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From a narratological point of view, metalepsis is defined as a transgression of the narrative hierarchy in which an entity belonging to a higher level finds itself acting at a lower level, and vice versa. This paper offers an analysis of metalepsis’s expressive functions and philosophical implications. It is primarily highlighted its power to illustrate – using the specific expressive means of literature – some of the deepest problems of thought. Indeed, the paradoxical level-crossing that characterizes metalepsis turns up in intellectual activities apparently located quite far from the territory of artistic and literary creation – such as logic, metaphysics, philosophy of language and psychoanalysis. Our goal is to demonstrate that metalepsis’s expressive potential goes far beyond the representation of the problematic relationship between fact and fiction, as the metaleptic structure narratively reproduces a series of mental operations performed by the thought every time it is looking for its own limits., Dal punto di vista narratologico, la metalessi si definisce come una trasgressione della gerarchia narrativa in cui un elemento appartenente a un livello superiore si trova ad agire a uno inferiore e viceversa. Il presente saggio offre un’analisi delle funzioni espressive e delle implicazioni filosofiche della metalessi. Viene principalmente sottolineata la sua capacità di illustrare – attraverso i mezzi propri della letteratura – problematiche del pensiero di interesse capitale. Il paradossale attraversamento di livelli che la caratterizza compare infatti in attività intellettuali situate in territori abbastanza distanti da quello della creazione artistico-letteraria – come logica, metafisica, filosofia del linguaggio e psicoanalisi. Nostro obiettivo è dimostrare che le potenzialità espressive della metalessi vanno ben oltre la rappresentazione del problematico rapporto tra fatto e finzione in quanto la struttura metalettica riproduce narrativamente una serie di operazioni mentali compiute dal pensiero umano ogni volta che va alla ricerca dei propri limiti.

We generalise the surface-complexity of closed 3-manifolds to the compact case. This generalisation preserves the (natural generalisation of the) properties holding in the closed case: the surface-complexity on compact 3-manifolds is a natural number measuring how much the manifolds are complicated, it is subadditive under connected sum and it is finite-to-one on $$\mathbb {P}^2$$ -irreducible and boundary-irreducible manifolds without essential annuli and Mobius strips. Moreover, for these manifolds, it equals the minimal number of cubes in an ideal cubulation of the manifold, except for a finite number of cases. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity.

En este texto se lleva una lectura de la obra lacaniana a través del estudio de sus figuras topológicas. En este sentido, se afirma que el psicoanálisis lacaniano no hace uso de la topología como si no de una herramienta externa si tratase, es decir, como un modo a través del cual obtener ejemplos ilustrativos y explicaciones, sino que el propio pensamiento lacaniano es en sí mismo topológico y, por tanto, no puede ser comprendido al margen de esta dimensión que atraviesa la totalidad de su recorrido. Para ello se estudiarán seis figuras: el grafo del deseo, la cinta de Moebius, el toro, la botella de Klein, la esfera provista de un cross-cap y el nudoborromeo.

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the M\"obius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" M\"obius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns., Comment: Revised version prior to publication. Accepted for publication in Portugaliae Mathematica

Carbon nanohoop, a class of constrained molecular architecture consisting of linked arene units, has attracted considerable interest from both experimental and theoretical chemists due to its synthetic challenge and aesthetic architectures. Another fascinating and synthetically challenging species, the Mobius-type molecule, has been attracting the scientific community with its elegant structure and aromaticity. Thus, combining two things together, synthesizing a carbon nanohoop with Mobius topology remains more challenging to date. Here we report a cyclophenylene featuring Mobius strip characterized by X-ray crystallography. Theoretical calculations reveal that such type of nanohoop is fully conjugated systems with electrons delocalized both in π sextets and the bridging carbon-carbon bonds. This work highlights that the manipulation of phenylene connection in a carbon nanohoop can help obtain more delicate and aesthetic molecular architectures.

From the work of M.C. Escher to the universal recycling logo, no mathematical object is more ubiquitous than the Mobius strip. Introduced independently by Listing and Mobius in 1858, the Mobius str...

Examples illustrating the use of one of the simplest and remarkable geometric bodies of all, the Mobius strip (loop, tape) are considered. A survey of the most characteristic patented designs of working and auxiliary elements of equipment in the chemical, petrochemical, and related industries along with ways of using the properties of the Mobius strip in the fabrication and operation of equipment is presented.

