# Geometries

In primary and secondary education, schools approach Geometry as a stable field of study, filled with trustworthy axioms and theorems that, when properly handled, help us understand space and the figures that, as result of algebraic calculations, occupy such space. Based on proportionality, Geometry is an efficient tool to solve a variety of problems, ranging from measuring minute objects to physics problems concerning vectorial information such as velocity and friction.

Geometry is also part of abstract calculations that involve graphs and deal with notions of limit, derivatives and integrals. Nevertheless, what is lost in Geometry’s appreciation is the understanding that its formulas have always depended on the progress of human thought and on historical circumstances through which premises gained or lost credibility. More importantly, turning a blind eye to this historical development erases the imprints that Geometry has left on human thought, on the way humans measure and organize the world and the knowledge they produce from it. Let’s remember, for the sake of illustration, that despite the existence of ancient representations of simple polyhedrons, dating back over 2000 B.C, mathematical theories concerning the representation of complex tridimensional forms were only formulated in the 19th century. Never before the advent of Geometry was it possible to predict with such precision the consequences of positioning cannons here or there in the battlefield and, therefore, it is no wonder that
the development of Descriptive Geometry by Gaspar Monge was classified as a military State secret for more than a decade during the French Revolution.

Furthermore, it would be possible to weave a genealogy of Geometry as war machine
with that of Geometry as cartography. Perhaps, it was because of the difficulties in representing the planet and the universe that humanity had many exciting debates involving geometry, in a trajectory that goes from Mercator’s cylindrical projection,
through Carl Friedrich Gauss’ theorem – which shows that a sphere can not be projected in a plane without distortion – to Bernhard Riemann’s studies that opened the way beyond Euclidean Geometry and allowed Albert Einstein to formulate his theory of relativity.

What could be said about the formulas that theorize the existence of many dimensions beyond the three or four dimensions we ordinarily use to understand space and time? Undetectable by current analytical tools, these unobservable dimensions could bear the key for the essential understanding of natural forces and of the ways matter and energy interact – facing them, what our common sense recognizes as the Universe is actually nothing more than a plane, flattened model. Moreover, how could we measure variables while conceiving the impacts of curvature notions on cosmological understandings? In this field, choosing closed, spherical, open or pseudo-spherical models can indicate, as a geometric analogy, the Universe’s ultimate destiny, whether towards total comprehension of matter, or its dissolution into sterile emptiness.

Regardless, it may be the artist’s mission is to revisit paradigms and central parts of history, and to emphasize models that represent thought and conceptions of the universe – H. G. Wells and Marcel Duchamp, for example, were deeply interested in notions concerning the fourth dimension and in theories of non-Euclidia Geometry, which, for the surrealists was a symbol of a renewed freedom from the tyranny of established laws. It is interesting that in many instances, artistic research and scientific/geometric studies meet around the concepts of models and maps, since they efficiently represent abstract ideas, as well as make visible aspects that could not be inferred by the simple accumulation of data. It is the possibility to generate models and maps that distinguishes Geometry among the fields of knowledge that work with abstract ideas and/or that escape the human capacity to visualize the space and time they live in. It was also this possibility that made Geometry the critical platform for the creation of avant-garde art and then the elaborate schemes conceived by conceptual artists such as Sol LeWitt.

Furthermore, this possibility is the foundation for this exhibit: a collective of works, chosen from an art collection that was put together having geometric art in mind, while simultaneously presenting a critique of the contemporary world, its institutions and abstract forces. This exhibit features works, by artists from different generations and different locales, that use Geometry – from the simple shapes that inhabit the Cartesian world to the complex forms from spherical Geometry – to give form to ideas and abstractions and/or to make visible that which goes beyond the limits of verbal discourse.

Having the Brazilian context in mind, such approach is rather welcome, implicating geometry in models and maps that influence perception, comprehension and disputation over reality. According to the paradigm implied in Brazilian art's master narrative, the speed in which the experiments involving geometric abstraction were introduced, questioned, accepted, consolidated and overcome is stunning, in a trajectory from
Mario Pedrosa's thoughts on international art to the Neoconcrete experiments, in less then one decade. Just in the beginning of the 50s (and, therefore, tardily) did this subject find its heroes – Waldemar Cordeiro, Amílcar de Castro and Ivan Serpa, who, in their own ways, outlined action programs that awarded art the positive nature of geometric reasoning... and, before the decade was over, Lygia Pape, Lygia Clark and Hélio Oiticica
had already experimented with the presence of abstract elements such as color, shape and line in the same space as the viewers' body, and from there stipulated the bases for what was then called environmental art, in which geometry loses its main role to the sphere of decisions, feelings and synthesis from a participating audience.

The problem with such a narrative built on the trajectory of an artist like Hélio Oiticica – whose first valued body of works consists of geometric paintings called Metaschemes – is that it suggests that Neoconcretism begins with the variables in geometric abstraction and
then ‘resolves’ and overcomes them, through experiments such as the Parangolés and the Penetráveis. Considering that all narratives have blind spots, we run out of alternatives for engaging and resolving the conflict that was a recurring subject in several parts of the western block after WWII, when the constructivism avant-gardes became simultaneously an oficial language promoted by the North-American cultural policies and a father-figure
to-be-destroyed. There is, in Brazilian history, a lack of personalities such as Eva Hesse, Franz Erhard Walther, and Gego, to name a few artists who promoted a relation of flirtation and desacralization of constructivism’s legacy. Instead of ‘overcoming’ geometry, these artists released it from a rational and scientific positivism, loosening its ideological and essentialist preconditions. Liberated from these markers, geometry became available for other ends, such as reorganizing everyday objects, replicating and parodying spatial models, or even choreographing the body’s movements.

