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September 2026
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20	21	22	23	24	25	26
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September 2026's Entries

The E6 Root Polytope
- Sep 8, 2026
- Counting and visualizing the faces of the E6 root polytope.

Random Past Entries

TeXnical Issues
- September 2, 2006
- Sage advice on viewing this blog and posting comments thereon.
QFT of Charged n-Particle: Sheaves of Observables
- Mar 6, 2007
- On the concepts of sheaves and nets of algebras of observables in quantum field theory.
Five Quickies
- Apr 6, 2015
- 1. Reflexive completion. 2. Mathematicians accepting and declining overtures from GCHQ. 3. The Euler characteristic of an algebra. 4. Review of Nick Gurski’s higher categories book. 5. Are lectures the best way to teach?
Quantum Computation and Symmetric Monoidal Categories
- Aug 28, 2006
- Ordinary computer scientists like cartesian closed categories. Quantum computer scientists like symmetric monoidal categories with duals.
Searching for a Video Proof of “Seven Trees in One”
- Jul 16, 2009
- James Propp is searching for a video proof that he saw once.
John McKay Visits Kent
- Oct 21, 2008
- What I took from John McKay’s talk on finite simple groups.

Recent Comments

Three Generations in E7 (6)
- John Baez
  - Sep 8, 2026 18:30
  - Among other things, no good argument is given here why no new fundamental fermion should, or can, decay faster than ...
- Jeffery Winkler
  - Sep 1, 2026 19:09
  - The following article talks about a chatbot developed at Argonne National Laboratory called ArgoLOOM, trained on physics papers and data, ...
- Jeffery Winkler
  - Sep 1, 2026 18:31
  - In comment #16, ohwilleke wrote the following. “For example, suppose that a mathematically consistent system requires that each generation of ...
- John Baez
  - Sep 1, 2026 11:34
  - Thanks. Percacci has done a lot of work on gravi-GUTs, many using Spin(3,11) to unify the double cover of the ...
- Jeffery Winkler
  - Aug 30, 2026 16:51
  - The following discussion thread gives another way of explaining three generations of fermions. https://www.physicsforums.com/threads/is-gravi-gut-a-candidate-theory-of-everything.1082728

Octonions and the Standard Model (Part 14) (8)
- Paul Schwahn
  - Aug 17, 2026 23:07
  - I indeed made a mistake that I managed to identify after Matthew sent me his notes. If we naively intersect ...
- John Baez
  - Jul 8, 2026 13:57
  - So you are disagreeing with Paul: you’re getting S(U(3)×U(2))\text{S}(\text{U}(3) \times \text{U}(2)) where he is getting SU(3)×SU(2)\text{SU}(3) \times \text{SU}(2) as the ...
- John Baez
  - Jul 7, 2026 17:19
  - I would love to help figure these things out — maybe Paul and I can work on it a bit ...
- Matthew Gratrix
  - Jun 25, 2026 22:42
  - Small correction: I should have stated the connected version (Stab(X)∩Stab(B) 0) 0≅S(U(3)×U(2)). \left(\operatorname{Stab}(X)\cap\operatorname{Stab}(B)_0\right)_0 \cong S(U(3)\times U(2)). The full intersection has ...
- Paul Schwahn
  - Jun 25, 2026 19:22
  - It is certainly interesting to look at other intersections of stabilizers. Let’s for now think about the “standard” subalgebras of ...

Octonions and the Standard Model (Part 15) (5)
- Bob MacGregor
  - Aug 8, 2026 22:14
  - A few comments using AI-assisted analysis: Exterior powers inside the Peirce descent. Fixing a minimal tripotent gives Peirce dimensions (1,10,5)(1,10,5). ...
- John Baez
  - Aug 6, 2026 18:52
  - It’s not a theory of physics, just a mathematical way of deriving the Standard Model gauge group and its representation ...
- Jeffery Winkler
  - Aug 4, 2026 18:29
  - Can you add the Higgs boson to this?
- John Baez
  - Aug 2, 2026 17:54
  - I apologize for taking so long to moderate this comment! It would be pretty easy to add gauge symmetry and ...
- Mitchell Porter
  - Jul 22, 2026 19:21
  - How do we add gauge symmetry and fermionic statistics to this?

