A constrained long-legged A is made from two rays with a common apex and a crossbar joining points at equal distance from that apex. We determine the maximum number of regions cut from the plane by n such objects, for every n.
Main theorem
01 · model
For an angle φ, write e(φ)=(cos φ,sin φ). Given an apex P∈ℝ², a radius r>0, and distinct unit vectors u,v, define
The first two pieces are the legs; the last is the crossbar. Equal leg lengths are a metric constraint, so arbitrary affine deformations do not preserve the problem.
Generic arrangement
A cross-shape intersection is proper when it lies strictly beyond every participating ray apex and strictly inside every participating crossbar. An arrangement is generic when supporting lines from different A's are nonparallel, all actual intersections are proper, no vertex lies on a foreign component, and no three shapes meet at one point.
Proposition
Here I is the number of finite proper crossings and F is the number of complementary regions.
To prove it, compactify the plane by one point at infinity. A single A becomes a connected graph with four vertices (apex, two bar endpoints, infinity) and five edges, hence first Betti number two. The union of n disjoint A's, with their infinity points identified, has first Betti number 2n. Every proper transverse crossing identifies two interior arc points and increases the first Betti number by one. The compactified union therefore has Betti number 2n+I, and Euler's formula on the sphere gives F=(2n+I)+1.
02 · extremal graph
Cutler, Karlsson, and Sloane proved two local facts: two constrained long-legged A's have at most eight proper finite intersections, and no three A's can be pairwise eight-crossing.
Form a graph G₈ with one vertex per A and an edge exactly when the pair has eight crossings. The three-object obstruction says that G₈ is triangle-free. If it has e edges, then every edge contributes at most eight crossings and every nonedge at most seven:
Mantel's theorem gives e≤⌊n²/4⌋. Combining this with the region identity yields
Equality is rigid: the eight-crossing graph must be the balanced complete bipartite graph K⌊n/2⌋,⌈n/2⌉, and every pair within a part must have seven crossings. The lower construction now has a precise target.
03 · lower construction
Fix part sizes p,q≥1, put m=max(p,q), and choose a sufficiently small ε>0. The construction uses O=(0,0), C=(1,0), and h=3/2.
Ni
For 1≤i≤p, set
Its leg directions are di±=e(αi±ε). Center the bar as
Then Bi={Mi+tni:−si≤t≤si} and its supporting line is e(αi)·X=cos αi cos ε.
Wj
For 1≤j≤q, set
The low and high directions are ℓj=e(λj) and uj=e(π−θj+λj). Put the high bar endpoint at Hj=C+ε(−Dj,h), then define
Thus (Qj)y=−1 and the bar joins Kj to Hj.
At ε=0, the relevant wide-family abscissae are
with Qj→(Aj,−1), Kj→(Bj,−1), and Hj→C.
Order each shape's components as first leg, second leg, crossbar. A 1 means that the indicated proper component intersection is required; a 0 makes no assertion.
110
110
111111
111
011111
011
01104 · incidence verification
Blow-up stability
If, after an invertible affine rescaling, two supporting lines and their directed component domains converge to nonparallel lines meeting in the relative interior of both limiting domains, then their original components intersect properly for all sufficiently small positive ε. This follows because the two line-intersection parameters are continuous rational functions and the limiting domain inequalities are strict.
Take i<k, write a=ai and b=ak, so b−a≥3. Under the blow-up (x,y)↦(x,y/ε) near O, the σ-leg of Ni tends to
The σ-leg of Ni and τ-leg of Nk meet at limiting abscissa x=(b−a)/(b−a+τ−σ)>0, because the denominator is at least (b−a)−2≥1. Hence all four leg–leg intersections are proper.
For the bar of Ni against the τ-leg of Nk, put A=aε, B=bε. Writing the point as Mi+t ni, direct line intersection gives
The endpoint-sensitive Taylor expansion is
Since a=3i≥3, all four margins are positive for small ε, so the earlier bar meets both later legs strictly inside.