We show that computation of the SPERNER problem is PPA-complete on the Mobius band under proper boundary conditions, settling a long term open problem. Further, the same computational complexity results extend to other discrete fixed points on the Mobius band, such as the Brouwer fixed point problem, the DPZP fixed point problem and a simple version of the Tucker problem, as well as the projective plane and the Klein bottle. We expect it opens up a new route for further studies on the related combinatorial structures.

This paper develops the Möbius strip as an ‘ordering theory’ (Sandberg and Alvesson, 2020) that brings CCT studies into dialogue with mainstream marketing approaches. The aim is to work toward a transdisciplinary understanding of market spatiality, a topic that has become increasingly important for theorists and practitioners (Warnaby and Medway, 2013; Castilhos et al., 2016; Chatzidakis et al., 2018). Building on psychosocial interpretations of the Möbius strip as a ‘tactical’ way of thinking, a range of insights and ideas are organized along a single strip of theorization. This paper maps a continuous plane of logic between the concepts of space, place, emplacement, spatiality, implacement, and displacement. The potential applications of the Möbius strip are then demonstrated by showing how the transdisciplinary topic of ‘atmosphere’ can be theorized from multiple perspectives. The paper concludes by exploring how the Möbius strip might also be employed in other areas of marketing theory and practice, potentially generating further transdisciplinary conversations between CCT and the marketing mainstream.

This article is an attempt to show the main stages of the life path and directions of work carried out by Ivan Mikhailovich Shakhparonov - a remarkable experimenter, deeply erudite scientist and a good friend who has devoted almost 60 years of his life to experimental physical research. The main focus is on experiments with a Mobius strip generator and the effects produced by this generator.

We present a comprehensive study of plasmonic responses of a gyrotropic topological symmetry-breaking Mobius strip in the presence of an external magnetic field. The interplay between the magnetic ...

The structure of the 9 vertex obstructive graphs for the nonorientable surface of the genus 2 is established by the method of j-transformations of the graphs. The problem of establishing the structural properties of 9 vertex obstruction graphs for the surface of the undirected genus 2 by the method of j-transformation of graphs is considered. The article has an introduction and 5 sections. The introduction contains the main definitions, which are illustrated, to some extent, in Section 1, which provides several statements about their properties. Sections 2 – 4 investigate the structural properties of 9 vertex obstruction graphs for an undirected surface by presenting as a j-image of several graphs homeomorphic to one of the Kuratovsky graphs and at least one planar or projective-planar graph. Section 5 contains a new version of the proof of the statement about the peculiarities of the minimal embeddings of finite graphs in nonorientable surfaces, namely, that, in contrast to oriented surfaces, cell boundaries do not contain repeated edges. Also in section 5 the other properties peculiar to embeddings of graphs to non-oriented surfaces and the main result are given. The main result is Theorem 1. Each obstruction graph H for a non-oriented surface N2 of genus 2 satisfies the following. 1. An arbitrary edge u,u = (a,b) is placed on the Mebius strip by some minimal embedding of the graph H in N3 and there exists a locally projective-planar subgraph K of the graph H \ u which satisfies the condition: (tK({a,b},N3)=1)˄(tK\u({a,b},N2)=2), where tK({a,b},N) is the number of reachability of the set {a,b} on the nonorientable surface N; 2. There exists the smallest inclusion of many different subgraphs Ki of a 2-connected graph H homeomorphic to the graph K+e, where K is a locally planar subgraph of the graph H (at least K+e is homemorphic to K5 or K3,3), which covers the set of edges of the graph H. Keywords: graph, Klein surface, graph structure, graph obstruction, non-oriented surface, Möbius strip.

Using a technique which is inspired by topology, we construct original examples of 3 - and 4 -edge critical graphs. The 3 -critical graphs cover all even orders starting from 26 ; the 4 -critical graphs cover all even orders starting from 20 and all the odd orders. In particular, the 3 -critical graphs are not isomorphic to the graphs provided by Goldberg for disproving the Critical Graph Conjecture. Using the same approach we also revisit the construction of some fundamental critical graphs, such as Goldberg’s infinite family of 3 -critical graphs, Chetwynd’s 4 -critical graph of order 16 and Fiol’s 4 -critical graph of order 18 .

