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Anisotropic isoperimetric double tilings of the plane

T0 review · 0 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read The cheapest way to tile the plane with two repeating cells under ℓ1 length is the Pythagorean arrangement of two axis-aligned squares that share a vertex.

desk verdict Clean L1 double-tiling classification: Pythagorean squares give the global profile, uniqueness holds for unequal areas, and the rectangular case is fully explicit. read the letter →

arxiv 2607.10636 v1 pith:GLOZBX4D submitted 2026-07-12 math.AP math.MG

Francesco Nobili , Matteo Novaga , Emanuele Paolini This is my paper · ORCID
classification math.APmath.MG MSC 49Q0552C2058E12
keywords tilingsanisotropicperimeterisoperimetricprofileℓ1-perimeterPythagoreantilingperiodicpartitionsWulffinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how to partition the plane into two families of identical cells so that the total length of interfaces, measured with the Manhattan (ℓ1) norm, is as small as possible for given cell areas. When the cells must repeat according to a fixed rectangular lattice, the optimal shapes switch from a square sitting inside a chipped rectangle (when one area is small) to a pair of adjacent rectangles (when the areas are comparable). Once every possible lattice is allowed, the global minimum is always achieved by two axis-aligned squares that share a single vertex—the classical Pythagorean tiling—and this arrangement is unique except when the two areas are equal. The same constructions remain locally optimal even after one cell area is sent to infinity, producing infinite-area “vortex” or “cross” partitions that cannot be improved inside any large box. The result supplies an explicit anisotropic counterpart to classical honeycomb and double-bubble theorems and shows that the ℓ1 geometry forces far more rigidity than the Euclidean perimeter.

What carries the argument

The (G,ℓ1)-isoperimetric profile IG,ℓ1(x) together with its global envelope Iℓ1(x); both are evaluated by combining the Wulff inequality for the ℓ1-perimeter with exhaustive case analysis on which generators are crossed by horizontal and vertical lines.

What would settle it

Exhibit any lattice of area 1 and any pair of generators of areas x and 1-x whose total ℓ1-perimeter is strictly smaller than 2√x + 2√(1-x), or show that a non-Pythagorean configuration saturates the bound for some x ≠ 1/2.

Watch

Extended reading notes

Core claim

For every area fraction x the global ℓ1-isoperimetric profile equals 2√x + 2√(1-x) and is realized uniquely (except at x = 1/2) by the Pythagorean double tiling of two axis-aligned squares sharing a vertex; for a fixed rectangular lattice the profile is piecewise explicit and the minimizers are completely classified as either a square-plus-chipped-rectangle or a pair of adjacent rectangles.

Load-bearing premise

The classification rests on the claim that every minimizer admits open, essentially bounded representatives whose topological boundaries coincide with their reduced boundaries, and that connectedness of one cell forces connectedness of the other.

Editorial extensions

If this is right

  • Any periodic double tiling that is not Pythagorean (or a rectangular pair when the lattice is fixed) can be improved by a pure lattice change or a shape change.
  • The infinite-area “vortex” and “cross” partitions obtained by sending one volume to infinity are locally ℓ1-minimizing, so they cannot be bettered inside any finite window.
  • When the two areas are equal the minimizers form infinite families of striped square packings, all sharing the same perimeter cost.
  • The squared excess (IG,ℓ1(x) - IG,ℓ1(0))² is concave, giving a quantitative concavity statement for rectangular lattices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Wulff-plus-line-crossing method should classify triple or higher anisotropic tilings once the appropriate combinatorial cases are enumerated.
  • Because ℓ1 forces axis-alignment, the uniqueness statements are stronger than their Euclidean counterparts and may extend to other crystalline norms whose Wulff shapes are polygons.
  • Local minimality of the infinite partitions suggests they could serve as building blocks for free-boundary anisotropic cluster problems with mixed finite and infinite chambers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies periodic double tilings of the plane that minimize the anisotropic ℓ₁-perimeter. For a fixed rectangular lattice G of area 1, Theorem 1.1 gives an explicit formula for the (G,ℓ₁)-isoperimetric profile IG,ℓ₁(x) and classifies all minimizers: a square plus chipped rectangle when one cell is small, and two adjacent rectangles otherwise. Minimizing further over all planar lattices, Theorem 1.2 shows that Iℓ₁(x)=2√x+2√(1-x), attained uniquely (for x eq1/2) by the Pythagorean double tiling of two axis-aligned squares sharing a vertex; when x=1/2 the minimizers are alternating strips of equal squares. Theorem 1.4 then proves that certain limiting non-periodic partitions obtained by sending one volume to infinity are locally ℓ₁-isoperimetric. The arguments combine the classical Wulff inequality for the ℓ₁-norm with elementary projection estimates and a finite case analysis on how horizontal and vertical lines meet the generators.

