A Cycle Five tutorial · Tropical geometry

Learn You a Tropical Geometry for Great Good!

Take ordinary geometry and replace addition with min and multiplication with plus. Lines fall apart into balanced tripods, cubics grow a cycle, and the Petersen graph turns out to be a moduli space. Everything here is computed — every chapter ends with a program whose assertions are the chapter's claims.

Start with Chapter 1 All chapters

Part I — Curves, tropicalised

Chapter 1 — Addition Is Min, Multiplication Is Plus (and a Line Is Not a Line)

The tropical semiring, the tie definition of a curve, and the tropical line: three rays, a Newton polygon, and balancing.

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Chapter 2 — A Conic Is a Line That Got Caught Twice

Weights and multiplicity, the doubled line hiding inside a conic, subdivisions from coefficients, and why the smooth conic has genus 0.

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Chapter 3 — The Cubic Has a Donut Inside It

The tropical cubic: one interior lattice point, one 6-cycle, genus 1 — predicted from the Newton triangle before it is computed.

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In preparation

The tutorial is being converted, part by part, from a 21-day research study (21 days, 22 figures, every claim asserted in code). Still to come:

Part II — Counting curvesFloor diagrams; the 12 cubics; genus is cycles

Part III — Interlude for programmersShortest paths are matrix multiplication

Part IV — Matroids & Hodge theoryBergman fans; the Chow ring; Huh's log-concavity

Part V — Moduli & tree spaceThe Grassmannian is a line; the Petersen graph is a moduli space

Part VI — Chip-firing & JacobiansSandpile groups; trees are invisible

Parts VII–VIII — Deep dives & open problemsThe defect ledger; the Vámós matroid; Dhar burning

For agents: serve the source

This site is built from Markdown, so it serves Markdown. Any page requested with an explicit Accept: text/markdown header returns its raw source — no HTML chrome, no conversion. Humans get the page; agents get the document.

curl -H 'Accept: text/markdown' https://minplus.xyz/chapters/chapter-01/