Large Sets 9 (28)
- Mike Shulman
  - Aug 6, 2026 23:16
  - By “integer-indexed” do you mean including negative indices as well? ⋯:U −2:U −1:U 0:U 1:U 2:⋯ \cdots : U_{-2} : ...
- Madeleine Birchfield
  - Aug 5, 2026 13:43
  - How does one model in membership based set theory a dependent type theory with an externally integer-indexed unverse hierarchy?
- Madeleine Birchfield
  - Aug 3, 2026 01:28
  - Most proofs shouldn’t need iterated powersets beyond like two or three iterations, yet mathematicians still believe in the truth of ...
- Madeleine Birchfield
  - Aug 2, 2026 23:57
  - As Mike Shulman describes below, a dependent type theory with a countable number of universes is the most common way ...
- Mike Shulman
  - Aug 2, 2026 21:51
  - Yeah, dependent type theories are most often presented with a countable hierarchy of universes 𝒰 0:𝒰 1:𝒰 2:⋯\mathcal{U}_0 : \mathcal{U}_1 ...

Integral Octonions (Part 9) (23)
- Jens Koeplinger
  - Jul 19, 2026 00:40
  - Hi - next to verifying pretty much everything posted here (from the original work by John Baez and Greg Egan, ...
- David Calderbank
  - Apr 2, 2023 20:04
  - The “we” who know this Turyn-inspired construction of the Leech lattice from 3 copies of L=E8 should include Tits 1980 ...
- Tim Silverman
  - May 23, 2017 06:57
  - Roger Howe suggested trying to construct the Leech lattice … [from] … the 3 cross-polytopes inside the 24-cell Yes, I’ve ...
- John Baez
  - May 21, 2017 11:36
  - One potentially relevant thing was pointed out nicely by Tim Silverman in a comment to another post. The weights of ...
- John Baez
  - May 21, 2017 04:19
  - Yesterday I gave a talk on The dodecahedron, the icosahedron and E8 at the Hong Kong Mathematical Society. Today at ...

Categorifying Riemann’s Functional Equation (1)
- Tim Kraus
  - Jul 14, 2026 09:47
  - Professor Baez, In light of your recent work on the Riemann Species generating ζ\zeta(ss) via structural cardinalities of finite commutative ...

Hoàng Xuân Sính (17)
- John Baez
  - Jun 23, 2026 14:56
  - Thanks! Someone should put this version of Sính’s thesis on the arXiv! However, I’ve put a link to it here: ...
- QIN Yuxuan (秦宇轩)
  - Jun 23, 2026 13:07
  - The above link is broken, and I found a working one from MathOverflow.
- Richard Kruel
  - Aug 14, 2024 14:27
  - Thanks, John !
- John Baez
  - Jun 4, 2024 09:55
  - The book in honor of Hoàng Xuân Sính is out. It’s mostly in French and Vietnamese. Here is the cover ...
- Richard
  - Jun 3, 2024 13:48
  - Dear John, Do you have news on the book in Hoàng Xuân Sính’s honor ? Is it in English?

Quantum Mechanics of the Inverse Cube Force Law (6)
- John Baez
  - Jun 22, 2026 11:24
  - Not that I know of, and I really doubt one exists. Not in 4d spacetime, that is. But 5d spacetime ...
- Dan Piponi
  - Jun 21, 2026 17:08
  - Is there a quantum field theory that gives rise to an inverse cube force the way an inverse square force ...
- John Baez
  - Jun 12, 2026 13:09
  - You mean it’s worth it to you for me to do more work. It’s a lot of work, but I ...
- Madeleine Birchfield
  - Jun 11, 2026 23:01
  - Think it’s worth writing another blog post on the number theory behind the cc values and how that relates to ...
- John Baez
  - Jun 10, 2026 18:45
  - The numbers that show up are not vastly more complicated than those you’d see in the spectrum of the hydrogen ...

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September 8, 2026

The E6 Root Polytope

Posted by John Baez

$MathML-enabled post (click for more details).$

I’ve been thinking about the exceptional Lie algebra E₆, as a spinoff of my project on E₇, so I want to get a good mental picture of the E₆ root polytope. This is 6-dimensional polytope with remarkable symmetry.

Continue reading “The E6 Root Polytope” …

Posted at 6:20 PM UTC | Permalink | Post a Comment

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