{"id":7392,"date":"2025-10-23T11:16:48","date_gmt":"2025-10-23T18:16:48","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=7392"},"modified":"2025-10-28T12:14:40","modified_gmt":"2025-10-28T19:14:40","slug":"visualizing-the-barnes-wall-lattice","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2025\/10\/23\/visualizing-the-barnes-wall-lattice\/","title":{"rendered":"Visualizing the Barnes-Wall and Leech Lattices"},"content":{"rendered":"\n

If you have access to Wolfram’s Mathematica, see my notebook with code and data that may be helpful here<\/a> (135Mb). A PDF is here<\/a> (65Mb). This has detail on the octonion (E82<\/sup> and E83<\/sup>) based constructions for both the 24D Leech lattices as well as the 16D Barnes-Wall lattice shown at the bottom of this post.<\/p>\n\n\n\n

The Leech lattice is a sub-quotient of the largest of the sporadic finite simple groups<\/em>, namely the Monster group<\/a>. It is related to (and can be constructed by) E8<\/sub> and the 23rd Niemeier lattice<\/a> of E8<\/sub>3<\/sup>. The 24D Leech lattice is interesting to the physics of sphere packing, error-correcting codes, and possibly unifying General Relativity (GR) with Quantum Mechanics (QM). String theory was founded on the ideas related to its relationship to the Monster group and how 24 dimensions relate to the bosonic energy levels of the partition function on a torus. For more on the Monster and Group Theory, see my post here<\/a>.<\/p>\n\n\n\n

But taking things down a 8D (Bott periodic) notch, roughly speaking the 16D Barnes-Wall lattice is related to E8<\/sub> in the same way by BW16<\/sub>=E8<\/sub>2<\/sup>.<\/p>\n\n\n\n

I found some very nice recent visualizations on the BW16<\/sub> WP article<\/a> by Misaki Ohta with related arXiv papers<\/a>. He provided links to related code and data here<\/a>. Below are my renderings using that provided information.<\/p>\n\n\n

\n
\"\"\/<\/a>
Rectified E8 with 6720 vertices<\/figcaption><\/figure>\n<\/div>\n\n
\n
\"\"\/<\/a>
4320 shortest vectors of BW16<\/sub> using orthogonal projection mapping \u211d\u00b9\u2076 \u2192 \u211d\u00b2<\/em> (i.e. to B<\/em>16<\/sub> basis vectors)<\/em><\/figcaption><\/figure>\n<\/div>\n\n
\n
\"\"<\/a>
As above with 604800 edges of 16D norm=4 with colors assigned by projected edge lengths<\/figcaption><\/figure>\n<\/div>\n\n
\n
\"\"<\/a>
61440 2nd shortest 16D vectors of BW<\/em>16<\/sub>  using orthogonal projection mapping \u211d\u00b9\u2076 \u2192 \u211d\u00b2<\/em> (i.e. to B<\/em>16<\/sub> basis vectors)<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n

It is interesting to note that if one projects the 61440 2nd shortest 16D vectors of BW<\/em>16<\/sub>  using orthogonal projection mapping \u211d\u00b9\u2076 \u2192 \u211d\u00b2<\/em> (i.e. to B<\/em>8<\/sub> basis vectors [x=Cos, y=Sin]@(0-15)\u03c0\/8 used for E8<\/sub> below vs. the proper (0-15)\u03c0\/16<\/em> B<\/em>16<\/sub> as done above), we get a similar <\/strong>result<\/em> as projecting the 2nd shortest 8D vectors of<\/em> E8<\/sub> using the same basis shown as grey vertices in Fig 2 of arXiv:2506.11725<\/a>:<\/p>\n\n\n

\n
\"\"<\/a>
61440 2nd shortest 16D vectors of BW16<\/sub> projected using the B<\/em>8<\/sub> basis vectors<\/em>. This looks to be the same result as Fig. 2 of arXiv:2506.11725<\/a> with grey vertices of the 2160 2nd shortest 8D E8<\/sub> vertices associated with the maximal magic states in the same projection.<\/figcaption><\/figure>\n<\/div>\n\n\n

Projecting the 2160 2nd shortest 8D vectors of<\/em> E8<\/sub> using the B<\/em>8<\/sub> basis vectors<\/em> including 69120 edges of 8D norm=\u221a8 with colors assigned by projected edge length (notice the similarity in the vertex locations of the figure above):<\/p>\n\n\n

