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Obstructions to Reality: Torsors & Visual Paradox

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A visual paradox is exactly a non-trivial network torsor, with the impossibility encoded as the non-triviality of a first cohomology class.

desk verdict A genuinely useful formalization of Penrose's cohomological view of impossible figures, with novel examples and real errors; the picture-to-cocycle bridge is not canonical, so the central 'precisely' overclaims. read the letter →

arxiv 2507.01226 v1 pith:J7C74JDS submitted 2025-07-01 math.AT

Robert Ghrist , Zoe Cooperband This is my paper · ORCID
classification math.AT MSC 55N3005C1092J30
keywords cohomologysheaftorsornonabelianholonomyvisualparadoximpossiblefiguresnetwork
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

This paper claims that a visual paradox—a figure whose parts are locally coherent but that cannot be realized globally—is precisely a non-trivial network torsor: a graph whose vertices and edges carry relative measurements but no absolute assignment of those measurements. Its central result, Proposition 4.5, says a torsor admits a consistent global assignment if and only if its class in the first cohomology set $H^1(X; G)$ is the identity, so the impossibility of a figure is exactly the non-triviality of that cohomology class. The authors demonstrate the claim on classic impossible staircases, on torus and Klein bottle variants (including what they call the first nonabelian visual paradox), and on paradoxes that emerge from boundary conditions rather than loops. If the claim is right, the study of impossibility becomes an algebraic classification: a figure is impossible when its locally consistent relations fail to globalize, and the failure has a numerical or group-theoretic shape.

What carries the argument

The central object is the network $G$-torsor, a discrete analogue of a principal homogeneous space: it assigns to each vertex and edge of a graph a non-empty set with a free and transitive right action of a group $G$, with equivariant restriction maps. The classification theorem identifies isomorphism classes of such torsors with elements of the first cohomology set $H^1(X; G)$; Proposition 4.5 then does the work, tying the absence of a global section—the paradox—to a non-trivial cohomology class. For boundary-induced paradoxes, the structure group is allowed to vary as a sheaf, so the same obstruction is measured by relative cohomology $H^1(X, A; G)$.

What would settle it

Find a figure that observers unanimously judge impossible but whose height-change or orientation cocycle is a coboundary (trivial in $H^1$), or a figure whose local cues do not fix the group element on some edge; either case would show the non-trivial torsor is not what makes a figure impossible.

Watch

Extended reading notes

Core claim

The central claim is the equivalence, used throughout, between a visual paradox and a non-trivial network torsor. The supporting theorem states that a network $G$-torsor over a graph admits a global section if and only if it is isomorphic to the trivial torsor; consequently, a visual paradox exists precisely when the torsor built from locally perceived relative changes carries a non-trivial element of $H^1(X; G)$. On a closed staircase the height changes around the circuit form a cocycle whose total is non-zero, and that non-zero total is the obstruction to a consistent global height assignment. The same construction classifies orientation-flip staircases on nonorientable surfaces, commuting-loop staircases on a torus, a nonabelian infinite-dihedral torsor on a Klein bottle, and boundary-pinned bistable figures treated with a nonconstant structure sheaf.

Load-bearing premise

The whole classification rests on the assumption that a figure's local geometric cues determine, without ambiguity, a group-valued cocycle on a graph, and that two figures whose cocycles are cohomologous share the same paradoxical essence.

Editorial extensions

If this is right

  • The classic impossible staircase and its variants are classified by an integer $k \in \mathbb{Z} \cong H^1(S^1; \mathbb{Z})$, the net height change around the loop; $k \neq 0$ means the figure cannot be realized.
  • For cubic staircases in three-dimensional translation space, isomorphism classes are distinguished by the greatest common divisor of the holonomy vector's components, putting the eight variants of Figure 4 into three classes.
  • Klein-bottle staircases are the first nonabelian visual paradoxes: their structure group is the infinite dihedral group, and traversing loop $ab$ yields a different result from $ba$.
  • Boundary-driven paradoxes, such as a row of alternating cubes with forced contradictory endpoints, are classified by relative cohomology $H^1(X, A; G)$, so a loop in the base space is not required for impossibility.