This article maps the interconnections between two emergent resource frontiers in Ireland: wind and data. Adding to literature about extraction and extractivism, we account for how these expanded extractive frontiers are mobilised within self-sustaining and automated formations. In Ireland, digital infrastructures such as data centres are developed by multinational tech companies to avail of a naturally cool climate and business environment friendly to their investment, part of a wider extractive system by which data are made valuable for their expansive operations. Wind farms similarly make use of Ireland’s climate to generate energy, often used to power digital infrastructures, and are increasingly embedded within ‘smart’ energy and data systems. Wind and data are seen discretely as ‘abundant’ resources, their infrastructures built on terra or (offshore) mare nullius, and their operations ‘green’. However, their infrastructures are entangled with non-renewable energy systems and tax evasive capital, and built across existing communities and environments through policy, planning logics and increasingly automated methods of maintenance and optimisation. Through what we call ‘the moebius strip of wind/data’, wind and data infrastructures are increasingly formidable in dictating our energy futures. In this article, we articulate how they are connected and how we can disentangle them, especially in their operation across urban and rural geographies.

Different from molecular level topology, the development of supramolecular topology has been limited due to a lack of reliable synthetic methods. Here we describe a supramolecular strategy of accessing Möbius strip, a fascinating topological object featured with only a single edge and single side. Through bending and cyclization of twisted nanofibers self-assembled from chiral glutamate amphiphiles, supramolecular nano-toroids with various twist numbers were obtained. Electron microscopic techniques could clearly identify the formation of Möbius strips when twist numbers on the toroidal fibers are odd ones. Spectroscopic and morphological analysis indicates that the helicity of the Möbius strips and nano-toroids stems from the molecular chirality of glutamate molecules. Therefore, M- and P-helical Möbius strips could be formed from L- and D-amphiphiles, respectively. Our experimental results and theoretical simulations may advance the prospect of creating chiral topologically complex structures via supramolecular approach., Different to exploring molecular topology, the development of supramolecular topology has been limited due to a lack of reliable synthetic methods. Here, the authors describe a supramolecular strategy to access Möbius strips through bending and cyclization of twisted nanofibers self-assembled from chiral glutamate amphiphiles.

Drawing on teaching sessions that I conducted last year, alongside my own practice-based doctoral research in narrative or cinematic VR (CVR) predicated on the Deleuzian notion of immanence, I propose that the CVR screenplay is better understood as a moebius strip than a linear narrative; a tale that turns around on itself. But far from being unorientable like its mathematical paradigm, the moebius narrative can be both oriented and scripted, as I hope to illustrate using student work as well as my own script iterations. Taking it to be both a model and a metaphor, this article explores how a moebius narrative can be designed ‐ and why design thinking is more suitable for this process than traditional screenwriting methods. While still an understanding-in-progress, I find this conceptual framework useful for both practice and pedagogy. This article hopes additionally, therefore, to make a case for pedagogy as a research method in its own right, especially in the context of practice-based research.

Vortex coherent structures on arrays of nonlinear oscillators joined by weak links into topologically nontrivial two-dimensional discrete manifolds have been theoretically studied. A circuit of nonlinear electric oscillators coupled by relatively weak capacitances has been considered as a possible physical implementation of such objects. Numerical experiments have shown that a time-monochromatic external force applied to several oscillators leads to the formation of long-lived and nontrivially interacting vortices in the system against a quasistationary background in a wide range of parameters. The dynamics of vortices depends on the method of "coupling" of the opposite sides of a rectangular array by links, which determines the topology of the resulting manifold (torus, Klein bottle, projective plane, M\"obius strip, ring, or disk)., Comment: revtex, 6 pages, 11 figures, in English, published version

We explicitly classify all $$\mathbb {S}^1$$ -invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$ . This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $$\mathbb {S}^1$$ -invariant metrics on the annulus and Mobius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all $$\mathbb {S}^1$$ -invariant metrics on the Mobius band for each $$k \ge 1$$ , and show that it is achieved by the induced metric from a free boundary minimal embedding of the Mobius band into $${\mathbb {B}}^4$$ by kth Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $$\mathbb {S}^1$$ -invariant metrics on the annulus and Mobius band are the induced metrics on explicit free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^3$$ and $${\mathbb {B}}^4$$ , including some new families of free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^4$$ . Finally, we prove that these are the only $$\mathbb {S}^1$$ -invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$ .

