---
title: "Triangles and quadrilaterals suffice"
authors:
  - "Mu-Tsun Tsai"
year: 2026
venue: "arXiv"
arxiv: "2610.13986"
doi: null
source: "https://arxiv.org/abs/2610.13986"
topics:
  - "computational-origami"
  - "circle-packing"
  - "medial-axis"
  - "geometric-construction"
seed_rank: 2298
seed_batch: "curiosity-2026-10-15"
reviewed: "2026-10-15"
pool: "maths"
relevance_score: 8
cites:
  - title: "Triangles and quadrilaterals suffice"
    url: "https://arxiv.org/abs/2610.13986"
    year: 2026
    arxiv: "2610.13986"
    doi: null
see: ["1470-monotile-kirigami"]
---

# Triangles and quadrilaterals suffice

## One-sentence takeaway

Proving Robert J. Lang's 20-year-old conjecture from circle-packed origami design: any axial polygon of five or more tangent circles can be reduced by adding one circle tangent to at least three others, so every base can be cut into triangles and quadrilaterals with straightedge-and-compass steps, using at most N − 4 added circles, a bound that is tight.

## Why it's lovely

In the circle/river method, each flap of an origami base is a circle, and the crease pattern inside each polygon of touching circles is easy only for triangles and quadrilaterals; larger polygons have needed numerical solvers. Lang noticed that adding a circle always seemed to help but never proved it. The proof is very neat: the free region inside an N-gon of circles has a medial axis (Blum's 1967 shape-recognition skeleton) that is a tree with exactly N leaves, and its internal vertices are exactly the circles you can add. So the geometry collapses into one elementary lemma about trees. Medial axes and skeletons turn up throughout graphics and navigation, which makes this a lovely reuse of a familiar tool.

## Key ideas

- **Setting.** Flaps are circles with radius proportional to flap length, packed in the square. Axial polygons join the centres of mutually tangent circles, and each one is filled with a "molecule" crease pattern.
- **Theorem 1.** For any N-gon of touching circles with N > 4, a circle touching at least three of them can be added so that the largest resulting polygon has at most N − 1 sides.
- **Medial-axis identification.** The medial axis of the enclosed free region is a tree with N leaves; its internal vertices are the centres of candidate added circles. Lang's conjecture then reduces to a purely combinatorial tree lemma.
- **Constructive.** Each added circle solves a classical Apollonius tangency problem, so the reduction stays within straightedge and compass. All the figures were generated by the author's implementation.
- **Tight count.** At most N − 4 added circles fully reduce an N-gon, and this cannot be improved. The result extends to polygons crossed by one or more rivers.

## Caveats

Single-author preprint (submitted 26 Sep 2026, math.HO/math.MG). As the paper itself notes, the point is theoretical: added circles introduce extra flaps that can make the actual fold harder. The implementation is described but no code link is given in the abstract.

## Links

- arXiv abstract: https://arxiv.org/abs/2610.13986
- PDF: https://arxiv.org/pdf/2610.13986