Reading between the lines

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

  • The same formalism suggests a definition of 'degree of impossibility' for any system with local relative measurements—robotic mapping, sensor networks, or multi-agent consensus—where a non-trivial $H^1$ class would predict a global inconsistency that no local correction can remove.
  • If the equivalence is taken literally, the framework predicts that two visually different figures with the same base space, structure group, and cohomology class are perceptually the same paradox; this is a concrete, untested claim about human perception.
  • The boundary-pinned construction suggests a mechanism for generating new illusions: render any local ambiguity with forced contradictory boundary conditions, and the relative cohomology class should tell you whether the figure will be impossible before anyone looks at it.
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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

4 major / 4 minor

Summary. The paper proposes a mathematical framework in which visual paradoxes such as the Penrose staircase are modeled by network torsors over graphs, with the existence of a global section of the torsor equivalent to global realizability. The central claim is that a visual paradox exists precisely when the associated torsor is non-trivial, i.e., when its class in H^1(X;G) is not the identity. The authors work through examples on the circle, cylinder, Moebius strip, projective plane, torus, and Klein bottle, including a claimed first nonabelian paradox built from an infinite dihedral structure group, and they treat boundary-condition paradoxes via non-constant structure sheaves and relative cohomology. They also introduce a category Par of network paradoxes to compare different paradoxes by morphisms that preserve the cohomological obstruction.

Significance. If the framework can be made precise, it would give a unified cohomological account of a broad class of impossible figures, extending Penrose's original observation with modern torsor language. The paper's systematic treatment of new examples on non-orientable surfaces, the attempt to formalize boundary-driven paradoxes, and the categorical comparison of paradoxes are genuinely useful ideas for the applied topology and perception communities. The paper builds on standard classification theorems for torsors and sheaf cohomology rather than deriving them, which is appropriate for its purpose. However, the central claim is currently undercut by a non-canonical figure-to-cocycle encoding and by several concrete mathematical errors in fundamental group computations and in the boundary-torsor construction; these issues must be resolved before the classification can be regarded as well-defined.

major comments (4)
  1. [Sec. 5.1 and 5.2] The central claim 'a visual paradox exists precisely when the resulting torsor is non-trivial' is not well-defined because the figure-to-cocycle bridge is not canonical. In Sec. 5.1 the structure group is called a modeling choice, and Sec. 5.2 explicitly permits projecting Z^3-torsors to Z-torsors. For the cubic holonomy h=(2,2,2), the unimodular matrix A=[[1,-1,0],[0,1,0],[0,0,1]] is an automorphism of Z^3 and sends h to (0,2,2); the Z^3-torsor class is unchanged, but the two coordinate projections send the new holonomy to 0 and 2, respectively. Thus the same figure is trivial under one permitted projection and non-trivial under another. Without a rule selecting a canonical encoding, the equivalence depends on the model, not the figure.
  2. [Sec. 6.3 and 6.4] The proposed G-torsor for boundary constraints does not encode the contradictory boundary values. Since the structure sheaf G has stalk {1} at boundary vertices, any G-torsor has singleton stalks there, and all singleton sets are isomorphic. Labeling P(v0)={+1} and P(terminal)={-1} is not part of the torsor data. The relative cohomology class may capture the obstruction, but the torsor reformulation is incomplete. The authors should either make boundary values part of the torsor data (for example, via a distinguished trivialization on the boundary) or revise the claim that boundary-driven paradoxes are non-trivial torsors in the sense of Definition 6.1.
  3. [Sec. 5.4, 5.7, 7.3] There are incorrect fundamental group computations. In Sec. 5.4, X=S1∨S1 has fundamental group F2, not Z×Z; the cohomology H^1(X;Z) still gives Z×Z, so the conclusion is not damaged. In Sec. 5.7, with the displayed relation ab=b^{-1}a, the Klein bottle group is a torsion-free semidirect product Z semidirect Z, not the infinite dihedral group, which has an involution. The homomorphism check remains valid for the relation, but the group identification should be corrected. In Sec. 7.3, Example 7.11 computes the abelianization of the group presentation as Z2×Z2; the correct abelianization is Z×Z2, so the proof that no morphism exists needs revision. The conclusion may still be true but requires a correct argument.
  4. [Sec. 5.6 and 8] For nonabelian structure groups, H^1(S1;G) is the set of conjugacy classes of elements of G, not the group G itself. The claim in Sec. 5.6 that two examples are classified by the same non-trivial element in H^1(S1; Z semidirect Z2) and the claim in Sec. 8 that H^1(S1;G) is isomorphic to G for any group G are both false for nonabelian G. Theorem 3.2 already gives the correct statement via Hom(pi1,G)/G. The nonabelian examples should cite the conjugacy class, not a specific element.
minor comments (4)
  1. [Sec. 3.3] The sentence that every 1-cochain is automatically a 1-cocycle should say 'for a graph with no 2-cells'; the cocycle condition is not vacuous once the graph is the 1-skeleton of a complex with 2-cells, as used in Sec. 5.7.
  2. [Sec. 6] The symbol G is used both for the structure sheaf with trivial stalks at the boundary and for the constant sheaf whose subsheaf GA vanishes there; please use distinct notation.
  3. [Sec. 7.2] In the reverse direction of Proposition 7.4, the definition of the inverse sheaf morphism requires a natural isomorphism relating the pullback of the pullback sheaf to the original sheaf; the proof should specify this naturality.
  4. [Throughout] There are many typos, including 'Shea ves' in the running title, missing spaces around H^1, and inconsistent blackboard bold; a careful copyedit is needed.

