http://www.metacafe.com/watch/331665/video

The Möbius strip is a mathematical construction demonstrating an evolution of a two-dimensional plane into a three-dimensional space; by merging the inner with the outer surface, it creates a single continuously curved surface. It allows returning to the point of departure after having completed a tour by following a path along its surface. This paradox can be explained by the fact that even though the strip has only one side, each point corresponds to two sides of its surface. 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 ouroboros and in contemporary consumer society as an icon of recycling.

Eight as a Symbol. The figure eight, also used as the symbol of infinity in mathematics, to graph relations between the numbers 5 and 4, 6 and 3, 7 and 2, 8 and 1 in an asymmetrical way. Recent studies of analemma which also mimics the figure eight show that its asymmetry is a result of the sun being projected to the curved surface of the Earth, but purging this deformation produces a symmetrical figure eight. Numbers between one and eight are paired as twins in multiplication tables: in this way, number nine is the sum of numbers 5 and 4, 6 and 3, 7 and 2, 8 and 1; presented in the sigma code (the sum of digits of a number), the pairs of numbers will always sum up to 9 even if both of the twins have been multiplied. These relations can be graphed in the form of a figure eight or a Möbius strip: when such a graphed numeric orbit flows through the inflection point (number 9) in a clockwise sense, it subsequently reverses itself into its own twin partner, continuing the flow through an anticlockwise orbit. Quite unlike the stationary circles, energy is released into the numbers so that they spin, one out of the other ... the bending and twisting in and out of separate energies, the big and the small, connected by a continuous movement through the eye at the centre of the storm of numbers."

Symbolically, the eternal revolving is traditionally symbolised by the ouroboros, representing a continuous circle of creation. The circumference completes the centre to suggest the idea of God. Being a symbol of the manifestation and cycles, the ouroboros represents unity, self-nourishment, union of matter and spirit; in hermetic tradition, it symbolises the union of Isis and Osiris - the Female and the Male principle represented also by two intertwining serpents related to the Sun and the Moon; as such, it has also been extensively used in Alberti's architecture. It symbolises a dialectics of life and death, the dynamics of circle, infinite movement, universal animation and is therefore extremely interesting as a subject of research in architectural animation. The ouroboros is a creator of time, duration and life and continuously returns to itself. The alchemists' Big Whole is a cosmic spirit, a symbol of eternity and cyclic time, also used by as a symbol of Divinity. An outstanding parallel can be drawn to the Zen tradition, based on the dynamic sphere of the two opposite principles in a perpetual interaction, the Yin and the Yang.

INTRODUCING THE DOUBLE MÖBIUS STRIP. A continuous topological entity results if the simple band with a continuous (yet still a composite) surface is duplicated and consequently transformed by applying a negative scale (x = -1, y = -1). This inversion results in a closed, continuous structure that we called the Double Möbius Strip. The same process can be applied to a simple twisted band describing a volume. 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. 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. 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.

The Möbius strip is a mathematical construction demonstrating an evolution of a two-dimensional plane into a three-dimensional space; by merging the inner with the outer surface, it creates a single continuously curved surface. It allows returning to the point of departure after having completed a tour by following a path along its surface. This paradox can be explained by the fact that even though the strip has only one side, each point corresponds to two sides of its surface. 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 ouroboros and in contemporary consumer society as an icon of recycling.

Eight as a Symbol. The figure eight, also used as the symbol of infinity in mathematics, to graph relations between the numbers 5 and 4, 6 and 3, 7 and 2, 8 and 1 in an asymmetrical way. Recent studies of analemma which also mimics the figure eight show that its asymmetry is a result of the sun being projected to the curved surface of the Earth, but purging this deformation produces a symmetrical figure eight. Numbers between one and eight are paired as twins in multiplication tables: in this way, number nine is the sum of numbers 5 and 4, 6 and 3, 7 and 2, 8 and 1; presented in the sigma code (the sum of digits of a number), the pairs of numbers will always sum up to 9 even if both of the twins have been multiplied. These relations can be graphed in the form of a figure eight or a Möbius strip: when such a graphed numeric orbit flows through the inflection point (number 9) in a clockwise sense, it subsequently reverses itself into its own twin partner, continuing the flow through an anticlockwise orbit. Quite unlike the stationary circles, energy is released into the numbers so that they spin, one out of the other ... the bending and twisting in and out of separate energies, the big and the small, connected by a continuous movement through the eye at the centre of the storm of numbers."

Symbolically, the eternal revolving is traditionally symbolised by the ouroboros, representing a continuous circle of creation. The circumference completes the centre to suggest the idea of God. Being a symbol of the manifestation and cycles, the ouroboros represents unity, self-nourishment, union of matter and spirit; in hermetic tradition, it symbolises the union of Isis and Osiris - the Female and the Male principle represented also by two intertwining serpents related to the Sun and the Moon; as such, it has also been extensively used in Alberti's architecture. It symbolises a dialectics of life and death, the dynamics of circle, infinite movement, universal animation and is therefore extremely interesting as a subject of research in architectural animation. The ouroboros is a creator of time, duration and life and continuously returns to itself. The alchemists' Big Whole is a cosmic spirit, a symbol of eternity and cyclic time, also used by as a symbol of Divinity. An outstanding parallel can be drawn to the Zen tradition, based on the dynamic sphere of the two opposite principles in a perpetual interaction, the Yin and the Yang.

INTRODUCING THE DOUBLE MÖBIUS STRIP. A continuous topological entity results if the simple band with a continuous (yet still a composite) surface is duplicated and consequently transformed by applying a negative scale (x = -1, y = -1). This inversion results in a closed, continuous structure that we called the Double Möbius Strip. The same process can be applied to a simple twisted band describing a volume. 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. 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. 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.