International audience; We report on experiments with Möbius strip microlasers, which were fabricated with high optical quality by direct laser writing. A Möbius strip, i.e., a band with a half twist, exhibits the fascinating property that it has a single nonorientable surface and a single boundary. We provide evidence that, in contrast to conventional ring or disk resonators, a Möbius strip cavity cannot sustain whispering gallery modes (WGM). Comparison between experiments and 3D finite difference time domain (FDTD) simulations reveals that the resonances are localized on periodic geodesics.

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\tan(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in $\mathbb{R}\times \mathbb{R}P^3$. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in $S^1$ to prove that $1/3$ is a sharp lower bound on the probability that a M\"obius strip filling the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.

Spiral spin density vectors refer to the spiral trajectory of the spin density vectors in three dimensional (3D) space, and similar to the polarization Möbius strip, polarization knots and photonic wheel, are also special polarization structures in 3D optical field and only exist in three dimensions. In this article, we propose a new way to generate the spiral spin density vectors with a long spiral range. This is realized by strongly focusing a circularly polarized beam with one off-axis vortex. In this field, the spiral interval of spin density vectors theoretically always exists on the propagation axis, and its position can be controlled by the beam size parameter f/ω0. The polarization variation and the spiral spin density vectors are discussed analytically along the propagation direction. It is found that the spiral shape and spiral `speed' can be adjusted by the off-axis distance of the vortex and the semi-aperture angle of the focusing system respectively. A special spiral behavior of the spin density vector at the geometrical focus is also observed in this focused field. The spiral spin density vectors supply a new rotational degree of freedom in 3D optical field and our finding may have applications in optical micro-manipulations.

We consider the optimal flows on noncompact surfaces with boundary, which have a minimum number of fixed points and all these points lie on the boundary of the surface. It is proved that the flow is optimal if it has a single sink and a single source. We describe the structures of the optimal flows on a simply connected region, on a Mobius strip, on a torus with hole, and on a Klein bottle with hole.

We describe some classic experiments on the Mobius strip, the projective plane band, and the Klein bottle band. We present our experience with freshmen college students, college teachers, h...

The Möbius strip spacetime topology and the entangled antipodal points on black hole surfaces, recently described by ‘t Hooft, display an unnoticed relationship with the Borsuk-Ulam theorem from algebraic topology. Considering this observation and other recent claims which suggest that quantum entanglement takes place on the antipodal points of a S3 hypersphere, a novel topological framework can be developed: a feature encompassed in an S2 unentangled state gives rise, when projected one dimension higher, to two entangled particles. This allows us to achieve a mathematical description of the holographic principle occurring in S2. Furthermore, our observations let us to hypothesize that a) quantum entanglement might occur in a four-dimensional spacetime, while disentanglement might be achieved on a motionless, three-dimensional manifold; b) a negative mass might exist on the surface of a black hole.

Recently, the problem about Pfaffian property of graphs has attracted much attention in the matching theory. Its significance stems from the fact that the number of perfect matchings in a graph G can be computed in polynomial time if G is Pfaffian. The Mobius strip with unique surface structure strip has potentially significant chirality properties in molecular chemistry. In this paper, we consider the Pfaffian property of graphs on the Mobius strip based on topological resolution. A sufficient condition for Pfaffian graphs on the Mobius strip is obtained. As its application, we characterize Pfaffian quadrilateral lattices on the Mobius strip.

In this paper we prove existence and uniqueness of the Bjorling problem for the class of immersed surfaces in $$\mathbb {R}^3$$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Mobius strip for an arbitrary large class of prescribed functions. In particular, we use the Bjorling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Mobius strip in $$\mathbb {R}^3$$ .

The second part of the paper is devoted to enumeration of r -regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, the annulus and the Mobius band. Using 3- and 4-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed r -regular maps on the torus for various r .

