Abstract
The shape enclosing a given area with the shortest boundary is, in the plane, a disc. This paper shows that on a flat torus the answer changes partway to a band, and measures numerically that at the threshold a descending method stops working. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, the threshold has a closed form. A disc of area $A$ has perimeter $2\sqrt{\pi A}$ and a band has perimeter $2$ whatever $A$ is, and the two cross at exactly $A=1/\pi=0.3183098862$ (Section 2).
Second, the tool was verified first. Feeding a disc and a band into a perimeter-decreasing flow, neither moves — the changes are $-2.16\times10^{-4}$ and $+3.33\times10^{-15}$ (Section 3).
Third, away from the threshold the flow answers correctly. At area $0.15$ the best of $12$ runs is $1.3811$ against a disc at $1.3729$, and at area $0.45$ it is $2.0189$ against a band at $2.0000$ (Section 4).
Fourth, this is the core. Near the threshold a descending method cannot choose. The gap to be resolved is only $0.0584$ at area $0.30$ and $0.0670$ at $0.34$, while the spread of $12$ runs is $1.1535$ and $0.9378$ — the spread is fourteen to twenty times the gap (Section 5).
Fifth, there are two switches. Past area $1-1/\pi=0.681690$ it is the hole that becomes a disc — the answer is symmetric about a half (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the isoperimetric inequality, the isoperimetric regions of a flat torus and threshold dynamics are all standard. It is not proved that the band and the disc are the only candidates — only those two, and the disc-shaped hole, were compared, and nothing shorter is ruled out. The numerical flow does not guarantee a global minimum — Section 5 is precisely a measurement of that absence of guarantee. The perimeter estimator is not exact — after calibration against known shapes a residue of about $0.7\%$ remains on a disc. Relation to Paper 198: 198 showed that on a flat cylinder “straight” takes the long way round — this paper shows that on a flat torus the shortest enclosure stops being a disc. In both, the curvature is innocent and the culprit is the global shape. Relation to Paper 193: 193 wrote that an estimate is wrong precisely where one would want to use it — Section 5 here has the same shape: a descending method cannot choose precisely where the answer switches. What is added is putting the crossing of the two candidates in closed form, verifying the perimeter-decreasing flow on known shapes before using it, confirming that the flow answers correctly on both sides away from the threshold, and measuring over $12$ runs each that near the threshold the spread exceeds the gap by more than fourteenfold.
1. Notation and Position
On a flat square torus of side $1$, a square with opposite edges glued, take a region of area $A$ and call the length of its boundary the perimeter. The perimeter is measured with an isotropic estimator — counting grid edges is not used, since it makes a diagonal boundary too long by a factor of $\sqrt2$. Every number below is computed here.
Position: Paper 198 showed that on a flat cylinder “straight” takes the long way round — the curvature was identically zero and the culprit was the global shape. This paper asks the same question in terms of area and perimeter: on something flat, from where does the shortest enclosure stop being a disc?
2. The Two Candidates Cross at Exactly $1/\pi$
On a torus there is a candidate the plane does not have: a band running all the way round. A band has no ends, so its boundary is only two cuts, of total length $2$ regardless of the area.
Check 2.1:
| Area $A$ | Disc, $2\sqrt{\pi A}$ | Band | Shorter |
|---|---|---|---|
| $0.100000$ | $1.120998$ | $2.000000$ | disc |
| $0.200000$ | $1.585331$ | $2.000000$ | disc |
| $0.300000$ | $1.941626$ | $2.000000$ | disc |
| $1/\pi=0.318310$ | $2.000000$ | $2.000000$ | equal |
| $0.350000$ | $2.097196$ | $2.000000$ | band |
| $0.450000$ | $2.377996$ | $2.000000$ | band |
Observation 2.2 Solving $2\sqrt{\pi A}=2$ gives $A=1/\pi$. The threshold is not an approximation but exactly that number. This paper does not prove that these two are the only candidates — only two were compared.
3. Verifying the Tool First
Check 3.1: calibrate the perimeter estimator on shapes whose answer is known. A residue remains even after calibration.
| Shape | True perimeter | Estimate after calibration |
|---|---|---|
| Disc of area $0.2$ | $1.585331$ | $1.597002$ |
| Band | $2$ | $1.984925$ |
The residue is about $0.7\%$, and that $0.7\%$ is the floor on what can be seen below.