\n
\"\"<\/a>
Projecting the 2160 2nd shortest 8D vectors of<\/em> E8<\/sub> using the B<\/em>8<\/sub> basis vectors<\/em> including 69120 edges of 8D norm=\u221a8 with colors assigned by projected edge lengths<\/figcaption><\/figure>\n<\/div>\n\n\n

Projecting the 4320 shortest 16D vectors of BW<\/em>16<\/sub>  using orthogonal projection mapping \u211d\u00b9\u2076 \u2192 \u211d\u00b2<\/em> using the B<\/em>8<\/sub> basis vectors [x=Cos, y=Sin]@<\/em>(0-15)\u03c0\/8 <\/em><\/em>gives the results shown in Fig. 2 as red and blue vertices:<\/p>\n\n\n

\n
\"\"\/<\/a>
Projecting the 4320 shortest 16D vectors of BW<\/em>16<\/sub>  using orthogonal projection mapping \u211d\u00b9\u2076 \u2192 \u211d\u00b2<\/em> using the B<\/em>8<\/sub> basis vectors (0-15)\u03c0\/8<\/em><\/em><\/figcaption><\/figure>\n<\/div>\n\n\n

This time with edges colored by projected edge lengths:<\/p>\n\n\n

\n
\"\"\/<\/a>
Same as above with 604800 edges of norm=4 with colors assigned by projected edge lengths<\/figcaption><\/figure>\n<\/div>\n\n\n

Now, getting creative with the 16D projections of the 4230 BW16<\/sub>, if we double the E8 Petrie basis vectors we get a similar result with 480+1 visible vertices with one at the origin w\/240 overlapping vertices. There are 240 yellow with no overlaps and 240 blue with 16 overlaps each. There are 604800 norm=4 edges using roughly the same color pallet as my WP E8<\/sub> Petrie image<\/a>. This makes it easier to see the difference between the 4230 BW16<\/sub> and the normal E8<\/sub> Petrie projection, which seem to share the grid-like pattern of the 24-cells within the folded E8<\/sub> = H4<\/sub> + \u03c6 H4<\/sub> :<\/p>\n\n\n

\n
\"\"\/<\/a>
16D projection of the 4230 BW16<\/sub> using the E8 Petrie basis vectors doubled<\/figcaption><\/figure>\n<\/div>\n\n\n

This is the same as above, but using the 61440 vertex BW16<\/sub> with no edges (given there are millions of them). Please note that the number of visible vertices is the same as E8241<\/sub> at 2160!<\/p>\n\n\n

\n
\"\"\/<\/a>
16D projection of the 61440 BW16<\/sub> using the E8 Petrie basis vectors doubled with 2160 visible vertices<\/figcaption><\/figure>\n<\/div>\n\n\n

Octonion based Barnes-Wall and Leech Lattice visualizations based on my implementation of work from Geoffrey Dixon and Robert Wilson:<\/p>\n\n\n

\n
\"\"\/<\/a>
Octonion defined Barnes-Wall to 16D B82<\/sup> projection<\/figcaption><\/figure>\n<\/div>\n\n
\n
\"\"\/<\/a>
Octonion defined 24D Leech lattice to 8D E8 Petrie projection with all 196560 vertices in 2D and 3D, noting the similarity to the 6720 vertex rectified E8 in the same projection (above).<\/figcaption><\/figure>\n<\/div>\n\n
\n
\"\"\/<\/a>
Octonion defined Leech lattice to 24D E83<\/sup> Petrie projection with 20 increments of 1000 vertices for every 10000 out of 196560<\/figcaption><\/figure>\n<\/div>\n\n
\n
\"\"\/<\/a>
Same as above with 50% of the inner edges filtered out<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

If you have access to Wolfram’s Mathematica, see my notebook with code and data that may be helpful here (135Mb). A PDF is here (65Mb). This has detail on the octonion (E82 and E83) based constructions for both the 24D Leech lattices as well as the 16D Barnes-Wall lattice shown at the bottom of this … Continue reading Visualizing the Barnes-Wall and Leech Lattices<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[8,13,4,34,14,16],"class_list":["post-7392","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-4d","tag-e8","tag-h4","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7392","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=7392"}],"version-history":[{"count":60,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7392\/revisions"}],"predecessor-version":[{"id":7492,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/7392\/revisions\/7492"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=7392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=7392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=7392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}