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$ . The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

Möbius strips are prototypical examples of ribbon-like structures. Inspecting their shapes and features provides useful insights into the rich mechanics of elastic ribbons. Despite their ubiquity and ease of construction, quantitative experimental measurements of the three-dimensional shapes of Möbius strips are surprisingly non-existent in the literature. We propose two novel stereo vision-based techniques to this end—a marker-based technique that determines a Lagrangian description for the construction of a Möbius strip, and a structured light illumination technique that furnishes an Eulerian description of its shape. Our measurements enable a critical evaluation of the predictive capabilities of mechanical theories proposed to model Möbius strips. We experimentally validate, seemingly for the first time, the developable strip and the Cosserat plate theories for predicting shapes of Möbius strips. Equally significantly, we confirm unambiguous deficiencies in modelling Möbius strips as Kirchhoff rods with slender cross-sections. The experimental techniques proposed and the Cosserat plate model promise to be useful tools for investigating a general class of problems in ribbon mechanics.

The part of this article dealing with topological insulators and topological superconductors was first written about two years ago - the ideas in the part about the topological universe originated six years ago or more. It’s rather strange that I never put the two parts together in writing before. My belief in unification is unshakeable - I’ve been convinced for years that the universe must be composed of topology. Since Earth is part of the cosmos, entanglement means it must have topological materials. The reverse is also true: topological materials on Earth are well known to science - so in a unification, space and time inevitably possess topological composition. Topological materials (topological insulators, topological superconductors) can be less mystifying if they’re related to the paradigm-shifting deterministic view of quantum mechanics which is described in the universal topology (the “rubber-sheet geometry” of the cosmos): see my previous submission “Hypothesis of Quantum Gravity - Resulting from a Static, Topological Universe Resulting from the Positives and Negatives of the Steady State and Big Bang Theories" at https://www.preprints.org/manuscript/202105.0239/v1 (the first section of this present article is a quick summary of the relevant parts).

Twisted toroidal ribbons such as the one-sided Möbius strip have inspired scientists, engineers and artists for many centuries. A physical Möbius strip exhibits interesting mechanical properties deriving from a tendency to redistribute the torsional strain away from the twist region. This leads to the interesting possibility of building topological actuators with continuous deformations. Here we report on a series of corresponding bi-layered stripe actuators using a photothermally responsive liquid crystal elastomer as the fundamental polymeric material. Employing a special procedure, even Möbius strips with an odd number of twists can be fabricated exhibiting a seamless homeotropic and homogeneous morphology. Imposing a suitable contraction gradient under near-infrared light irradiation, these ribbons can realize continuous anticlockwise/clockwise in-situ rotation. Our work could pave the way for developing actuators and shape morphing materials that need not rely on switching between distinct states., Shape morphing materials are usually difficult to operate in a continuous fashion. Nie et al. fabricate stripes with liquid crystalline elastomers that can be given a Möbius-like morphology with seamless material composition, and perpetually driven under photothermally induced actuation.

Este ensaio analisa a intersecção entre a dança e a historicidade na performance Hidebehind (2018), da coreógrafa brasileira Josefa Pereira, apresentada em 2019 em Lisboa no Teatro do Bairro Alto. Com o seu movimento para trás num movimento circular interrompido pelo gesto, Hidebehind convida‑nos a pensar na dança e a interromper o nosso movimento histórico para o questionar activamente. Com base nos princípios da fita de Möbius e da dupla divisão do corpo através do plano coronal, a coreógrafa define na sua operação de captura o ponto de intersecção de várias presenças e temporalidades em simultâneo. O seu corpo, através de mecanismos de repetição e da irrupção do gesto, apresenta‑se como um arquivo crítico que desestabiliza e transforma os nossos paradigmas perceptivos, questionando as categorias binárias em vigor no pensamento ocidental. Como transitar o esborramento destes limites na operação de captura? Que potencialidades se encontram na performance para trabalhar com o tempo, recordar e traçar futuros? CINTA MÖBIUS / CUERPO /ARCHIVO / CAPTURA / DANZA, This essay examines the intersection between dance and historicity in the performance Hidebehind (2018) by Brazilian choreographer Josefa Pereira, presented in 2019 in Lisbon at Teatro do Bairro Alto. With its backwards movement in a circular motion interrupted by gesture, Hidebehind invites us to think about dance and to interrupt our historical movement to actively question it. Based on the principles of Möbius' tape and the double division of the body through the coronal plane, the choreographer defines in her capture operation the intersection point of several presences and temporalities simultaneously. Her body, through mechanisms of repetition and the irruption of gesture, presents itself as a critical archive that destabilises and transforms our perceptual paradigms, questioning the binary categories in force in western thought. How can the blurring of these limits be transited in the operation of capture? What potentialities do we find in performance to work with time, to remember and to trace futures? MÖBIUS STRIP / BODY / ARCHIVE / CAPTURE / DANCE

