PreprintarXiv licencearXiv
Triangles and quadrilaterals suffice
Abstract
We prove a conjecture proposed by Robert J. Lang over 20 years ago, stating that in a circle-packed origami design, any axial polygon with 5 or more sides can be reduced by properly adding a new circle. The proof is based on the structure of the medial axis of the region enclosed by the circles, and yields a concrete algorithm for finding the added circle. The result extends to configurations involving rivers.
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BibTeX
@misc{tsai2026triangles,
title = {{Triangles and quadrilaterals suffice}},
author = {Mu-Tsun Tsai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13986},
eprint = {2610.13986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13986},
}
RIS
TY - UNPB
TI - Triangles and quadrilaterals suffice
AU - Mu-Tsun Tsai
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13986
UR - https://arxiv.org/abs/2610.13986
AB - We prove a conjecture proposed by Robert J. Lang over 20 years ago, stating that in a circle-packed origami design, any axial polygon with 5 or more sides can be reduced by properly adding a new circle. The proof is based on the structure of the medial axis of the region enclosed by the circles, and yields a concrete algorithm for finding the added circle. The result extends to configurations involving rivers.
ER -