Significance. The work supplies a complete, explicit solution of the anisotropic double-tiling problem for the Manhattan perimeter, both for rectangular lattices and after free minimization over all lattices. The profile formula of Theorem 1.2 is parameter-free and is realized by a classical geometric construction (the Pythagorean tiling), while uniqueness follows from the rigidity of the Wulff shape. The classification for rectangular lattices and the local-minimality statements for infinite-volume partitions extend the authors’ earlier isotropic results and fit naturally into the ongoing program on anisotropic clusters and periodic partitions. The proofs are elementary once the Wulff inequality and the projection lemmas are in place, and the geometric models are fully explicit.

minor comments (4)
  1. In the proof of Theorem 3.5 the density constants are obtained by covering a fundamental domain with finitely many balls and taking the worst-case constants from Theorem 3.4; a one-sentence remark that the resulting c,ρ₀ depend only on G and φ (and not on the particular tiling) would make the uniformity claim fully transparent.
  2. Lemma 3.7 asserts connectedness of E₂ whenever R^{2}\(Ē₁+G) is connected. The argument is short and correct, but a brief parenthetical note that the same conclusion holds after replacing E₁ by any finite union of its G-translates would clarify why the construction (3.3) preserves the property.
  3. Figure 4 and the accompanying text in the introduction describe a continuum of energetically equivalent placements of the small square; it would help the reader if the precise range of admissible (h,l) were stated once in the body of Theorem 1.1 rather than only in the caption.
  4. A few typographical inconsistencies appear (e.g., “prescrived” on p. 15, “horizonal” on p. 15, and the occasional missing space before parentheses). A light copy-edit pass would remove them.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: profile lower bound is classical Wulff, upper bound explicit competitor; self-citations only for existence/context.

full rationale

The central claim of Theorem 1.2 follows by the classical Wulff inequality (2.2) giving I_ℓ1(x) ≥ 2√x + 2√(1-x), matched by the explicit Pythagorean competitor of two axis-aligned squares; uniqueness is a short geometric rigidity argument forcing lattice vectors of the form (a,h),(w,a) with w=h=√x (or strips when x=1/2). Theorem 1.1 is obtained by case analysis on line-touching properties plus elementary projection estimates (Lemmas 4.1–4.5) and the same Wulff bound, with equality cases classified by connectedness and rectangle characterization (Lemma 4.2). Existence (Theorem 3.1) and density/regularity (Theorems 3.4–3.5, Corollary 3.6) cite the authors’ prior isotropic work [NN24] and standard GMT references, but these supply only the existence of minimizers and open representatives; the numerical value of the profile and the geometric models are derived independently and do not reduce to those citations. No fitted parameters, self-definitional identities, or load-bearing uniqueness theorems imported from the authors appear. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper works entirely inside classical geometric measure theory. The only external inputs are the Wulff inequality for the L1 norm (standard) and the existence/regularity theory for anisotropic isoperimetric tilings developed in the authors’ earlier papers; no free parameters or new physical entities are introduced.

assumptions (3)
  • standard math Wulff inequality for the L1 norm: Per_ℓ₁(E)≥4√|E| with equality precisely for axis-aligned squares (Theorem 2.1 / (2.2)).
    Classical result used as the lower bound for every generator; equality case supplies uniqueness of squares.
  • domain assumption Existence of a φ-isoperimetric N-tiling with essentially bounded generators when d=2 (Theorem 3.1, citing NN24).
    Guarantees that minimizers exist and can be taken open and bounded so that the subsequent topological arguments apply.
  • domain assumption Density estimates for volume-constrained anisotropic partitions (Theorem 3.4).
    Used to obtain open representatives with H^{1}( abla E\nabla*E)=0; proof is omitted as standard.

