According to Peterson [1], the astronomer and mathematician August Ferdinand Möbius (1790-1868) studied the development of a geometrical theory of polyhedra and identified surfaces in terms of flat polygonal pieces variously glued together. The curious single continuous surface named after him has only one side and one edge, and is made by twisting the band by 180° then joining the two ends. When following the path of its surface, one can reach any other point without ever crossing an edge. The phenomenal one-sidedness of the band can be explained if its surface is divided into a row of triangles and subsequently twisted and joined at its ends. Möbius published his discoveries in 1865 in his paper "On the Determination of the Volume of a Polyhedron", also revealing that there are polyhedra to which no volume can be assigned. But it was only in the past century that attention in mathematics was drawn to studies of hyper- and fractal dimensions.
Another interesting situation appears when a band with joined
ends is cut in half lengthwise until getting back to the starting
point: a single band twice as long as the original is produced
if its ends have been rotated for 180°, but rotating its
ends for 360° forms two interlocking rings. Yet the Möbius
strip is far more than just a mathematical abstraction. Symbolically,
its two-dimensional projection forming the figure eight represents
infinity and cycles, but can also be found in many natural phenomena
related to fluid dynamics and the analemma. The latter is known
as a representation of the virtual path of the sun projected
to the surface of the Earth. It reveals the dynamics of sunlight
as a source of our vision and an instrument of construction of
our space-time perception. Representing temporality, the cyclical
nature of processes and eternity, it is no wonder that the twisted
ring is an archetype, a symbol of infinity, present both in alchemistic
iconography as the serpent biting its tail, named the
Symbolically,
the eternal revolving is traditionally symbolised by the
Hartmann [8] argues that the Möbius strip subverts the normal, i.e. Euclidean way of spatial and temporal representation, seemingly having two sides, but in fact having only one. At one point the two sides can be clearly distinguished, but when you traverse the strip as a whole, the two sides are experienced as being continuous. Lacan also employs the Möbius strip as a mode to conceptualise the "return of the repressed" as well as to illustrate the way psychoanalysis reconceptualises certain binary oppositions (inside/outside, before/after, signifier/signified, etc.). In Lacanian terminology, it is by suturing off the real that the reality of the subject remains a coherent illusion that prevents him or her from falling prey to the real. Similarly, reversible images that have first been studied in Gestalt psychology and explored by many artists, carry a double meaning, their significance shifting from one figure to the other in loops just like moving through a Möbius strip topology. If reversible images are said to demonstrate the dynamics of human visual perception and visual thinking by opening a two-dimensional surface into a three-dimensional space, introducing time into spatial dimensions, a similar principle could be identified in Möbius strip: it exists between dimensions, i.e. in fractal dimensions. Introducing topology into architectural space, its curious geometry has made architects rediscover the Möbius strip in the age of information society, topological architecture and hypersurfaces. As a consequence of the paradigmatic shift, generated by the electronic revolution, both science and culture began to question their Cartesian foundations. As a result of the introduction of non-linear dynamics, relativity, indeterminacy and topological geometry, architecture started to reflect dynamics and animation as a cultural condition rather than the principle of stasis, establishing a new formal vocabulary. Within the physical sciences, topology theory is regarded as the rubbermath or rubber sheet geometry. It is also a conceptual and qualitative paradigm - deforming shapes without destroying their essential properties. Topology theory entails deformation processes such as pulling, twisting, stretching, turning and contorting, but no configuration of shapes using cutting, tearing or pasting. The idea of a structure containing a loop appears in projects for a Möbius House by Stephen Perrella, by Architectonics or by UN Studio and the Möbius strip design at Laguna Beach by Perrella, Robert and Petresin. Here, the Möbius strip is used as a diagram for post-Cartesian dwelling and is neither an interior space nor an exterior form: it is a transversal membrane reconfiguring these binary notions into a continuous, non-linear form, thus creating a basis of a complex, temporal experience. It is interesting how the dynamics of dwelling can be described as a cyclical activity or a process in the interior and exterior space, tied to fixed trajectories on a torus. Attractors in a vertical and a horizontal direction can move in loops or figures of eight around such a torus. By analogy, the dualities in contemporary dwelling, e.g. the concepts of inside/outside, below/above, physical/virtual, private/non-private, material/media can be represented as such attractors; these form twisted tracks as on a Möbius strip where one end is twisted and then joined to the other end to create a single continuous surface. The Möbius diagram adapts a metaphorical looping and by its conversion of binary notions in architecture, it offers new possibilities for architecture where the inside of the object becomes the outside; where the media penetrates the private sphere, the unique is tied to the general through a trajectory of this virtual form, where floor become walls become floor or where a place is reversed into a non-place. This is why the Moebius strip has become a geometrical structure par excellence of topological architecture, being a relatively adequate representation of dwelling activities.
