Beloch, named after Margherita Piazzola Beloch, is a declarative language for origami, built on the Huzita-Justin axioms, that compiles source models into folded states, crease patterns, and step-by-step folding diagrams.

Status: the evaluator implements all seven Huzita-Justin axioms over an exact real-algebraic number kernel and emits FOLD. See decisions/ for the architecture, spec/SPECIFICATION.md for the language, and notes/ for the design journal. More at beloch.toph.so.

Bases

Two traditional bases, each a handful of statements. mark only scores a crease, fold moves paper, and flatten collapses a vertex flat — deriving the crease that closes it when no axiom can construct one. Every program emits a FOLD file, which renders into the pairs below.

The crease pattern shows the sheet as it is folded: mountains and valleys are derived from the layer order, and the construction lines that were only scored stay flat.

Fish base

Two long flaps from opposite corners. Each half of the diagonal is collapsed with flatten, which derives the crease that closes the vertex — the one crease here that no axiom constructs from the named points.

paper square

mark --diag = map .a onto .c
mark --ray = through .a .c

mark --l1 = map --ab onto --diag
mark --l2 = map --da onto --diag
flatten (--l1) (--l2) (--ray) {toward .d}

mark --l3 = map --cd onto --diag
mark --l4 = map --bc onto --diag
flatten (--l3) (--l4) (--ray) {toward .d}

crease pattern	folded

Swivel rabbit ear

A rabbit ear whose hinges sit at a free height on the side edges rather than at the triangle's angle bisectors, so the crease that flattens the vertex is not constructible by any Huzita axiom from the named points — flatten solves for it, and binds it to --ear:

--ear = flatten (--ba \ .a) (--bb \ .b) (--v \ .m) {toward .c}

crease pattern	folded

The full program, with the scaffolding that locates the hinge height, is in examples/bases/swivel-rabbit.bel. More programs, from simple midline folds to Messer's cube-root-of-two construction (axiom 7), are in examples/.

Development

The evaluator core is OCaml; tooling lives at the edges in TypeScript (see decision 0001). A Nix flake provides the OCaml toolchain (dune, menhir, sedlex, ocaml-lsp, …):

direnv allow          # or: nix develop
dune build
dune exec beloch -- --version

Layout

packages/core/      evaluator core + `beloch` CLI (OCaml)
packages/multifold/ multifold axiom enumeration (Alperin-Lang reproduction)
packages/render-2d/ FOLD→SVG render engine (bun)
packages/www/       landing + docs + Playground site
packages/eval-web/  js_of_ocaml browser eval bundle
packages/grammar/   tree-sitter grammar
packages/vscode/    editor extension
spec/         human-readable language specification (grows per increment)
decisions/    architecture decision records (ADRs)
notes/        dated design journal (+ antipatterns.md dead ends)
examples/     .bel programs, tagged works / aspirational / anti
paper/        the eventual write-up (arXiv / JOSS / OSME) + bibliography.md