Psychology is the science of behavior and mind, including conscious and unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense scope and diverse interests that, when taken together, seek an understanding of the emergent properties of brains, and all the variety of epiphenomena they manifest. As a social science it aims to understand individuals and groups by establishing general principles and researching specific cases.[1][2]
Interdisciplinarity or interdisciplinary studies involves the combining of two or more academic disciplines into one activity (e.g., a research project).[1] It draws knowledge from several other fields like sociology, anthropology, psychology, economics etc. It is about creating something by thinking across boundaries. It is related to an interdiscipline or an interdisciplinary field, which is an organizational unit that crosses traditional boundaries between academic disciplines or schools of thought, as new needs and professions emerge. Large engineering teams are usually interdisciplinary, as a power station or mobile phone or other project requires the melding of several specialties. However, the term “interdisciplinary” is sometimes confined to academic settings.
Interdisciplinarity or interdisciplinary studies involves the combining of two or more academic disciplines into one activity (e.g., a research project).[1] It draws knowledge from several other fields like sociology, anthropology, psychology, economics etc. It is about creating something by thinking across boundaries. It is related to an interdiscipline or an interdisciplinary field, which is an organizational unit that crosses traditional boundaries between academic disciplines or schools of thought, as new needs and professions emerge. Large engineering teams are usually interdisciplinary, as a power station or mobile phone or other project requires the melding of several specialties. However, the term “interdisciplinary” is sometimes confined to academic settings.
Transdisciplinarity connotes a research strategy that crosses many disciplinary boundaries to create a holistic approach. It applies to research efforts focused on problems that cross the boundaries of two or more disciplines, such as research on effective information systems for biomedical research (see bioinformatics), and can refer to concepts or methods that were originally developed by one discipline, but are now used by several others, such as ethnography, a field research method originally developed in anthropology but now widely used by other disciplines. The Belmont Forum [1] elaborated that a transdisciplinary approach is enabling inputs and scoping across scientific and non-scientific stakeholder communities and facilitating a systemic way of addressing a challenge. This includes initiatives that support the capacity building required for the successful transdisciplinary formulation and implementation of research actions.
Cognitive science is the interdisciplinary, scientific study of the mind and its processes.[2] It examines the nature, the tasks, and the functions of cognition (in a broad sense). Cognitive scientists study intelligence and behavior, with a focus on how nervous systems represent, process, and transform information. Mental faculties of concern to cognitive scientists include language, perception, memory, attention, reasoning, and emotion; to understand these faculties, cognitive scientists borrow from fields such as linguistics, psychology, artificial intelligence, philosophy, neuroscience, and anthropology.[3] The typical analysis of cognitive science spans many levels of organization, from learning and decision to logic and planning; from neural circuitry to modular brain organization. The fundamental concept of cognitive science is that “thinking can best be understood in terms of representational structures in the mind and computational procedures that operate on those structures.”[3]
Simply put: Cognitive Science is the interdisciplinary study of cognition in humans, animals, and machines. It encompasses the traditional disciplines of psychology, computer science, neuroscience, linguistics and philosophy. The goal of cognitive science is to understand the principles of intelligence with the hope that this will lead to better comprehension of the mind and of learning and to develop intelligent devices. The cognitive sciences began as an intellectual movement in the 1950s often referred to as the cognitive revolution.
1580s, “pertaining to cognition,” with -ive + Latin cognit-, past participle stem of cognoscere “to get to know, recognize,” from assimilated form of com “together” (see co-) + gnoscere “to know,” from PIE root *gno- “to know.”
Taken over by psychologists and sociologists after c. 1940. Cognitive dissonance “psychological distress cause by holding contradictory beliefs or values” (1957) apparently was coined by U.S. social psychologist Leon Festinger, who developed the concept. Related: Cognitively.
Figure it out: A non-dual cognitive neuroscience perspective on the Möbius band.
The Möbius band is an extraordinary geometrical figure. The band is eponymously named after the German mathematician August Ferdinand Möbius who described it in 1885, contemporaneously with another German mathematician named Johann Benedict Listing. It is a so called ruled surface with only one side and one boundary and it possesses the mathematical property of non-orientability (viz., a non-orientable manifold). In fact, the Möbius band is the simplest possible non-orientable surface. A Gedankenexperiment is helpful to understand this property intuitively: Imagine walking on the surface of a giant Möbius band. If you would travel long enough you would end up at the very starting point of the journey, only mirror-reversed. This journey can be repeated ad infinitum. Therefore, the Möbius band can also be interpreted as a metaphor for infinity, i.e., the beginningless and the endless. A similar principle can be found in the interpretation of the Ouroboros serpent (which eats its own tail), a gnostic symbol which originated in ancient Egyptian iconography (ἓν τὸ πᾶν – “The all is one”) in the 10th century. A similar symbolism could later also be found in the western Greek magical tradition (Ancient Greek: οὐροβόρος). The psychoanalytic meaning of the Ouroboros was discussed as an archetype by the depth-psychologist C.G. Jung.