Check 3.2: feed a disc and a band into a perimeter-decreasing flow, an area-preserving threshold dynamics. Both are already candidates for the optimum, so neither should move.
| Shape fed in | Perimeter before | After | Change |
|---|---|---|---|
| Disc of area $0.20$ | $1.597002$ | $1.596787$ | $-2.16\times10^{-4}$ |
| Disc of area $0.10$ | $1.129082$ | $1.128550$ | $-5.32\times10^{-4}$ |
| Band of area $0.40$ | $1.984925$ | $1.984925$ | $+3.33\times10^{-15}$ |
| Band of area $0.60$ | $1.984925$ | $1.984925$ | $+2.22\times10^{-16}$ |
Observation 3.3 Both are fixed points. The flow does not destroy good shapes — and only then is it used where the answer is unknown.
Observation 3.4 (the first attempt failed) The first run of this paper was under-converged — the diffusion length was too short, the flow stopped before the noise had gathered into one blob, and the final perimeters came out between $2.5$ and $3.9$, matching neither a disc nor a band. Without the verification placed first, those numbers would have been written down.
4. Away from the Threshold, the Flow Answers Correctly
Check 4.1: run the flow from smooth random initial shapes, $12$ times at each area, and take the best.
| Area | Disc | Band | Best of the flow | Nearer to |
|---|---|---|---|---|
| $0.1500$ | $1.3729$ | $2.0000$ | $1.3811$ | disc |
| $0.4500$ | $2.3780$ | $2.0000$ | $2.0189$ | band |
Observation 4.2 On both sides of the threshold the flow agrees with the closed form, to $0.6\%$ at $0.15$ and to $0.9\%$ at $0.45$.
5. The Core: Near the Threshold a Descending Method Cannot Choose
Check 5.1: repeat near the threshold, and measure the spread of the $12$ runs as well.
| Area | Gap to resolve | Best | Median | Spread |
|---|---|---|---|---|
| $0.1500$ | $0.6271$ | $1.3811$ | $1.9339$ | $1.0008$ |
| $0.2600$ | $0.1924$ | $1.8503$ | $2.4933$ | $2.5780$ |
| $0.3000$ | $\mathbf{0.0584}$ | $1.9757$ | $2.1500$ | $\mathbf{1.1535}$ |
| $1/\pi$ | $\mathbf{0.0000}$ | $2.0120$ | $2.2993$ | $0.9454$ |
| $0.3400$ | $\mathbf{0.0670}$ | $2.0161$ | $2.2155$ | $\mathbf{0.9378}$ |
| $0.4500$ | $0.3780$ | $2.0189$ | $2.8598$ | $5.2529$ |
Observation 5.2 At area $0.30$ the gap to be resolved is only $0.0584$, while the $12$ runs scatter over $1.1535$ — the spread is twenty times the gap. At $0.34$ it is $0.9378$ against $0.0670$, fourteen times.
Check 5.3: look at the gap itself, finely, around the threshold.
| Area | Disc | Band | Gap as a percentage |
|---|---|---|---|
| $0.250000$ | $1.772454$ | $2.000000$ | $11.377\%$ |
| $0.290000$ | $1.908991$ | $2.000000$ | $4.550\%$ |
| $0.310000$ | $1.973721$ | $2.000000$ | $1.314\%$ |
| $1/\pi$ | $2.000000$ | $2.000000$ | $\mathbf{0\%}$ |
| $0.330000$ | $2.036394$ | $2.000000$ | $1.820\%$ |
| $0.350000$ | $2.097196$ | $2.000000$ | $4.860\%$ |
Observation 5.4 Just either side of the threshold the two shapes differ by only about one per cent — the same order as the $0.7\%$ residue of the estimator measured in Section 3. So even if the flow worked perfectly, the measuring could not tell them apart.
Observation 5.5 And at the threshold itself the gap is exactly zero. The two shapes have the same length, and a procedure that only decreases the perimeter has no reason to prefer either. “Where the answer switches” is also where descent stops being a method. Paper 193 wrote that an estimate is wrong precisely where one would want to use it — here descent falls silent precisely where one most wants to ask.
6. There Are Two Switches, Symmetric About a Half
Check 6.1: as the area grows, it is the hole that becomes a disc. Set the three candidates side by side.
| Area | Disc | Band | Hole is a disc | Shortest |
|---|---|---|---|---|
| $0.100000$ | $1.120998$ | $2.000000$ | $3.362995$ | disc |
| $0.250000$ | $1.772454$ | $2.000000$ | $3.069980$ | disc |
| $1/\pi$ | $2.000000$ | $2.000000$ | $2.926836$ | threshold |
| $0.500000$ | $2.506628$ | $2.000000$ | $2.506628$ | band |
| $1-1/\pi$ | $2.926836$ | $2.000000$ | $2.000000$ | threshold |
| $0.800000$ | $3.170662$ | $2.000000$ | $1.585331$ | hole is a disc |
Observation 6.2 The switches are at $1/\pi=0.318310$ and $1-1/\pi=0.681690$, symmetric about a half. At area one half the disc and the disc-shaped hole are exactly equal in length — and both lose to the band.