Circularity Check

3 steps flagged · score 6.0 of 10

The central equivalence 'visual paradox iff non-trivial torsor' is definitional: the paradox is encoded as the cocycle whose non-triviality is then read off, with the allowed modeling choices making the result encoding-dependent.

  1. self definitional [Section 1.1 and Section 5 (four-step approach, after Construction 4.7/Proposition 4.5)]
    "Relative attributes are described mathematically by local data-transition functions between adjacent parts of the figure formalized as cocycles. A visual paradox emerges when these cocycles cannot be integrated into a globally consistent assignment for the attribute in question ... By Proposition 4.5 a visual paradox exists precisely when the resulting torsor is non-trivial, indicating an obstruction to assigning globally consistent values to the locally defined relative attributes."

    The paper builds the torsor P_eta from the same 1-cochain eta that encodes the perceived local changes, and then defines the paradox as the failure of this cocycle to have a global section. Proposition 4.5 is the torsor-theoretic tautology that a torsor has a global section iff its H^1 class is trivial. Therefore the statement that 'a visual paradox exists precisely when the resulting torsor is non-trivial' is an immediate restatement of the framework's own definition of paradox; no independent visual property is derived or predicted. The classical examples illustrate the encoding but do not test it against an external criterion.

  2. self definitional [Section 5.2 'Three-Dimensional Cubic Staircases' and footnote 4]
    "The numerical values are subject to a choice of basis. ... We note that one could project these Z3-torsors onto a single dimension (such as height) to obtain simpler Z-torsors. This dimensional reduction loses information about the spatial translation."

    Because the structure group, basis, and projection are explicitly left as modeling choices, the 'resulting torsor' is not determined by the figure. For h=(2,2,2) in Z^3, a unimodular change of basis sends h to (0,2,2); projecting onto the first coordinate gives the trivial Z-torsor (no paradox), while projecting onto the second gives a non-trivial one. The permitted encodings therefore classify the same visual object both ways. The claimed equivalence is not an invariant statement about the figure; it is a property of the freely chosen cocycle encoding, confirming that the paradox/non-triviality link is built into the modeling step rather than derived from it.

1 more flagged steps
  1. self definitional [Section 6.4 'Cohomological Analysis for General Structure Groups']
    "Thus, for any boundary data β = (g0, g1) with g0 ≠ g1, the corresponding obstruction class δ∗(β) ∈ H^1(X; GA) is non-trivial. This obstruction class precisely captures the paradoxical nature of the configuration."

    In the boundary case the paper constructs a structure sheaf that pins trivial (singleton) values at the boundary and then encodes contradictory endpoint states as a torsor; the 'paradoxical nature' is asserted to be exactly the non-trivial relative cohomology obstruction. Since the boundary data were chosen to be the visual cues whose inconsistency defines the paradox, the non-triviality of the obstruction is a restatement of that choice, not a separate consequence. The isomorphism H^1(X;GA) ≅ H^1(X,A;G) is imported machinery, but the identification of its non-trivial classes with 'genuine paradox' is definitional.