The geometry of the Möbius band (also referred to as “Möbius strip”) has far-reaching interdisciplinary implications. The principles of its peculiar topology have been applied to a broad array of scientific disciplines including mathematics, cosmology, computer science, physics, chemistry, biology, psychology, et cetera. Practical applications include, for instance, superconductors with high transition temperatures, molecular engines, and bandpass filters (see exemplary references below).
In addition to its scientific relevance, the Möbius band can be found as a leitmotif in multifarious artworks across various cultures (for an example see the ancient mosaic depicted below).1 Moreover, the abstract principles derived from its topological structure have been applied to music theory (e.g., the space of all two-note chords, referred to as dyads, resembles the shape of a Möbius band).
The Möbius band is a very interesting visual percept in the context of perceptual cognitive psychology and neuropsychology, as it helps researchers to investigate the cognitive and neuronal mechanism which undergird cognition and perception. (Besides, in the first half of the 20th century magicians used the Möbius band for “magical” tricks.)2 Interestingly, a recent series of fMRI neuroimaging studies focused on the topic of ego-dissolution which is associated with non-dual states of consciousness in which the border between self and other (the dichotomy between inside and outside) temporarily dissolves. The default-mode network3 appears to be an important neuroanatomical correlate in this context.
Next to its neuropsychological aspects, the Möbius band inspires profound philosophical contemplations concerning the relationship between mind & matter (e.g., the “Pauli-Jung conjecture”4 in the context of dual aspect monism)5. In the classical 17th century Cartesian framework (which is still highly influential), mind & matter (psyche & physis – or res extensa & res cogitans)6 are two separate phenomena (this dichotomisation is known as Cartesian dualism or the Cartesian split). However, alternative ontological theories postulate that mind & matter are complementary with respect to each other (in the quantum physical sense of complementarity), i.e., they are different aspects of the same underlying “substance” (hence the term “monism” as opposed to “dualism”).
Currently, a dualistic mind/matter conception is the (mostly implicitly accepted) reigning scientific paradigm (cf. Thomas Kuhn)7, particularly within the neurosciences (e.g., epiphenominalism/emergence theories of consciousness)8. However, this dualistic working hypothesis9 can be challenged on various logical grounds and has not been empirically validated (e.g., correlation ≠ causation; viz., the “cum hoc ergo propter hoc” logical fallacy of implied causality).
Therefore, dual-aspect monism is a viable conceptual alternative worth considering – particularly given recent empirical data obtained in the domain of experimental quantum physics which deeply challenges our intuitive quasi-Newtonian notions of reality which are ubiquitously (prima facie) taken for granted without deeper critical reflection on their logical validity and empirical evidential foundation. The dual-aspect monism perspective is therefore iconoclastic towards the reigning dualistic psychological and neuroscientific status quo paradigm.
It is argued that the Möbius band can be interpreted as a visual metaphor for “chiastic convergence” a coincidentia oppositorum (Latin for “coincidence of opposites”; cf. C.G. Jung), i.e., the non-duality of psyche and physis, internal and external, subject and object, inside and outside, mind and matter, the knower and the known, the “seer and the seen” (Sanskrit: Drg-Drsya; as analyzed in the ancient Advaita Vedānta text “Drg-Drsya-Viveka”). William James eloquently summarized this non-dual view:
“Granted that a definite thought, and a definite molecular action in the brain occur simultaneously, we do not possess the intellectual organ, nor apparently any rudiment of the organ, which would enable us to pass by a process of reasoning from the one phenomenon to the other. They appear together but we do not know why.”