Finally every narrow bar line passes through Xε=(cos ε,0). Its bar coordinate is sin(aε)(1−cos ε)=O(ε³), while the half-length is si=ε+O(ε³). Therefore two narrow bars meet in both interiors. Total: 4+2+1=7.
Fix Ni,Wj, and abbreviate a=ai, L=Lj, D=Dj, μ=tan θj=2qj/(1−qj²), and ν=tan(θj/2)=qj.
The determinant between the wide low direction and the σ-leg of the narrow A is sin((L−a−σ)ε), and L−a−σ≥(4m+1)−(3m+1)=m>0. The two ray parameters satisfy
so both narrow legs meet the wide low leg properly.
For the other six incidences, use the chart (X,Y)=((x−1)/ε,y/ε) at C. The narrow legs tend to Y=±1 and the narrow bar tends to X=0, −1<Y<1. The wide high leg and bar tend to
The line Y=σ meets the wide bar with X+D=(h−σ)/ν>0, since h=3/2>1. Thus both narrow legs meet the wide bar in its interior. The wide high-ray apex escapes to infinity opposite the ray, so both finite intersections with the narrow legs lie inside the ray.
At X=0, the high leg and bar have heights h−Dμ=3/2−2/(1−qj²) and h−Dν=1/2. Since 0<qj≤1/4, the first lies in [−19/30,−1/2); both heights are strictly between −1 and 1. The narrow bar therefore meets the wide high leg and wide bar properly. Total: 2+6=8.
Let j<k and write μr=2qr/(1−qr²), νr=qr. Because qk≤qj/4 and qj≤1/4,
In the chart at C, every high leg or crossbar of Wr limits to a line through (−Dr,h) of slope −s, with s=μr or νr. For a j-line of slope −s and k-line of slope −t, s>t. With Δ=Dk−Dj>0, their displacements past their high endpoints are
These strict positive displacements give all four high/bar incidences: high–high, high–bar, bar–high, and bar–bar.
It remains to make the low leg of Wj meet all three components of Wk. In the chart (x,Y)=(x,(y+1)/ε), the low leg of Wr tends to Y=Lr(x−Ar), x>Ar. The k-th high leg tends locally to the half-line x=Ak,Y>0; its bar tends locally at the low endpoint to x=Bk,Y>κk, where
Since Aj<Ak<Bk, the low j-leg hits the high k-leg forward. For the bar hit, Dj≤Dk/4 and Lk/Lj<5/4 give
Hence the low j-leg hits inside the k-bar. Finally the two low limiting lines meet at
so the low–low crossing is forward on both rays. These three low incidences plus the four high/bar incidences give 7.
05 · general position
The transparent base construction has intentional degeneracies; for example all narrow crossbar supporting lines concur at (cos ε,0). This is removed without losing any required crossing.
Choose ε small enough that all strict component-domain inequalities above hold simultaneously. They define an open neighborhood U in the parameter space of apices, radii, and leg directions. Use the rational chart for each unit direction
After clearing denominators, every unwanted degeneracy—support-line parallelism, a foreign component through an apex or bar endpoint, or coincidence of two distinctly labelled finite crossings—is contained in the zero set of a polynomial in the rational chart parameters. Only finitely many labels occur. None of these polynomials vanishes identically on U, because a sufficiently small independent change of one participating apex, radius, or direction breaks the event. The bad locus is therefore a finite union of proper algebraic zero sets and has empty interior.
Rational points are dense in the chart, so U contains a rational point outside every bad set. This gives a generic rational arrangement with all strict required incidences preserved.
Completion
Take p=⌊n/2⌋ and q=⌈n/2⌉. There are pq=⌊n²/4⌋ cross-family pairs with eight crossings and all remaining pairs have seven. Thus
The upper bound gives equality, and F=I+2n+1 proves the theorem.