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Pith. "Pith review of Anisotropic isoperimetric double tilings of the plane." pith.science (2026). https://pith.science/paper/GLOZBX4D

@misc{pith2026260710636,
  author       = {Pith},
  title        = {Pith review of: Anisotropic isoperimetric double tilings of the plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLOZBX4D}},
  note         = {Machine review of arXiv:2607.10636}
}
abstract

We study periodic partitions of the plane into two distinct cells minimizing the anisotropic $\ell_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,\ell_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the $\ell_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a vertex. This configuration is unique unless the two cells are assigned the same area. Finally, we prove that the limiting (non periodic) partitions obtained by sending one volume to infinity are locally $\ell_1$-isoperimetric.

Figures

Figures reproduced from arXiv: 2607.10636 by the authors.

Figure 1
Figure 1. Isoperimetric profile IG,ℓ1 (x) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The ℓ1-isoperimetric double tiling generated by a square and a chipped rectangle [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The ℓ1-isoperimetric double tiling generated by two adjacent rectangles. Instead, when x ∈ (0, 1), the minimizers are also completely classified, and the minimizing models are depicted in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Given x ≤ b 2 4 , any choice of E1 (red) and E2 (white) with h, l ∈ [0, √ x] generates a double tiling with perimeter equal to IG,ℓ1 (x) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Example of generators with same total ℓ1-perimeter. profile function defined by [0, 1] ∋ x 7→ Iℓ1 (x) := inf{IG,ℓ1 (x): G lattice with Area(G) = 1}. For the standard perimeter (i.e. for the choice of anisotropic norm given by φ(x) = ∥x∥ℓ2 ), the above was considered by…
Figure 6
Figure 6. Figure 6: The Pythagorean double tiling [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Possible partitions obtained by sending |E2| ↑ ∞ while keeping |E1| = 1 on the isoperimetric double tilings: the first three partitions are obtained from Case i) in Theorem 1.1, the last one (the vortex ) is obtained from Theorem 1.2. induced by any of the ℓ1-isoperime…
Figure 8
Figure 8. Figure 8: A possible ℓ1-isoperimetric double tiling composed of two cells of equal area and with the same perimeter of the Pythagorean double tiling. 2. Preliminaries In this section, we recall some useful facts around the classical and anisotropic perimeters. 2.1. Sets of finit…

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Reference graph

Works this paper leans on

63 extracted references · 1 canonical work pages

  1. [1]

    The standard Lens cluster in \( R ^2\) uniquely minimizes relative perimeter

    Stan Alama, Lia Bronsard, and Silas Vriend. The standard Lens cluster in \( R ^2\) uniquely minimizes relative perimeter. Trans. Am. Math. Soc., Ser. B , 12:516--535, 2025

    2025

  2. [2]

    Connected components of sets of finite perimeter and applications to image processing

    Luigi Ambrosio, Vicent Caselles, Simon Masnou, and Jean-Michel Morel. Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. (JEMS) , 3(1):39--92, 2001

    2001

  3. [3]

    Functions of bounded variation and free discontinuity problems

    Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems . Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

    2000

  4. [4]

    F. J. Almgren, Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc. , 4(165):viii+199, 1976

    1976

  5. [5]

    A stability inequality for planar lens partition

    Marco Bonacini, Riccardo Cristoferi, and Ihsan Topaloglu. A stability inequality for planar lens partition. Proceedings of the Royal Society of Edinburgh: Section A Mathematics , page 1–34, 2025

    2025

  6. [6]

    The lens cluster and triod cluster uniquely minimize the anisotropic perimeter in R ^2

    Paula Benitez. The lens cluster and triod cluster uniquely minimize the anisotropic perimeter in R ^2 . arXiv:2604.03510, 2026

    arXiv 2026

  7. [7]

    Brothers and Frank Morgan

    John E. Brothers and Frank Morgan. The isoperimetric theorem for general integrands. Mich. Math. J. , 41(3):419--431, 1994

    1994

  8. [8]

    On the non-uniqueness of locally minimizing clusters via singular cones

    Lia Bronsard, Robin Neumayer, Michael Novack, and Anna Skorobogatova. On the non-uniqueness of locally minimizing clusters via singular cones. arXiv:2507.13995, 2025

    arXiv 2025

Show all 63 references
  1. [9]

    Lattice tilings minimizing nonlocal perimeters

    Annalisa Cesaroni, Ilaria Fragalà, and Matteo Novaga. Lattice tilings minimizing nonlocal perimeters . arXiv:2310.01054, to appear in Comm. Contemp. Math , https://doi.org/10.1142/S0219199724500433, 2023

