Leonardo
My Transformable Structures Based on the Möbius Strip
Author(s): Enrique Carbajal G. Sebastián
Source: Leonardo, Vol. 8, No. 2 (Spring, 1975), pp. 148-149
Published by: The MIT Press
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Leonardo, Vol. 8, pp. 148-149. Pergamon Press 1975. Printed in Great Britain
MY TRANSFORMABLE STRUCTURES
BASED ON THE MOBIUS STRIP
Sebastian(EnriqueCarbajalG.)*
In 1966 I made a number of small sculptures of surface of the elements to add interest to the
folded paper and cardboard. These objects, which various forms into which it can be transformed
I call desplegables (Spanish for 'unfolding'), were [5].
responsible for directing my attention to the branch My earlier studies of crystallography [6] and my
of geometry called topology [1, 2]. I read about experience with the Mobius strip influenced the
topological spaces with the hope of finding some- design of my next object. I made of paper a ring
thing to serve me in my art. In this quest I became of tetrahedrons that can turn on its central axis.
acquainted with and much fascinated by the I then devised a scheme (called 'structural strategy'
Mobius strip. It can be described as follows: A long
rectangle (e.g. a strip of paper, 2 x 30 cm.) is held
fixed at one end and turned through 180? at the
other end. The two ends are then joined yielding
a one-sided surface [1]. On such a surface, if one
places one's finger at any point on it and then
moves the finger along the strip, the finger will
return to the starting point after traversing the
surface on both sides of the untwisted strip.
My first object, a folding sculpture, (1967)
utilizing the idea of twisting and joining in a ring
(40 cm. diameter) was made from a paper strip
containing creases delineating a chain of equi-
lateral triangles. This strip was given three twists
(540?), instead of just one (180?), as in the M6bius
strip. The resulting one-sided surface can be
transformed, by folding, into five different 3-
dimensional forms and one 2-dimensional shape
that has a hexagonal perimeter; hence the name Fig. 1. 'Estructura Articulada', transformable sculpture
that I gave to this type of construction is 'Hexa- (one of its possible forms), wood, lacquer, 30 x 30 x 30 cm.
flexagono'. Art objects made of elements that can (1970).
be rearrangedby a viewer are called transformables
[3] and paintings and sculptures of this kind have
been devised by numerous artists in recent years [4].
I then wished to see whether other objects with
flexion characteristics similar to those of a 'Hexa-
flexagono' could be made and whether other
polygons might be formed. I found that they could
and I devised a procedure in which the folding and
twisting were varied. Although the formation of a
Mobius strip represents one stage in their fabrica-
tion, the topological property of the Mobius strip
is not preserved in the final folded objects. The
'Tetraflexagono', which I constructed in this way,
can be arranged into a 2-dimensional square and
into other forms including that of a cube. I have
prepared a printed edition of a 'Tetraflexagono',
with colored circular and linear stripes on the
Fig. 2. 'Cubo Hexac6nico', transformable sculpture (one of
* Artist living at Cda. Protasio Tagle 33, Mexico City, its possible forms), cardboard, acrylic paint, 30 x 30 x 30
18 D.F., Mexico. (Received 19 Nov. 1973.) cm. (1970). (Cf. Figs. 3 and 4.)
148
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Sebastian (Enrique Carbajal G.) 149
by Jiirgen Claus [7] for cutting, folding and twisting
a chain that permits the making of transformables
with geometrical and architectonic forms (Figs. 1
and 2) [8]. Figs. 2 and 3 show two of the trans-
formations obtained with the object 'Cubo Hexa-
gonico'.
Fig. 4 shows the final layout for 'Cubo Hexa-
gonico', which represents the first step in making
one of my paper transformables. To arrive at a
layout that leads to a feasible 3-dimensional trans-
formable is quite complicated and requires con-
siderable experience.
My ideas about architectonic structures were in-
fluenced by reading a book on architecture by
Michel Ragon [9] and by the sculptures and archi-
tectural work of Mathias Goeritz [10, 11]. I am
much interested in the housing of people and, in Fig. 4. Layout for the transformable sculpture 'Cubo
particular, in the need to adapt housing to radically Hexac6nico'. (Cf. Figs. 2 and 3.)
different ways of living. Thus, I have begun to
investigate habitable underground, underwater and air-borne structures. I am aware of the many
obstacles (technical, psychological, economic and
political) that confront the execution of such pro-
jects. However, I hope that my studies, including
the construction of transformables, will contribute
to the development of the architecture of the future.
REFERENCES
1. Hu Sze-Tsen, Homotopy Theory (New York: Academic
Press, 1959).
2. A. H. Wallace, An Introduction to Algebraic Topology
(Oxford: Pergamon Press, 1957).
3. Terminology, Leonardo 2, 81 (1969).
4. K. Martin, Construction and Change: Notes on a
Group of Works Made between 1965 and 1967,
Leonardo 1, 363 (1968).
5. E. C. G. Sebastian, Tetraflexagono, Eduque jugando
(Mexico City: Universidad Nacional Autonoma de
Mexico, 1969).
6. E. Flint, Principios de Cristalografia (Moscow: Paz,
1966).
7. J. Claus, Expansion del Arte (Mexico City: Extem-
poraneos, 1970).
8. Homenaje a los 5 cuerpos regulares, exhibition cata-
logue (Mexico City: Instituto Nacional de Bellas Artes
y Literatura, 1974).
9. M. Ragon, Los Visionarios de la Arquitectura (Mexico
City: Siglio XXI, 1969).
10. M. Goeritz, The Sculpture 'The Serpent of El Eco':
A Primary Structure of 1953, Leonardo 3, 63 (1970).
Fig. 3. 'Cubo Hexac6nico', transformable sculpture (one of 11. C. B. Smith, Builders in the Sun (New York: Archi-
its possible forms). (Cf. Figs. 2 and 4.) tectural Book Publ. Co., 1967).
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