Eight ways to multiply, and where they get you: the octonions, triality, the magic square and E8 — drawn from the arithmetic and told plain. A reading of one paper, with everything in it recomputed here.
Live: https://nanobotco.github.io/exceptional-magic/
A. Garrett Lisi, Division Algebras, Triality, and Exceptional Magic, arXiv:2609.12112v1 [math-ph], 10 September 2026 — https://arxiv.org/abs/2609.12112 — licensed CC BY-NC-ND 4.0.
That licence permits no derivatives, and this site is not one. No text, equation image or
figure from the paper is reproduced. The mathematics is recomputed from the definitions in
tools/; the paper is cited by section wherever it is the reason a topic appears; its 34
references are transcribed with links on the credit page. Read the paper first.
- Numbers — the four division algebras, the Cayley–Dickson ladder that builds them, what each doubling costs (measured on random elements, not asserted), and a multiplier that breaks associativity on demand.
- Wheel — the octonion table is the Fano plane. Seven lines found by walking the table; click two units and the line lights up.
- Triality — the cyclic form T(v, ψ, χ), checked three ways round on thousands of random triples; the 8v → 8c → 8s cycle of so(8), built from the diagram symmetry and verified; su(3, 𝔻) = tri(𝔻) + 3𝔻 for all four algebras.
- Square — the magic square, all forty-nine boxes from
dim = tri(A) + tri(B) + 3·dim A·dim B, each checked against the algebra it names. - E8 — G2, D4, F4, E6, E7, E8 built from their definitions and projected onto the Coxeter plane (eight rings of thirty), turnable in the browser; the triality split 92 + 78 + 78 found by search and checked on 13,440 root sums; the two-grading 120 + 128.
- Physics — what it is all supposed to be for, with the objections on the same page.
- Checks — the 139 arithmetic checks that run before the site will publish, listed.
- Gallery · Credit — every figure, downloadable; every source, linked.
python3 tools/algebra.py # the algebra checks, printed
python3 tools/roots.py # the root-system checks, printed
python3 tools/figures.py # every figure (SVG, JPG), build/facts.json, build/data/
python3 tools/site.py # the pages, into build/site/
./publish.sh # all of it, gated, into docs/
python3 tools/serve.py 8849 # a local preview, mounted at /exceptional-magic/
Python 3.9+, numpy, Pillow. Static HTML with the stylesheet inlined; the scripts in js/
read the same JSON the figures were drawn from.
Text and figures CC BY 4.0 — credit line Nan · hongdam.net · CC BY 4.0 with a link back.
Code MIT. See NOTICE.txt.
- Mot Dang — city directory for Chiang Mai and Chiang Rai
- The Mae Hong Son Loop — motorcycling the 600 km loop out of Chiang Mai — curves counted, air measured
- Muay Thai — the eight limbs, the thirty named techniques, the ceremony, and every gym on the map
- Roads of Chiang Mai — the square of 1296, four rings, and what each one did to the city — counted from the map
- wichaa — Lanna manuscripts, the amulet market, and the traditions around them
- Hand Poke — 28 traditions of marking skin by hand — the leg-tattoo zone of Burma, the Shan States and Lanna, counted
- Black Holes, Drawn — black holes modelled and drawn from the equations — generators, the past, present and future, the legends
- Quantum Computing, plainly — the history and theory of quantum computing in plain words, with demos; refreshed weekly
- Goin' Fast — a dirt-simple explainer about speed — twenty measured speeds from the ground under the house to light, and what each one costs
- The Three-Body Problem — the mathematics of the three-body problem in plain words, with the orbits found rather than copied
- Amulet Atlas — amulets, charms and talismans worldwide
- Carolina Barbecue — barbecue in North and South Carolina
- Wing Country — the American chicken wing
- Pink Box — the American mom-and-pop donut shop
- Basque Tables — Basque dining rooms of California, Nevada and Idaho
- Pinot Country — pinot noir: the vine, the regions, the cellars
- Care Abroad — treatment across borders, with published prices and their dates
- Thai Roots — a root dictionary of Thai, with a word decomposer
- The index — every corpus, site and repository, counted
- Uptake — a field manual on publishing for machines that copy
- NaNoBotCo — the portal
- ฮักฝรั่ง — เรื่องเงิน วีซ่า และชีวิตกับแฟนฝรั่ง
- Offrampt — turning crypto into spendable local money, Thailand first
All of it, counted: https://nanobotco.github.io/index/ · roster as JSON: https://nanobotco.github.io/index/fleet.json