Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles
Abstract
For a dissection D of the unit square into n nondegenerate triangles, let R(D)=maxi ai−mini ai, Δ(n)=infD R(D). We prove that this infimum is attained for every n≥2, and determine the exact minima for n=5 and n=7, allowing T-junctions. For five triangles, Δ(5)=(5√5−11)/8; equality holds precisely when three areas equal (3−√5)/4 and two equal (3√5−5)/8. For seven triangles, Δ(7)=r7, where r7 is the unique root in $(0,1/4900)$ of 864r4+2160r3−6060r2+4972r−1. Every minimizer has four areas (1+3r7)/7 and three areas (1−4r7)/7, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound Δ(9)≤ 0.0001273496861283553341…, which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than 1/3500; hence a global minimizer must lie outside that family. The exact value of Δ(9) remains open.
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BibTeX
@misc{li2026exact,
title = {{Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles}},
author = {Muxi Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22693},
eprint = {2609.22693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22693},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles
AU - Muxi Li
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22693
UR - https://arxiv.org/abs/2609.22693
AB - For a dissection D of the unit square into n nondegenerate triangles, let R(D)=max_i a_i−min_i a_i, Δ(n)=inf_D R(D). We prove that this infimum is attained for every n≥2, and determine the exact minima for n=5 and n=7, allowing T-junctions. For five triangles, Δ(5)=(5√5−11)/8; equality holds precisely when three areas equal (3−√5)/4 and two equal (3√5−5)/8. For seven triangles, Δ(7)=r_7, where r_7 is the unique root in $(0,1/4900)$ of 864r^4+2160r^3−6060r^2+4972r−1. Every minimizer has four areas (1+3r_7)/7 and three areas (1−4r_7)/7, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound Δ(9)≤ 0.0001273496861283553341…, which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than 1/3500; hence a global minimizer must lie outside that family. The exact value of Δ(9) remains open.
ER -