full rationale

The imported classification theorems (Theorem 3.2, Theorem 4.4, Theorem 6.2) are standard external results cited to Giraud, Mac Lane–Moerdijk, etc., and are not themselves circular; the self-citations to Cooperband's thesis and prior graphic-statics papers are not load-bearing for the central torsor classification. The circularity lies in the interpretive bridge: Section 1.1 defines a visual paradox as the failure to globalize a cocycle of relative attributes, Section 5 constructs the torsor from that same cocycle, and Proposition 4.5 states that non-triviality is exactly the absence of a global section. The headline characterization is therefore true by construction, not by prediction. The non-canonicality of the encoding (free choice of group, basis, projection; GL(3,Z) can trivialize a projected §5.2 holonomy) reinforces that the criterion is a property of the modeler's chosen cocycle rather than of the figure, so the claimed equivalence carries no independent empirical content. This is partial circularity: the algebraic framework and novel examples have independent mathematical content, but the central claim that visual paradoxes are precisely non-trivial network torsors reduces to the definition of the encoding. Score 6.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard torsor classification plus a modeling assumption that figures determine cocycles. No parameters are fitted to data; Δ is an input read from the figure. The least supported axiom is the nonabelian relative-cohomology isomorphism in the boundary section.

free parameters (1)
  • height change Δ (cocycle value) = nonzero integer, unspecified
    In §5.6 and §5.7, η_e2=(Δ,+1) and ρ(a)=(Δ,-1) take a nonzero integer Δ read off the staircase figure. Non-triviality holds for any Δ≠0, but isomorphism classes can depend on Δ mod 2; the paper never fixes or derives Δ from independent data.
assumptions (4)
  • standard math Classification of network torsors: H^1(X;G) bijects with isomorphism classes of network G-torsors (Theorem 4.4).
    Imported from Giraud's nonabelian cohomology and standard for principal bundles; cited [10].
  • standard math For a constant sheaf of groups G on a graph, H^1(X;G) is naturally isomorphic to Hom(π1(X),G)/conjugacy (Theorem 3.2).
    Standard classification of flat G-bundles on a 1-dimensional complex.
  • domain assumption Every visual paradox under study can be encoded as a graph X, a structure group G, and a cocycle η capturing local relative attributes.
    Section 5's four-step approach assumes the figure determines the cocycle; §2 footnote 2 narrows the scope to stair-like forced-perspective images.
  • ad hoc to paper For boundary-constrained paradoxes, H^1(X;G_A) is isomorphic to H^1(X,A;G) and the low-degree nonabelian cohomology sequence is exact.
    Stated without proof in §6.4-6.5; nonabelian relative cohomology is delicate, and no reference is given for this isomorphism.

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Cite this review

Pith. "Pith review of Obstructions to Reality: Torsors & Visual Paradox." pith.science (2026). https://pith.science/paper/J7C74JDS

@misc{pith2026250701226,
  author       = {Pith},
  title        = {Pith review of: Obstructions to Reality: Torsors & Visual Paradox},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7C74JDS}},
  note         = {Machine review of arXiv:2507.01226}
}
abstract

Visual paradoxes like the Penrose staircase present a fundamental tension: locally coherent geometric relationships that cannot be realized globally. Inspired by Penrose's observations connecting such paradoxes to cohomology, we develop a mathematical framework that precisely characterizes this phenomenon through network torsors and sheaf cohomology. Network torsors capture the essential nature of visual paradoxes by formalizing relative geometric attributes (height changes, orientation flips) without requiring absolute measures. We demonstrate that a significant class of visual paradoxes can be rigorously characterized as non-trivial network torsors, with their obstruction to global consistency quantified by elements of $H^1$. This framework enables analysis of classical paradoxes and construction of novel examples on various topological spaces. Key contributions include: (1) the first nonabelian visual paradox, classified by an infinite dihedral torsor on a Klein bottle; (2) paradoxes driven by boundary conditions rather than loops, analyzable via non-constant structure sheaves; and (3) a categorical framework for comparing paradoxes that reveals unexpected connections between visually distinct figures. Our approach unifies diverse visual paradoxes under a single mathematical principle: the obstruction to globalizing locally consistent geometric relationships.

Figures

Figures reproduced from arXiv: 2507.01226 by the authors.