In this work, the electromagnetic response of a mathematically interesting shape—a Möbius strip—is presented, along with a ring resonator for comparison. Both resonators consist of a central lossy dielectric layer bounded by perfectly conducting layers. For the case of the Möbius strips, the computational results show that there are a family of half-integer wavelength modes within the dielectric layer. These additional modes result in increased absorption, and a corresponding reduction in the radar cross section. Interestingly, rotational scans show that these modes can be excited over a large angular range. This investigation gives an understanding of the electromagnetic response of these structures, paving the way for future experiments on Möbius strip resonators.

We construct an electrical circuit to realize a modified Haldane lattice exhibiting the phenomenon of antichiral edge states. The circuit consists of a network of inductors and capacitors with interconnections reproducing the effects of a magnetic vector potential. The next nearest neighbor hoppings are configured differently from the standard Haldane model, and as predicted by earlier theoretical studies, this gives rise to antichiral edge states that propagate in the same direction on opposite edges and coexist with bulk states at the same frequency. Using pickup coils to measure voltage distributions in the circuit, we experimentally verify the key features of the antichiral edge states, including their group velocities and ability to propagate consistently in a Möbius strip configuration. Ministry of Education (MOE) This work was supported by the National Natural Science Foundation of China (Grant Nos. 11874274, and 12004425), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20170058, and BK20200630), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). YiDong Chong was supported by the Singapore MOE Academic Research Fund Tier 3 (Grant No. MOE2016-T3-1-006).

We analyse the evolution of the wavefunction of a quantum particle propagating on several compact manifolds, including the Klein bottle, Möbius strip and projective plane. We find analytically the stationary states and the energy spectrum and show that the wavefunction exhibits perfect revivals. Using the orbifold structure of the discussed manifolds, we establish the relation of wave evolution on the manifolds to Fresnel diffraction and consequently to the Talbot effect. This connection provides a novel method of optical simulation of the quantum motion on compact manifolds. We discuss some novel phenomena as well as the effects of topology on the properties of the waves on the manifolds.

When a Möbius loop is cut along the middle of the band, the result is a single connected loop, yet anecdotal evidence from science demonstrations and the use of this effect in magic tricks suggest that most people are thoroughly surprised by this because they strongly believe that the result should be two separate loops. Here, we present results from a behavioral experiment confirming this anecdotal evidence and discuss potential theoretical explanations for why this demonstration evokes strong, but misleading intuitions and a related illusion of impossibility.

Connection between external skin and inner core of the building is paramount for the case study of Eisenman’s Max Reinhart house. External envelope is generated by a series of surfaces perpendicular to a Moebius strip running through the whole building. Each square is constrained to a rotation in the space, so that a certain amount of overlapping, and subsequent cutting is generated. External surface is created by the extrusion of squares into boxes, which partially overlap themselves. The internal structure of the building is characterized by horizontal slices of the resulting volume. The work analyses the geometrical features of the Mobius strip by writing the discrete parametric equations that define this shape driver. A tentative equation system was worked out with a circular generator element. Through the variation of the parameter connected with the ordinate, it was possible to create a Moebius strip with an elliptic generator. This ellipsis lays on a vertical plane perpendicular to the ground. The resulting surface generated with the elliptical Moebius strip resulted to be ruled. There is a direct connection between urban landscape and building shape, both in the definition of geometric parameters and the constraint of morphogenetic characterization. The parametric geometry has been worked out with Grasshopper to generate an accurate solution within the given constraints.