    2023 doi arXiv

  2. [10]

    On the existence and regularity of fundamental domains with least boundary area

    Jaigyoung Choe. On the existence and regularity of fundamental domains with least boundary area. J. Differential Geom. , 29(3):623--663, 1989

    1989

  3. [11]

    Sharp stability inequalities for planar double bubbles

    Marco Cicalese, Gian Paolo Leonardi, and Francesco Maggi. Sharp stability inequalities for planar double bubbles. Interfaces Free Bound. , 19(3):305--350, 2017

    2017

  4. [12]

    Minimal periodic foams with equal cells

    Annalisa Cesaroni and Matteo Novaga. Minimal periodic foams with equal cells. In Anisotropic Isoperimetric Problems and Related Topics , pages 15--24, Singapore, 2024. Springer Nature Singapore

    2024

  5. [13]

    Periodic partitions with minimal perimeter

    Annalisa Cesaroni and Matteo Novaga. Periodic partitions with minimal perimeter. Nonlinear Anal. , 243, 2024. Paper No. 113522

    2024

  6. [14]

    Proof of the planar double bubble conjecture using metacalibration methods

    Rebecca Dorff, Gary Lawlor, Donald Sampson, and Brandon Wilson. Proof of the planar double bubble conjecture using metacalibration methods. Involve , 2(5):611--628, 2009

    2009

  7. [15]

    Connected perimeter of planar sets

    Fran c ois Dayrens, Simon Masnou, Matteo Novaga, and Marco Pozzetta. Connected perimeter of planar sets. Adv. Calc. Var. , 15(2):213--234, 2022

    2022

  8. [16]

    The standard double soap bubble in R ^2 uniquely minimizes perimeter

    Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba. The standard double soap bubble in R ^2 uniquely minimizes perimeter. Pacific J. Math. , 159(1):47--59, 1993

    1993

  9. [17]

    A quantitative version of the isoperimetric inequality: the anisotropic case

    Nicola Fusco, Luca Esposito, and Cristina Trombetti. A quantitative version of the isoperimetric inequality: the anisotropic case. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) , 4(4):619--651, 2005

    2005

  10. [18]

    The double-bubble problem on the square lattice

    Manuel Friedrich, Wojciech G \'o rny, and Ulisse Stefanelli. The double-bubble problem on the square lattice. Interfaces Free Bound. , 26(1):79--134, 2024

    2024

  11. [19]

    A characterization of \( _1\) double bubbles with general interface interaction

    Manuel Friedrich, Wojciech G \'o rny, and Ulisse Stefanelli. A characterization of \( _1\) double bubbles with general interface interaction. Adv. Calc. Var. , 18(3):609--637, 2025

    2025

  12. [20]

    The \( _1\) double-bubble problem in three dimensions

    Manuel Friedrich, Wojciech G \'o rny, and Ulisse Stefanelli. The \( _1\) double-bubble problem in three dimensions. J. Geom. Anal. , 35(10):27, 2025. Id/No 323

    2025

  13. [21]

    Fortes, P

    M. Fortes, P. Graner, and P. Teixeira. Mixing and sorting of bidisperse two-dimensional bubbles . Eur. Phys. J. E , 9, 161--169 (2002)

    2002

  14. [22]

    Figalli, F

    A. Figalli, F. Maggi, and A. Pratelli. A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. , 182(1):167--211, 2010

    2010

  15. [23]

    On the Steiner property for planar minimizing clusters

    Valentina Franceschi, Aldo Pratelli, and Giorgio Stefani. On the Steiner property for planar minimizing clusters. The isotropic case. Commun. Contemp. Math. , 25(5):29, 2023. Id/No 2250040

    2023

  16. [24]

    Symmetric double bubbles in the Grushin plane

    Valentina Franceschi and Giorgio Stefani. Symmetric double bubbles in the Grushin plane. ESAIM, Control Optim. Calc. Var. , 25:37, 2019. Id/No 77

    2019

  17. [25]

    Fortes and P

    M. Fortes and P. Teixeira. Minimum perimeter partitions of the plane into equal numbers of regions of two different areas . Eur. Phys. J. E , 6, 133--137, (2001)