The instant field of the present is at all times what I call the ‘pure’ experience. It is only virtually or potentially either object or subject as yet. For the time being, it is plain, unqualified actuality, or existence, a simple that. […] Just so, I maintain, does a given undivided portion of experience, taken in one context of associates, play the part of the knower, or a state of mind, or “consciousness”; while in a different context the same undivided bit of experience plays the part of a thing known, of an objective ‘content.’ In a word, in one group it figures as a thought, in another group as a thing. […] Things and thoughts are not fundamentally heterogeneous; they are made of one and the same stuff, stuff which cannot be defined as such but only experienced; and which one can call, if one wishes, the stuff of experience in general. […] ‘Subjects’ knowing ‘things’ known are ‘roles’ played, not ‘ontological” facts’. ~ William James (1904)
The whole duality of mind and matter […] is a mistake; there is only one kind of stuff out of which the world is made, and this stuff is called mental in one arrangement, physical in the other. ~ Bertrand Russell (1913)
There is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination. ~ Daniel Dennett (1995)
Formulaic notation specifying a Möbius band in 3-dimensional Euclidean space
\[
{\displaystyle x(u,v)=\left(3+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u}
\]
\[
{\displaystyle y(u,v)=\left(3+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u}
\]
\[
{\displaystyle z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}}
\]
where \[{\displaystyle 0\leq u<2\pi }\] and \[{\displaystyle -1\leq v\leq 1}\]
This parametrization produces a single Möbius band with a width of 1 and a middle circle with a radius of 3. The band is positioned in the xy plane and is centred at coordinates (0, 0, 0). The Möbius band can be plotted in R (an open-source software environment for statistical computing and graphics). The associated code to create the graphic is based on the packages “rgl” (Murdoch, 2001, 2018) and “plot3D” (Soetaert, 2014, 2017) and can be found below. The appended R code creates an interactive plot that allows to scale and rotate the Möbius band in three dimensional space.
You can plot an interactive Möbius band by using the open-source software “R” which you can download using the URL below. Simply copy & paste the provided code into R and it will produce an interactive scaleable and rotatable 3-dimensional Möbius band. You have to install the “rgl” and “plot3D” package for this to work. cran.r-project.org/mirrors.html
#Source URL: https://r.prevos.net/plotting-mobius-strip/
library(rgl) #RGL: An R Interface to OpenGL (Murdoch, 2001)
library(plot3D) #plot3D: Plotting multi-dimensional data (Soetaert, 2014)
# Define parameters
R <- 3
u <- seq(0, 2 * pi, length.out = 100)
v <- seq(-1, 1, length.out = 100)
m <- mesh(u, v)
u <- m$x
v <- m$y
# Möbius strip parametric equations
x <- (R + v/2 * cos(u /2)) * cos(u)
y <- (R + v/2 * cos(u /2)) * sin(u)
# Visualise in 3-dimensional Euclidean space
bg3d(color = "white")
surface3d(x, y, z, color= "red")
import plotly.plotly as py
import plotly.figure_factory as FF
import plotly.graph_objs as go
import numpy as np
from scipy.spatial import Delaunay
u = np.linspace(0, 2*np.pi, 24)
v = np.linspace(-1, 1, 8)
u,v = np.meshgrid(u,v)
u = u.flatten()
v = v.flatten()
tp = 1 + 0.5*v*np.cos(u/2.)
x = tp*np.cos(u)
y = tp*np.sin(u)
z = 0.5*v*np.sin(u/2.)
points2D = np.vstack([u,v]).T
tri = Delaunay(points2D)
simplices = tri.simplices
fig1 = FF.create_trisurf(x=x, y=y, z=z,
colormap="Portland",
simplices=simplices,
title="Mobius Band")
py.iplot(fig1, filename="Mobius-Band")
#Source URL: https://plot.ly/python/trisurf/
How to create a real Möbius band manually
It is easy to create a Möbius band manually from a rectangular strip of paper. One simply needs to twist one end of the strip by 180° and then join the two ends together (see also Starostin & Van Der Heijden, 2007).
Aion – the Greek God of eternity standing in a celestial Möbius band
Figure 1. The antique mosaic shows a central part of a large floor mosaic, from a Roman villa in Sentinum (now Sassoferrato, in Marche, Italy), ca. 200–250 C.E. The Hellenistic deity named Aion (Greek: Αἰών), the god of time and eternity, is standing inside a celestial sphere (presumably the orb circle encompassing the universe) decorated with zodiac signs, in between a green tree and a bare tree (symbolizing summer and winter, respectively). Sitting in front of him is the mother-earth goddess, Tellus (the Roman counterpart of Gaia) with her four children, who possibly represent the four seasons. Conceptions of “time” play an essential rôle in ancient philosophical schools of thought and consequently in Greek mythology and modern science (subjects which are deeply interwoven).