Figure 1
Figure 1. [left] The impossible triangle of Reutersvärd (1934) is locally but not globally consistent; [right] The classic Penrose triangle (1954-56) similarly cannot be lifted to a 3-D structure, despite local coherence. 2020 Mathematics Subject Classification. 55N30, 05C10, 92J30. Key words and phrases. cohomology, sheaf, torsor, nonabelian, holonomy, paradox. 1 arXiv:2507.01226v1 [math.AT] 1 Jul 2025 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. The Penrose staircase (1954-56) has paradoxical height as one traverses the loop. Impossible figures represent a fascinating intersection of visual perception, geometry, and topology. Epitomized by the Penrose triangle or staircase and elaborated in the works of Reutersvärd, Escher, and others, these structures present locally coherent geometric infor￾mation – segments of stairs appear correctly joined, perspectives… view at source ↗
Figure 3
Figure 3. The Necker cube (1832) [left] and the Schröder staircase (1858) [right] each have a bistable gestalt switch that can serve as the basis for impossible figures, see, e.g., Figures 13 and 14. such figures. Our central thesis is that a significant class of geometric visual paradoxes can be precisely characterized as non-trivial network torsors, with sheaf cohomology providing a natural classification scheme that quanti… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Eight variants of the classic Penrose staircase that exhibit local relative coordinate translations but do not possess corresponding global coordinates. These appear similar, but the precise mismatches of coordinates vary [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: [left] A variant of the classic Penrose staircase that has local but not global height. [right] A cylindrical version of an impossible stair: identify left and right sides to obtain a staircase which has well-defined height changes locally but not globally. Magenta ste…
Figure 6
Figure 6. Figure 6: An impossible object on a cylinder with left and right sides identified. that of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: [left] A paradoxical staircase on a Möbius strip, with left and right identified with a vertical reflection. [right] This reflection reverses which “side” of the steps one walks on as one passes the left/right edge. Remark 2.1. The Möbius strip reveals a subtle geometr…
Figure 8
Figure 8. Figure 8: The double cover of the Möbius strip is an annulus, and the staircase of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Impossible staircase on a real projective plane RP 2 as a hemisphere with antipodal boundary points identified with orientation reversal [left], where the north and south poles are identified with a reflection. Note the resemblance to the classical Penrose triangle – t…
Figure 10
Figure 10. Figure 10: Zigzag depth paradox: A path with alternating right-angled turns [top] can be realized as a loop (identifying left and right sides) only when the number of corner points is even. When viewed from the side marking only straight-vs-corner points [bottom], such a loop is…
Figure 11
Figure 11. Figure 11: Impossible staircase on a torus T 2 represented as a square with opposite edges identified: [left] a simple example with a single height change; [right] a complex example from a different perspective, where the height change is indicated by a vertical “ladder” in mage…
Figure 12
Figure 12. Figure 12: [left] A paradoxical staircase on a Klein bottle, with top and bottom identified normally, left and right identified with a vertical reflection. This reflection reverses which “side” of the steps one walks on. The universal cover of the Klein bottle [right] is drawn p…
Figure 13
Figure 13. Figure 13: The gradient Necker paradox is a sequence of cubes with forced con￾tradictory interpretations at the endpoints, creating an ambiguous transition zone where no consistent interpretation is possible. across the interval that makes the illusion stick [PITH_FULL_IMAGE:fi…
Figure 14
Figure 14. Figure 14: Impossible Bars: Two long bars with forced contradictory perspectives indicated via right-angle cues at the ends. The interior of the bar suggests continuity, causing a conflict with the endpoints. Covering one end of the bar then the other triggers a gestalt switch. …
Figure 15
Figure 15. Figure 15: Classification of Penrose triangle interpretations by fiber equiva￾lence. Double arrows denote fiber equivalence, with the solid double arrow indi￾cating isomorphism. The paradoxes partition into two distinct, non-equivalent families. These geometric interpretations c…

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Forward citations

Cited by 1 Pith paper

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    math.AT 2026-01 unverdicted novelty 7.0 of 10

    Contradictory preference cycles in Condorcet's paradox are non-orientable surfaces: the Klein bottle (unrealised) or the real projective plane (realised), yielding an orientability reformulation of Arrow's theorem.