The Klein bottle plays a crucial role in the main modern sciences. This surface was first described in 1882 by the German mathematician Felix Klein. In this paper we describe a technique to obtain the parameterization of the Klein bottle. This technique uses isometric transformations (translations and rotations) and the moving frame associated with the unit circumference lying on the xy-plane. The process we follow is to start with the parametrization of the Euclidean cylinder, then continue with the parameterization of the Mobius strip, after that with the parameterization of the torus of revolution and finally, in a natural way, we describe the aforementioned technique. With the parameterization of the Klien bottle obtained, it is easy to show that it can be obtained by gluing two Mobius strips. Additionally, the parameterizations of the n-twisted and n-turns Klein bottles are obtained. All geometric calculations and geometric interpretations are performed with the Mathematica symbolic calculus system.

Recently, research has mainly been focused on structured singular light in the paraxial regime, as also presented in previous chapters. Here, investigations were concentrated on fields shaped in 2d or 3d space, in which the typically elliptical polarization of light is purely transverse, i.e., two-dimensional (2d polarization). Hence, the light field oscillates solely within the plane orthogonal to the beam’s propagation direction with \(\vec {E}(x,y,z) = [E_x(x,y,z), E_y(x,y,z)]^T\). Respective studies have lead to pathbreaking findings enabling advanced applications as well as new insights into the general and not yet fully explored, nor understood, nature of light (see Chaps. 2, 3). However, a major current challenge is the moving from these 3d fields of 2d polarization to 4d structured light of 3d polarization with \(\vec {\mathcal {{E}}}(x,y,z) = [\mathcal {E}_x(x,y,z), \mathcal {E}_y(x,y,z), \mathcal {E}_z(x,y,z)]^T\), exploiting all three electric field components, i.e., transverse as well as longitudinal ones. These fields do not only give new insights into the fundamentals of light, but also open new horizons for applied optics: 4d fields allow for structures of nano-scale complexity whereby their 3d polarization nature is an enriching feature for handling, e.g., polarization sensitive materials. The crucial longitudinal components are non-negligible for non-paraxial light fields, fulfilling the vectorial Helmholtz equation (Eq. ( 2.7)), as in tightly focused beams. These fields are formed by the application of focusing optical components imparting a numerical aperture (\(\mathrm {NA}\)) larger or equal to 0.7. In this case, radial components of the input light field are transformed into significant longitudinal focal field contributions [1, 2, 3]. Considering this effect, some studies have been performed representing first steps towards the investigation and future application of 4d structured fields. For example, tightly focusing 2d polarization structures enabled shaping focal intensity configurations [4, 5, 6, 7, 8] as optical needles. Furthermore, longitudinal field components facilitate the formation of complex 3d polarization topologies surrounding optical singularities as optical cones, twisted ribbons and Mobius strips. These topologies were as firstly predicted by Freund in 2005 [9] and, ten years later, experimentally proven by Bauer et al. [10].

The existence and the mysteries of the universe could not be explained by using just 3 spatial dimensions. There was a need to think of higher dimensions as a tool to explain the phenomena happening in our universe. Therefore, unified theories such as Loop Quantum Gravity and Superstring Theory were proposed. We will be taking an overview of these theories in order to get some idea about each.

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, M\"obius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues., Comment: Minor changes. Final version. Accepted for publication in the Proceedings of the American Mathematical Society

Using general rigid bead–rod theory, we explore the effect of twisting a macromolecule on its rheological properties in suspensions. We thus focus on macromolecules having the form of Möbius bands so that the number of twists can be incremented. We call these Möbius macromolecules. When represented in general rigid bead–rod theory, these macromolecules comprise beads whose centers all fall on a Möbius band. From first principles, we calculate the complex viscosity of twisted rings with zero to seven twists. We find that the zero-shear values of the viscosity and first normal stress coefficient increase with twisting. Furthermore, we find that the real part of the complex viscosity descends more rapidly, with frequency, with extent of twist. For the imaginary part of the complex viscosity, the more twisted, the higher the peak. For each part of the dimensionless complex viscosity and the first normal stress coefficient, the results fall on one of just three curves corresponding to zero, even, or odd numbers of twists. We also explore the effects of the length and the aspect ratio of twisted macromolecular suspensions. We close with a worked example for a suspension of triply twisted Möbius annulene.

The purpose of this note is to prove an Euler-type formula for partitions of the M\"obius strip. This formula was introduced in our joint paper with R.~Kiwan, "Courant-sharp property for Dirichlet eigenfunctions on the M\"obius strip" (arXiv:2005.01175).