On Phil's post, M theorist kneemo provides this link. I have been waiting for kneemo to mention it again, so that I can post a couple of its diagrams, namely one $M2$ brane and one $M5$ brane, in the stringer's jargon. Here they are:

The triangle is used to describe the complex projective plane. To each interior point, imagine gluing a little torus. Exterior points get circles, which are degenerate tori. An interior trivalent vertex is then like a (circle over a) pair of pants, which is a projective line. We recognise the dual tetractys on the right hand diagram, where all $4$-valent vertices have been resolved into trivalent ones.

Tropical geometers know that lines are trivalent vertices. Their lines come from considering the maximum (or minimum) function as a product on the positive real number set. This product is distributive with respect to addition, giving us two products to define a semiring. Polynomials are then defined using these two products. Hexagons appear for higher degree curves.

In ternary geometry, there is no need to stop at two products. Since distributivity can be weakened, we retain the multiplicative product. Then the positive real numbers come with three natural operations. $M$-branes can be taught to children.

7 years ago

Please see the comment on page 20 of the "mysterious duality" paper. The configuration which maps to your dual tetractys would be an "M8-brane", not an M5-brane. The diagram for an M5-brane is simpler.

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