    2001

  18. [26]

    u cksichtigung der Anwendungsgebiete . Springer-Verlag, Berlin-G\

    L. Fejes T\' o th. Lagerungen in der E bene, auf der K ugel und im R aum , volume Band LXV of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber\" u cksichtigung der Anwendungsgebiete . Springer-Verlag, Berlin-G\" o ttingen-Heidelberg, 1953

    1953

  19. [27]

    Fejes T\' o th

    L. Fejes T\' o th. Isoperimetric problems concerning tessellations. Acta Math. Acad. Sci. Hungar. , 14:343--351, 1963

    1963

  20. [28]

    Fejes T\' o th

    L. Fejes T\' o th. Regular figures . The Macmillan Company, New York, 1964

    1964

  21. [29]

    Fejes T\' o th

    L. Fejes T\' o th. What the bees know and what they do not know. Bull. Amer. Math. Soc. , 70:468--481, 1964

    1964

  22. [30]

    Branko Gr \"u nbaum and G. C. Shephard. Tilings and Patterns . W. H. Freeman and Co. Ltd., Oxford, 1987. IX + 700 p., 1395 figs

    1987

  23. [31]

    T. C. Hales. The honeycomb conjecture. Discrete Comput. Geom. , 25(1):1--22, 2001

    2001

  24. [32]

    Some theorems on the free energies of crystal surfaces

    Conyers Herring. Some theorems on the free energies of crystal surfaces. Phys. Rev., II. Ser. , 82:87--93, 1951

    1951

  25. [33]

    The double bubble conjecture

    Joel Hass, Michael Hutchings, and Roger Schlafly. The double bubble conjecture. Electron. Res. Announc. Amer. Math. Soc. , 1(3):98--102, 1995

    1995

  26. [34]

    Proof of the double bubble conjecture

    Michael Hutchings, Frank Morgan, Manuel Ritor\'e, and Antonio Ros. Proof of the double bubble conjecture. Ann. of Math. (2) , 155(2):459--489, 2002

    2002

  27. [35]

    Double bubbles minimize

    Joel Hass and Roger Schlafly. Double bubbles minimize. Ann. of Math. (2) , 151(2):459--515, 2000

    2000

  28. [36]

    Sets of finite perimeter and geometric variational problems , volume 135 of Cambridge Studies in Advanced Mathematics

    Francesco Maggi. Sets of finite perimeter and geometric variational problems , volume 135 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory

    2012

  29. [37]

    Wulff clusters in R ^2

    Frank Morgan, Christopher French, and Scott Greenleaf. Wulff clusters in R ^2 . J. Geom. Anal. , 8(1):97--115, 1998

    1998

  30. [38]

    Multi-bubble isoperimetric problems

    Emanuel Milman. Multi-bubble isoperimetric problems. arxiv:2510.07078, 2025

    2025

  31. [39]

    Unilateral tilings of the plane with squares of three sizes

    Horst Martini, Endre Makai, and Valeriu Soltan. Unilateral tilings of the plane with squares of three sizes. Beitr\"age Algebra Geom. , 39(2):481--495, 1998

    1998

  32. [40]

    The G aussian double-bubble and multi-bubble conjectures

    Emanuel Milman and Joe Neeman. The G aussian double-bubble and multi-bubble conjectures. Ann. of Math. (2) , 195(1):89--206, 2022

    2022

  33. [41]

    The Structure of Isoperimetric Bubbles on R ^n and S ^n , 2024

    Emanuel Milman and Joe Neeman. The Structure of Isoperimetric Bubbles on R ^n and S ^n , 2024. arXiv:2205.09102, Acta Math. to appear

    2024 arXiv

  34. [42]

    Spines of minimal length

    Bruno Martelli, Matteo Novaga, Alessandra Pluda, and Stefano Riolo. Spines of minimal length. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 17(3):1067--1090, 2017

    2017

  35. [43]

    The hexagonal honeycomb conjecture

    Frank Morgan. The hexagonal honeycomb conjecture. Trans. Amer. Math. Soc. , 351(5):1753--1763, 1999

    1999

  36. [44]

    Geometric measure theory

    Frank Morgan. Geometric measure theory. A beginner's guide . Elsevier/Academic Press, Amsterdam, fourth edition, 2009

    2009

  37. [45]

    The standard double bubble is the unique stable double bubble in R^2

    Frank Morgan and Wacharin Wichiramala. The standard double bubble is the unique stable double bubble in R^2 . Proc. Amer. Math. Soc. , 130(9):2745--2751, 2002