In order to model a surface, a minimum of two curves are needed to define its edges; the problem of the Möbius strip is that its single continuous surface only involves one curve along the band's edges in a Cartesian space - a paradox that discontinues processes in the modeling programme. A solution would therefore be to use a minimum of two curves: let us cut the Möbius curve into two isolated curves and reintegrate them so the computer can recognise them as a composed object, thus enabling its calculation and building a surface. A regular surface being a flat, two-dimensional construction in a Cartesian space, its normals are perpendicular to its surface having a common orientation (positive or negative). A digitally constructed surface is divided into faces, their direction being defined by their normals. A surface can only be closed when the normals of its two extreme faces have identical orientation. But the surface of a Möbius strip that we tried to create by lofting the composite curve resulted in a paradoxical formation: the normals of the faces at the adjoining edges of the band have opposite direction that makes it impossible to join the band's borders thus closing the surface. Modeling an infinitely continuous surface in a virtual environment that is limited to being an open-ended structure means that it can not be recreated in a Cartesian space. Assigning a curve having a singular orientation to the surface of the Möbius strip to create a volume (i.e. a corridor) results in a similar paradox: if the curved profile of the structure following the path of the Möbius surface has positive direction at the starting point of the translation, it will have a negative direction after having completed the path and revolving to the starting point. Obviously, studying a form that exists between the second and the third dimension relates to a fractal quality, but is there a way of taming the computer one more time? Instead of using a single profile, the curve can be copied, applied at the scale of -1 and translated along the path of the Möbius surface to define a new volume. The section of this new structure contains rotational or mirror symmetry regarding the normals of the Möbius strip faces, meaning that its adjoining edges can be glued together thus forming a single continuous tubular volume or a warped torus.
The fractal geometry of the double Möbius strip becomes
evident by identifying its characteristics as applied to architectural
space. If the two intertwining bands of the double Möbius
strip represent the structure of a wall and a floor, the two
architectural articulations can become mutually entangled and
exchanging while following a path along the two surfaces. Starting
on a horizontal surface representing the floor results in moving
along the surface of the wall become floor after having completed
a tour and vice versa. It is also interesting to notice that
entering the structure from a particular band always results
in revolving to the starting point of the identical band. If
the double Möbius structure is transparent, its users can
walk along the paths of both strips representing floor-become
walls-become floor and never interfere with the users starting
from a different entrance placed at the other strip as the surfaces
never intersect ( These interesting characteristics do not only indicate the non-Euclidean nature of the double Möbius strip, thus making it an ideal polygon for formal and conceptual research in post-Cartesian architecture, they also refer to a possibility of reversing and even uniting archetypal binary notions of surface/volume, space/time, inside/outside, matter/media etc. In architecture based on topological geometry, the traditional articulation of surfaces forming ceilings, walls and floors is absent as the structural members known in classical composition melt into a fluid, dynamic entity. Let us step back and consider another interesting analogy that can be established between the double Möbius strip, its principle of unity through perpetual duality, and the DNA (discovered by Watson, Crick and Wilkins): the model of the double helix is composed of two serpent-like intertwining spirals, representing a biological reflection of the archetypal idea of time as a spiral, creating a reunion of the linear and the cyclical aspects of time as a perpetual flow. The idea of unity through continuity also appears as the most important characteristic trait of structure in Lacan's lecture on Structure & Reality.[9] If the previous Gestalt notion of good form has been related to its function of joining and producing the "unifying unity", Lacan on the other hand reconsiders good form as a "countable unity" one, two, three etc., as an integer that can be used for counting with the formula (n + 1). In this way, the question of "one more" actually becomes the key to the genesis of numbers and has been applied in our attempt to generate the double Möbius strip.