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages · cited by 1 Pith paper

  1. [1]

    The sheaf-theoretic structure of non-locality and contex- tuality.New Journal of Physics, 13:113036, 2011

    Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contex- tuality.New Journal of Physics, 13:113036, 2011

    work page 2011

  2. [2]

    Contex- tuality, cohomology and paradox

    Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield. Contex- tuality, cohomology and paradox. In24th EACSL Annual Conference on Computer Science Logic (CSL 2015), volume 41 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 211–228. Schloss Dagstuhl, 2015

    work page 2015

  3. [3]

    On the cohomology of contextuality

    Giovanni Carù. On the cohomology of contextuality. In Ross Duncan and Chris Heunen, editors,Pro- ceedings of the 13th International Conference on Quantum Physics and Logic (QPL 2016), volume 236 of Electronic Proceedings in Theoretical Computer Science, pages 25–41, 2017

    work page 2016

  4. [4]

    Cervantes and Ehtibar N

    Víctor H. Cervantes and Ehtibar N. Dzhafarov. Contextuality analysis of impossible figures.Entropy, 22(9):981, 2020

    work page 2020

  5. [5]

    Cellular Cosheaves, Graphic Statics, and Mechanics

    Zoe Cooperband. Cellular Cosheaves, Graphic Statics, and Mechanics. PhD thesis, University of Penn- sylvania, 2024

    work page 2024

  6. [6]

    Towards homological methods in graphic statics

    Zoe Cooperband and Robert Ghrist. Towards homological methods in graphic statics. InProceedings of the IASS Annual Symposium 2023, 2023

    work page 2023

  7. [7]

    A cosheaf theory of reciprocal figures: Planar and higher genus graphic statics.arXiv preprint arXiv:2311.12946, 2023

    Zoe Cooperband, Robert Ghrist, and Jakob Hansen. A cosheaf theory of reciprocal figures: Planar and higher genus graphic statics.arXiv preprint arXiv:2311.12946, 2023

    arXiv 2023

  8. [8]

    Thaddeus M. Cowan. The theory of braids and the analysis of impossible figures.Journal of Mathematical Psychology, 11(3):190–212, 1974

    work page 1974

Show all 19 references
  1. [9]

    Thaddeus M. Cowan. Supplementary report: Braids, side segments, and impossible figures.Journal of Mathematical Psychology, 16(3):254–260, 1977

    1977

  2. [10]

    Springer-Verlag, Berlin, 1971

    Jean Giraud.Cohomologie Non Abélienne, volume 179 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1971

    1971

  3. [11]

    Sheaf neural networks

    Jakob Hansen and Thomas Gebhart. Sheaf neural networks. InNeurIPS 2020 Workshop on Topological Data Analysis and Beyond, 2020

    2020

  4. [12]

    Opinion dynamics on discourse sheaves.SIAM Journal on Applied Mathematics, 81(5):2033–2060, 2021

    Jakob Hansen and Robert Ghrist. Opinion dynamics on discourse sheaves.SIAM Journal on Applied Mathematics, 81(5):2033–2060, 2021

    2021

  5. [13]

    W. F. Harris. Perceptual singularities in impossible pictures represent screw dislocations.South African Journal of Science, 69:10–13, 1973. TORSORS & VISUAL PARADOX 35

    1973

  6. [14]

    Springer-Verlag, Berlin, 1990

    Masaki Kashiwara and Pierre Schapira.Sheaves on Manifolds, volume 292 ofGrundlehren der mathema- tischen Wissenschaften. Springer-Verlag, Berlin, 1990

    1990

  7. [15]

    Are impossible figures possible?Signal Processing, 5:201–220, 1983

    Zenon Kulpa. Are impossible figures possible?Signal Processing, 5:201–220, 1983

    1983

  8. [16]

    Springer-Verlag, New York, 1992

    Saunders Mac Lane and Ieke Moerdijk.Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, New York, 1992

    1992

  9. [17]

    L. S. Penrose and R. Penrose. Impossible objects: A special type of illusion.British Journal of Psychology, 49:31, 1958

    1958

  10. [18]

    On the cohomology of impossible figures.Leonardo, 25(3-4):245–247, 1992

    Roger Penrose. On the cohomology of impossible figures.Leonardo, 25(3-4):245–247, 1992

    1992

  11. [19]

    Diffusion of information on networked lattices by gossip

    Hans Riess and Robert Ghrist. Diffusion of information on networked lattices by gossip. InProceedings of the 2022 IEEE Conference on Decision and Control (CDC), 2022. Departments of Mathematics and Electrical & Systems Engineering, University of Pennsyl- v ania, Philadelphia, ...

    2022

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