thanks again for your patience here - sorry to have bothered you. $\endgroup$ Dec 11, 2017 at 10:38
  • $\begingroup$ @DavidHalitsky It's fine, just let it be. Congrats again. $\endgroup$
    – shalop
    Dec 11, 2017 at 13:16

Apology accepted.

Yes, there are things that could have been done better, and they are mentioned by the truck load in the comments above. But, to err is human, to forgive divine.

Besides, the questions you asked were interesting. At least I went and read about $E_8$ after I had read the above post of yours(of course, I did not know about it beforehand). Granted you were a little excited, but you did give me something new to read, and that I appreciate, and I hope others do despite the fact that the question may have not have been presented as per norms, or as per people's expectations.

Now, what you can do, if you are indebted to this site, is to contribute back. When you do have time, you can ask and answer questions and help raise the quality of the site. You may have something to offer that no one else has here, and MSE is the place if you want to contribute and feel good about it. Remember, there are many like you who come to this site initially for the purpose of getting answers to some question(I was an exception) and they want to go home happy. But we want to make them feel at home in MSE. I hope you did feel so.

It's a big community, a very friendly family. You are welcome back(and to say and play on) anytime.

(If anybody sees this post, please leave a comment below so that I can improve this answer. I know it's bad, and anything bad can become better with your help).

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    $\begingroup$ Actually, I don't think an apology, on meta, was necessary. The asker has been very responsive to suggests, pointers, clarification, and has already expressed sincere gratitude many times. The truth is, most of us start here on this site a little clueless or naive about how the site works, its protocol, and its culture/etiquette norms. Learning about this site takes a lot of time, and invariably, needs one participation in it to climb a rather, initially, steep learning curve. So I would just say "No apology needed, but thank you for your sincerity." $\endgroup$
    – amWhy
    Dec 4, 2017 at 17:33

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