Similarities in form and movement of the double Möbius
strip structure can be established with the structure and the
nature of formation of a vortex. This fractal phenomenon of fluid
dynamics has brought forward a connection with our previous research
in applications of processes of particle physics to create and
control deformations (
Using the principles of particle physics in animation software allows architectural composition to exceed its Cartesian limitations as well as enable extremely complex topological architecture a quantifiable descent into the built environment. In fact, topological surface-based forms bring forward the issue of mathematically determined space, be it Euclidean or Riemannian, and its relation to organic form. Cartesian and Euclidean notions of space seem to be no longer
accurate to describe topological architecture. Classical architectural
composition relies on mathematical tools to create or analyse
the generation of form. In a computer generated architectural
environment, the language of mathematics can once more become
a powerful instrument of non-linear composition and a way to
avoid purely random search for beautiful images; forces within
a field can be programmed to simulate processes known from fluid
dynamics and particle physics ( Dynamic systems theory mainly relates to non-linear processes in motion and their behaviour. Such systems are known to have inseparable space-time, a process-dependent, contingent observer and non-local causality. Fields that give an input to the system function as complex, non-linear, multidimensional networks, where every particular action is linked to that of another entity and every moment is irreversible. They are composed of delocalised entities, i.e. the particles with mutually entangled space-times. The particles are dynamic, fluid and can change in the course of development either as a result of feedback or directly as a response to the environment; these changes influence subsequent behaviour of the system. It is obvious that such premises are highly relevant for a creation of a responsive (interactive) environment with an entanglement of the observer and the observed. Our research was oriented towards operations in which the object (or a particle within a field) is immersed as agent within general conditions of creating forms, processes, and fluxes. Processes of metamorphosis such as the transformation, the
dynamic principle in which destruction of a particular state
is necessary in order to proceed discretely to the next level
or a different form of manifestation have been the key elements
of our research. In animation and creation of special effects,
principles of particle physics are integrated within the software
to simulate natural phenomena, but it is also possible to use
them to demonstrate particular processes and their effects in
a virtual environment as applied to the structure of a double
Möbius strip. The reactions of the construction points of
a surface or a volume within a field can be controlled and traced
individually as well as exposed to various physical processes.
Fields are forces used to animate the motion of particles. We
have defined every construction point or vertex of a surface
as a particle with its own identity and characteristics, subsequently
exposing them to processes of physical and mathematical force
fields ( In a zero frame (0,0), proper identities can be defined for
each particle and their relative force fields. We have not been
operating from a point of view of a surface deformation, but
from singular dimensionless entities activated in a space-time
system that allow its evolution. A series of independent movements
causing system transformations within the time line has been
created ( In physical architectural space, mechanical and electronic systems such as joints and sensors can be applied to structures to create a kinetic architecture to simulate particles and field activities. A great example of a computer generated process translated to a built structure is the Aegis Hyposurface by dECOi (10). The kinetic potential of inert architectural form can be studied using animation software and motion capture. The studies in Möbius strip raise further questions on our perception of space and the mechanisms of optical illusions, but they have also been aimed at re-establishing the central role of the observer/user as known from the relativity theory, thus integrating the human scale within a culturally and technologically rapidly evolving environment. Human activities could be monitored as force fields creating an input to the space and its meaning. Therefore the role of the observer/user adapting to his environment can be compared to the idea of developing intelligent spaces interacting with their users.
return to text[2] G. Jobin and F. Treichler. [3] Cecil Balmond, [4] E. Petresin, [5] Balmond, [6] Nicholas of Cusa, I [7] R.R. Baldino and T.C.B. Cabral, "Lacan's
Four Discourses in Mathematics Educational Credit System",
http://s13a.math.aca.mmu.ac.uk/Chreods/Issue_13/4Discourses.
[8] Bernd Herzogenrath, "Digression 2: How to Make
a Moebius Strip", in "On
the Lost Highway: Lynch and Lacan, Cinema and Cultural Pathology."
[9] J. Lacan, "On
Structure & Reality" (lecture), http://lacan.com/hotel.htm.
[10] dECOi: Aegis Hyposurface, Birmingham Hippodrome Foyer Artwork Competition (1st prize), 1999.
decoi)and
Tom Kovac and has worked for Stephen Perrella's Hypersurface
Systems. With Laurent-Paul Robert, they formed Rubedo in 2001,
and have published and exhibited internationally. Her current
research interests include visual perception and design theories,
as well as the impact of emergent technologies on architecture.Laurent-Paul
Robert is a French/Swiss artist and
computer animator based in London. After having worked as an
artist, performer and musician, he has continued in 3D animation
since 1994, becoming the lead art director and senior animator
for Arxel Tribe. He also worked in architecture with Stephen
Perrella's Hypersurface Systems, and has continued his design
and research with Vesna Petresin as Rubedo since 2001; he has
exhibited and published internationally. He is currently working
as a consultant for clients such as Arup and
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