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September 2026
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20	21	22	23	24	25	26
27	28	29	30

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.
nLab - Latest Changes
- Sep 17, 2009
- Trying to install better software support for logging latest changes to the nLab.
2-Branes in 11 Dimensions
- Jun 15, 2009
- A question about 2-branes in 11 dimensions.
Interactive Science Symposium
- Sep 8, 2009
- A sympositum on the use of web technology for communication among scientists.
The Earliest Stars?
- Dec 25, 2006
- Have people caught sight of the earliest stars in the Universe?
Compositional System Dynamics for Public Health
- Jun 24, 2023
- Nathaniel Osgood will give a talk about category theory for public health software on Thursday June 29, 10:00-11:00 Pacific Time.

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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