“Aromaticity is a key concept in chemistry, dating back to faraday’s discovery of benzene in 1825 and kekulé’s famous alternating-double-bond structure of 1865. in 1858, the möbius strip was discovered by möbius and listing. the hückel rules for predicting aromaticity, stating that [4n + 2] π electrons result in an aromatic system, work for planar molecules. although molecules with möbius geometry are not found in nature, chemists have tried to synthesize such molecules since the first theoretical prediction by heilbronner in 1964 and the prediction of möbius aromaticity for suitable compounds with [4n] π electrons. however, möbius-aromatic molecules have proved difficult to synthesize, and sometimes even to identify. here we summarize recent contributions of several research groups that have succeeded in synthesizing möbius-type molecules. the results of this survey lead us to suggest that the generation of möbius topologies in expanded porphyrins is easier than hitherto appreciated.”
Ajami, D., Oeckler, O., Simon, A., & Herges, R.. (2003). Synthesis of a Möbius aromatic hydrocarbon. Nature
“The defining feature of aromatic hydrocarbon compounds is a cyclic molecular structure stabilized by the delocalization of pi electrons that, according to the hückel rule, need to total 4n + 2 (n = 1,2, em leader ); cyclic compounds with 4n pi electrons are antiaromatic and unstable. but in 1964, heilbronner predicted on purely theoretical grounds that cyclic molecules with the topology of a möbius band-a ring constructed by joining the ends of a rectangular strip after having given one end half a twist-should be aromatic if they contain 4n, rather than 4n + 2, pi electrons. the prediction stimulated attempts to synthesize möbius aromatic hydrocarbons, but twisted cyclic molecules are destabilized by large ring strains, with the twist also suppressing overlap of the p orbitals involved in electron delocalization and stabilization. in larger cyclic molecules, ring strain is less pronounced but the structures are very flexible and flip back to the less-strained hückel topology. although transition-state species, an unstable intermediate and a non-conjugated cyclic molecule, all with a möbius topology, have been documented, a stable aromatic möbius system has not yet been realized. here we report that combining a ‘normal’ aromatic structure (with p orbitals orthogonal to the ring plane) and a ‘belt-like’ aromatic structure (with p orbitals within the ring plane) yields a möbius compound stabilized by its extended pi system.”
Chang, C. W., Liu, M., Nam, S., Zhang, S., Liu, Y., Bartal, G., & Zhang, X.. (2010). Optical Möbius symmetry in metamaterials. Physical Review Letters
“We experimentally observed a new topological symmetry in optical composites, namely, metamaterials. while it is not found yet in nature materials, the electromagnetic möbius symmetry discovered in metamaterials is equivalent to the structural symmetry of a möbius strip, with the number of twists controlled by the sign change of the electromagnetic coupling between the meta-atoms. we further demonstrate that metamaterials with different coupling signs exhibit resonance frequencies that depend only on the number but not the locations of the ‘twists,’ thus confirming its topological nature. the new topological symmetry found in metamaterials may enable unique functionalities in optical materials.”
Fan, Y. Y., Chen, D., Huang, Z. A., Zhu, J., Tung, C. H., Wu, L. Z., & Cong, H.. (2018). An isolable catenane consisting of two Möbius conjugated nanohoops. Nature Communications
“Besides its mathematical importance, the möbius topology (twisted, single-sided strip) is intriguing at the molecular level, as it features structural elegance and distinct properties; however, it carries synthetic challenges. although some möbius-type molecules have been isolated by synthetic chemists accompanied by extensive computational studies, the design, preparation, and characterization of stable möbius-conjugated molecules remain a nontrivial task to date, let alone that of molecular möbius strips assembling into more complex topologies. here we report the efficient synthesis, crystal structure, and theoretical study of a catenane consisting of two fully conjugated nanohoops exhibiting möbius topology in the solid state. this work highlights that oligoparaphenylene-derived nanohoops, a family of highly warped and synthetically challenging conjugated macrocycles, can not only serve as building blocks for interlocked supermolecular structures, but also represent a new class of compounds with isolable möbius conformations stabilized by non-covalent interactions.”