The same argument works for arbitrary p,q≥1: there is a generic rational arrangement with eight crossings on every cross-family pair and exactly seven on every same-family pair. The exact region count is
Colored extremal graphs also play a central role in the related published paper A Two-Graph Refinement of Paulsen's Lollipop Bounds.
06 · the 55-region witness
Four A's need 46 globally distinct proper crossings: four cross-part pairs contribute eight each and two within-part pairs contribute seven each. The region identity then gives 46+2·4+1=55.
n04.json. Horizontal
scale is enlarged to separate its extremely compressed geometry.
The colors distinguish the four A's and the face coloring makes the
55 complementary regions visible.
A later rational simplification found a second witness with small, human-readable fractions. It proves the same 7/8 crossing pattern, but it is not the coordinate set used in the two primary figures above.
07 · exact finite checks
These thirteen files are independent reproducibility checks, not
premises of the all-n proof. Every coordinate and direction
parameter is rational. The verifier uses only Python's standard
library and fractions.Fraction; it does not make
floating-point or visual decisions.
| n | parts | crossings | a(n) | exact certificate |
|---|---|---|---|---|
| 4 | 2+2 | 46 | 55 | n04.json |
| 5 | 2+3 | 76 | 87 | n05.json |
| 6 | 3+3 | 114 | 127 | n06.json |
| 7 | 3+4 | 159 | 174 | n07.json |
| 8 | 4+4 | 212 | 229 | n08.json |
| 9 | 4+5 | 272 | 291 | n09.json |
| 10 | 5+5 | 340 | 361 | n10.json |
| 11 | 5+6 | 415 | 438 | n11.json |
| 12 | 6+6 | 498 | 523 | n12.json |
| 13 | 6+7 | 588 | 615 | n13.json |
| 14 | 7+7 | 686 | 715 | n14.json |
| 15 | 7+8 | 791 | 822 | n15.json |
| 16 | 8+8 | 904 | 937 | n16.json |
a(0),…,a(16) 1, 3, 13, 30, 55, 87, 127, 174, 229, 291, 361, 438, 523, 615, 715, 822, 937.
08 · research provenance
The path was a sequence of exact computational discoveries followed by an analytic compression. This record is included to make the model-assisted provenance explicit.
The 55-region discovery and its first proof were developed in the ChatGPT web UI with GPT-5.6 sol pro. This produced the complicated rational witness shown above and the exact target pattern: four 8-crossing pairs and two 7-crossing pairs.
Local Codex running GPT-5.6 sol xhigh, using the n=4 manuscript as its starting point, generated and checked rational equality certificates for every n from 5 through 16. Those experiments made the balanced complete bipartite 8-crossing graph impossible to miss.
A zip containing the certificates, proof directions, and discovery artifacts was supplied to the ChatGPT web UI running GPT-5.6 sol pro. It found the narrow/wide parametric construction, converted the finite pattern into the all-n analytic proof on this page, and closed the sequence.
Local Codex audited the proof, including an exact symbolic expansion of the endpoint-sensitive narrow–narrow calculation, an independent rational verifier for the finite certificates, and regression checks of the parametric base through balanced size 64. The models also assisted in preparing the manuscript and this HTML presentation.
09 · reproducibility
The complete research-paper version of the proof and certificate appendix.
Download PDF · 119 KBSubmission-ready source files.
main.tex references.bib main.bblChecks all rational certificates n=4,…,16 using only the Python standard library.
verify_certificates.py archived outputChecks the three exact incidence masks for balanced sizes through n=64.
verify_parametric_base.py archived outputOriginal minimal-package notes and SHA-256 ledger.
README.md SHA256SUMS
Preserve the scripts/, certificates/, and
verification/ sibling folders shown on this page, then
run:
python3 scripts/verify_certificates.py \
--output verification/exact_certificates_n4_n16.json
python3 scripts/verify_parametric_base.py \
--max-n 64 \
--output verification/parametric_base_n2_n64.jsonPython 3.10 or later is recommended. No third-party package is used.
10 · sources