    2002

  38. [46]

    A strong form of the quantitative Wulff inequality

    Robin Neumayer. A strong form of the quantitative Wulff inequality. SIAM J. Math. Anal. , 48(3):1727--1772, 2016

    2016

  39. [47]

    Lattice tilings with minimal perimeter and unequal volumes

    Francesco Nobili and Matteo Novaga. Lattice tilings with minimal perimeter and unequal volumes. Calc. Var. Partial Differential Equations , 63(9):Paper No. 246, 26, 2024

    2024

  40. [48]

    Periodic double tilings of the plane

    Francesco Nobili, Matteo Novaga, and Emanuele Paolini. Periodic double tilings of the plane. Trans. Am. Math. Soc. , 379(6):4323--4348, 2026

    2026

  41. [49]

    Minimizing clusters with prescribed asymptotic geometry

    Robin Neumayer, Michael Novack, and Anna Skorobogatova. Minimizing clusters with prescribed asymptotic geometry. arXiv:2606.07468, 2026

    2026 arXiv

  42. [50]

    Isoperimetric clusters in homogeneous spaces via concentration compactness

    Matteo Novaga, Emanuele Paolini, Eugene Stepanov, and Vincenzo Maria Tortorelli. Isoperimetric clusters in homogeneous spaces via concentration compactness. J. Geom. Anal. , 32(11):Paper No. 263, 23, 2022

    2022

  43. [51]

    Isoperimetric planar clusters with infinitely many regions

    Matteo Novaga, Emanuele Paolini, Eugene Stepanov, and Vincenzo Maria Tortorelli. Isoperimetric planar clusters with infinitely many regions. Netw. Heterog. Media , 18(3):1226--1235, 2023

    2023

  44. [52]

    Novaga, E

    M. Novaga, E. Paolini, and V. M. Tortorelli. Locally isoperimetric partitions. Trans. Am. Math. Soc. , 378(4):2517--2548, 2025

    2025

  45. [53]

    Existence of a non-standard isoperimetric triple partition

    Matteo Novaga, Emanuele Paolini, and Vincenzo Maria Tortorelli. Existence of a non-standard isoperimetric triple partition. arXiv:2507.14112, 2025

    2025 arXiv

  46. [54]

    Paolini and A

    E. Paolini and A. Tamagnini. Minimal cluster computation for four planar regions with the same area. Geom. Flows , 3(1):90--96, 2018

    2018

  47. [55]

    Minimal clusters of four planar regions with the same area

    Emanuele Paolini and Andrea Tamagnini. Minimal clusters of four planar regions with the same area. ESAIM Control Optim. Calc. Var. , 24(3):1303--1331, 2018

    2018

  48. [56]

    Paolini and V

    E. Paolini and V. M. Tortorelli. The quadruple planar bubble enclosing equal areas is symmetric. Calc. Var. Partial Differential Equations , 59(1):Paper No. 20, 9, 2020

    2020

  49. [57]

    Reichardt

    Ben W. Reichardt. Proof of the double bubble conjecture in R^n . J. Geom. Anal. , 18(1):172--191, 2008

    2008

  50. [58]

    Sullivan and Frank Morgan

    John M. Sullivan and Frank Morgan. Open problems in soap bubble geometry. Int. J. Math. , 07(06):833--842, 1996

    1996

  51. [59]

    Jean E. Taylor. Crystalline variational problems. Bull. Am. Math. Soc. , 84:568--588, 1978

    1978

  52. [60]

    On the division of space with minimum partitional area

    William Thomson. On the division of space with minimum partitional area. Acta Math. , 11(1-4):121--134, 1887

  53. [61]

    Proof of the planar triple bubble conjecture

    Wacharin Wichiramala. Proof of the planar triple bubble conjecture. J. Reine Angew. Math. , 567:1--49, 2004

    2004

  54. [62]

    Weaire and R

    D. Weaire and R. Phelan. A counter-example to K elvin's conjecture on minimal surfaces. Forma , 11(3):209--213, 1996. Reprint of Philos. Mag. Lett. 69 (1994), no. 2, 107--110

    1996

  55. [63]

    o sung der krystallfl \

    Georg Wulff. Zur frage der geschwindigkeit des wachsthums und der aufl \"o sung der krystallfl \"a chen. Zeitschrift f \"u r Kristallographie - Crystalline Materials , 34:449 -- 530, 1901

    1901

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