Marchionini, G., Wildemuth, B. M., & Geisler, G.. (2006). The open video digital library: A möbius strip of research and practice. Journal of the American Society for Information Science and Technology
“The open video digital library (ovdl) provides digital video files to the education and research community and is distinguished by an innovative user interface that offers multiple kinds of visual surrogates to people searching for video content. the ovdl is used by several thousand people around the world each month and part of this success is due to its user interface. this article examines the interplay between research and practice in the development of this particular digital library with an eye toward lessons for all digital libraries. we argue that theoretical and research goals blur into practical goals and practical goals raise new research questions as research and development progress—this process is akin to walking along a möbius strip in which a locally two-sided surface is actually part of a globally one-sided world. we consider the gulf between the theories that guide current digital library research and current practice in operational digital libraries, provide a developmental history of the ovdl and the research frameworks that drove its development, illustrate how user studies informed its implementation and revision, and conclude with reflections and recommendations on the interplay between research and practice.”
Goldstein, R. E., Moffatt, H. K., Pesci, A. I., & Ricca, R. L.. (2010). Soap-film Mobius strip changes topology with a twist singularity. Proceedings of the National Academy of Sciences
“It is well-known that a soap film spanning a looped wire can have the topology of a möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. experimental studies presented here show that this process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film’s plateau border and the centerline of the wire. we conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. the change in linking number is shown to be a consequence of a viscous reconnection of the plateau border at the moment of the singularity. high-speed imaging of the collapse dynamics of the film’s throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity. a number of challenging mathematical questions arising from these observations are posed.”
Walba, D. M., Richards, R. M., & Haltiwanger, R. C.. (1982). Total Synthesis of the First Molecular Möbius Strip. Journal of the American Chemical Society
Cador, O., Gatteschi, D., Sessoli, R., Larsen, F. K., Overgaard, J., Barra, A. L., … Winpenny, R. E. P.. (2004). The magnetic möbius strip: Synthesis, structure, and magnetic studies of odd-numbered antiferromagnetically coupled wheels. Angewandte Chemie – International Edition
“Understanding frustration: the control of structure through the choice of the template has allowed the synthesis of nonanuclear metal wheels that contain {cr8ni} or {cr7(vo)2} cores. magnetic studies (see picture) of one of these wheels shows that it behaves as a magnetic möbius strip. these are the first detailed magnetic studies of an odd-numbered ring larger than trinuclear and should help in the understanding of spin-frustrated systems.”
Pond, J. M.. (2000). Möbius dual-mode resonators and bandpass filters. IEEE Transactions on Microwave Theory and Techniques
“It is shown that a topological surface known as the mobius strip has applications to electromagnetic resonators and filters. using identical rectangles to construct a cylindrical loop and a mobius strip results in the path length of the edge of the mobius strip being twice the path length of an edge of the cylindrical loop. this path-length advantage is consistent with the electromagnetic analog of a mobius strip resonating at half the resonant frequency of the electromagnetic analog of the cylindrical loop even though b) they have the same mean diameter. dual-mode mobius resonators have been demonstrated in planar format and as wire-loaded cavities. two-pole bandpass filters have been constructed using these resonators. it is shown that these bandpass filters possess intrinsic transmission zeros that can be adjusted to enhance filter response. an equivalent circuit, which demonstrates excellent agreement with measured data, is presented and discussed”
Leweke, T., Thompson, M. C., & Hourigan, K.. (2009). Motion of a Möbius band in free fall. Journal of Fluids and Structures
Liu, W. M.. (1997). Is there a Möbius band in closed protein beta-sheets?. Protein Engineering
Show/hide publication abstract
“Protein beta-strands can form beta-barrels and other complicated structures. this paper defines bifurcations and pseudobifurcations of beta-sheets. they are important structural elements for protein folding. this paper also presents a characteristic number that can be used to test whether the surface of a closed beta-sheet is one- or two-sided. searching the whole protein data bank released in april 1997 with the definition of beta-structures given by the dssp program, we do not find any one-sided beta-möbius band. however, there are interesting structures such as beta-bands with odd number of antiparallel ladders and odd number of bifurcations. there are also beta-structures that are closed only at a singular point. adding a small patch near the singular point in different ways can make it a one- or two-sided surface. the catalytic triad of a gmp synthetase (1gpm) is near the singular point of such a beta-sheet.”
Cartwright, J. H. E., & González, D. L.. (2016). Möbius Strips Before Möbius: Topological Hints in Ancient Representations. Mathematical Intelligencer
“August m”obius discovered his eponymous strip — also found almost contemporaneously by johann listing — in 1858, so a pre-1858 m”obius band would be an interesting object. it turns out there were lots of them.”
Todres, R. E.. (2015). Translation of W. Wunderlich’s “On a Developable Möbius Band”. Journal of Elasticity
This project was funded by the 