7. Changing the Sides Moves the Threshold
Check 7.1: keep the area at $1$ and change the sides of the torus. The shortest band cuts across the shorter side twice, so its perimeter is $2\min(a,b)$.
| Sides | Shortest band | Threshold area | Times $1/\pi$ |
|---|---|---|---|
| $1.00\times1.00$ | $2.000000$ | $0.318310$ | $1.0000$ |
| $0.80\times1.25$ | $1.600000$ | $0.203718$ | $0.6400$ |
| $0.50\times2.00$ | $1.000000$ | $0.079577$ | $0.2500$ |
| $0.25\times4.00$ | $0.500000$ | $0.019894$ | $0.0625$ |
Observation 7.2 The threshold is $\min(a,b)^2/\pi$ — the more the torus is squeezed, the sooner the band wins. At $0.25\times4.00$ the switch comes at barely $2\%$ of the area.
8. The Separator and the Fence
Observation 8.1 (the separator) The answer does not change because of curvature — the torus is flat. What changes it is whether one can go all the way round, which is the global shape. The same shape as the flat cylinder of Paper 198: the curvature was innocent both times.
Observation 8.2 (the fence) The fence around “the shortest enclosure is a disc” is being in the plane — more precisely, that no other candidate exists. On a torus a band appears, and that alone changes the answer. While the area is small the plane's answer survives: below $1/\pi$ the disc wins.
Observation 8.3 And the other fence this paper measured lies on the side of the method. Descent is usable only while one answer is clearly better, and it does not work where two are level. That place can be written in one line in closed form — $A=1/\pi$.
9. The Census and the Close
What this paper checked:
- The threshold is $1/\pi$ in closed form (Check 2.1).
- The tool was verified first (Check 3.2): disc and band are fixed points. The first run failed, under-converged (Observation 3.4).
- Away from the threshold the flow is right (Check 4.1): to $0.6\%$ and $0.9\%$.
- Near it, it cannot choose (Check 5.1): the spread is fourteen to twenty times the gap.
- At the threshold the gap is exactly zero (Check 5.3): descent has no reason to prefer either.
- Two switches, symmetric about a half (Check 6.1).
- Squeezing makes the band win sooner (Check 7.1): the threshold is $\min(a,b)^2/\pi$.
References
- Steiner, J. (1838). Einfache Beweise der isoperimetrischen Hauptsaetze. Journal fuer die reine und angewandte Mathematik, 18.
- Hauswirth, L., Perez, J., Romon, P., Ros, A. (2004). The Periodic Isoperimetric Problem. Transactions of the American Mathematical Society, 356(5).
- Ros, A. (2005). The Isoperimetric Problem. In: Global Theory of Minimal Surfaces. Providence: American Mathematical Society.
- Merriman, B., Bence, J. K., Osher, S. (1992). Diffusion Generated Motion by Mean Curvature. UCLA Computational and Applied Mathematics Report 92-18.
- Osher, S., Sethian, J. A. (1988). Fronts Propagating with Curvature-Dependent Speed. Journal of Computational Physics, 79(1).
- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. Zenodo. 10.5281/zenodo.22104170
- Yamagishi, Y. (2026). Two Coupled Costs Cannot Be Added [Paper 193]. Zenodo. 10.5281/zenodo.22106668
- Yamagishi, Y. (2026). “Straight” Does Not Mean “Shortest”, and There Are Two Fences [Paper 198]. Zenodo. 10.5281/zenodo.22107954
- Yamagishi, Y. (2026). The Two Hundredth Census [Paper 200]. Zenodo. 10.5281/zenodo.22108829
- Yamagishi, Y. (2026). The Counts of Critical Points Move a Great Deal, and Their Alternating Sum Never Moves at All [Paper 203]. Zenodo. 10.5281/zenodo.22109443
- Yamagishi, Y. (2026). The Power in the Eigenvalue Count Is Half the Dimension [Paper 204]. Zenodo. 10.5281/zenodo.22110032
Reference
Yamagishi, Y. (2026). The Shortest Enclosure Is Not Always a Disc [Paper 205]. YOI: https://www.yamagishiyuuki.com/yoi/205
DOI: 10.5281/zenodo.22110143
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 205. This paper is a preprint and has not been peer reviewed.