The exhibition 'Geometries' organizes, within the restricted space of an apartment, a panorama that intersects Brazilian and international contemporary artists who don't directly resolve the impasse of geometric abstraction that the constructive avant-garde movements left us, but who actually liberate geometry from the narrative intrinsic to modern art history and deepen the bond with other possibilities for geometry in today's world. Let's start with the works by Gabriel Sierra, Jorge Macchi, Cristiano Lenhardt and Roberto Winter, exhibited in the first space of the apartment, interwoven visually and
spatially. All of their pieces are structured as simple Euclidian geometry shapes, such as cubes, pyramids and circumferences. (Untitled) The Donkey and the Carrot, by Gabriel Sierra, is the one that most closely references the constructive avant-garde movements,
formalizing a known dialectic tension between geometry and organicity, here represented by the play between precariously orthogonal planes and an apple. Corner Pyramid, by Cristiano Lenhardt and Mars Attack by Jorge Macchi echo this principle, but underline the understanding of geometric shapes possibly as a gesture that marks space and its qualities – the print folded into three planes embodies the corner as the gordian nod of ordinary architectonic space, the intersection of vertical and horizontal planes; while the pencils bent over themselves, and twirled to draw circles, dramatize the tension between the planarity of geometric representation and the materiality of the device that draws it.

Autonomy, by Roberto Winter, turns the cube into a model for demonstrating the incompleteness of purely visual perception, and the role of memory and reasoning in constructing an understanding of an object – the colored translucent acrylic piece shelters inside a printed X, which cannot be fully read from any point of view, since one of its segments is always hidden by the optical play between the colors of the cube's facets and the colors of the lines that form the X. Lastly, Map by Jorge Macchi gathers
line fragments in an orthogonal grid with different heights, resembling the structure of unfolded paper, which suggests an open map on the table. In each of theses works, shown at the very entrance of the apartment, the universally simple geometric shapes and volumes become test objects for equations between two or more physical variables within
the exhibition space, or between models that help visualize previously formulated ideas. In either case, the human mind's easiness to conceive and comprehend these shapes is a condition to the conversion between the models and its perception.

In the exhibition's opposite end, we find works by Damian Ortega, Tomás Saraceno and Cinthia Marcelle. While studying cases of non-Euclidian geometry, Pedro Barbosa recognized in some pieces of his collection shapes that could just as easily be tridimensional mock-ups of a knotted torus, a pseudo-sphere and a north-south meridian.
We can assume that the poetic intentions behind these works are not specific reflections on the discovery of geometric models, however, the similarity of their shapes with these models reveals something about their formal structure. In a time of fragmentation, simultaneity and simulacrum, the adoption of morphologies that resemble the discoveries of non-Euclidian geometry can be taken as a kind of reaffirmation of the appreciation for
the discourse complexity of several contemporary thinkers. More than an attack on positivist science, as was the case for modern artists, the incorporation of such shapes by contemporary artists seems to suggest the necessity to establish slower-reading models, incompatible with the comprehension dexterity typical of Euclidian shape and volumes. Thus, this group preserves the unity and cohesion found in the exhibition's first group of
works, while the fast nominal association when we face a cube, a sphere or a circle is delayed.

Crossing between these two geometric approaches, there are several cartographies designed by artists. As previously mentioned, making of maps and establishing frontier definitions represent some of the activities that make geometry a power instrument
over the world. The initiatives brought together in this exhibition explore such feature. Again, Jorge Macchi's Map synthesizes the morphology of a map in its reticular skeleton, evidenced by the paper's creases, while also suggesting an allegory to the imperial Spanish urbanization in Latin America, which repeatedly and systematically imposed the
orthogonal grid model as a paradigm for all the cities it founded in the American colonies. Not so subtle in its hints, Memory Link, the map of Nicosia, Cyprus' capital, by Rosangela Rennó, was created by asking people from different ethnic and political
groups to draw the urban frontiers over the on an image of the city's aerial view – the discrepancy between the overlapping lines marks how the territorial and sovereignty disputes contaminate the supposedly objective task of mapping. Between both cases, Moral census in the city of Recife, by Jonathas de Andrade, spatializes a map from the
pages of a street guide, and marks on it a series of addresses from which horizontally stacked lines lead to a questionnaire on good manners (appropriated from a good manners book from 1980 and administered by the artist to people from different
neighborhoods he visited in 2008). The underlying content of this mapping surpasses the usual socio-economic data used on urban planning, since it collects private opinions, precepts and prejudice.

Even so, the private reactions to the questions asked by the artist end up revealing a spatial and, above all, temporal cartography of the city, providing elements for a panorama of the political life – in its original meaning, the life of and in the polis
– facing the inequalities of multiple restructuring processes in the Recife urban environment.

1. As aproximações citadas neste parágrafo são devidas ao estudo perspicaz da pesquisadora da Divisão de Ciências Naturais e Matemática da Lesley University, Angela Vierling-Claassen, responsável pelo artigo "Models of Surfaces and Abstract Art in the Early 20th Century", e que, por sua vez, consultou Linda Henderson. "The Fourth Dimension and Non-Euclidean Geometry in Modern Art". Princeton University Press, 1983.

ARTISTAS: Ana Dias Batista, André Komatsu, Camila Sposati, Carla Chaim, Cinthia Marcelle, Cristiano Lenhardt,  Damián Ortega, Deyson Gilbert, Gabriel Orozco, Gabriel Sierra, Giovanni Ozzola, Gisela Motta, Jonathas de Andrade, Jorge Macchi, José Damasceno, Lawrence Weiner, Leandro Lima, Lia Chaia, Marcelo Cidade, Marcius Galan, Marila Dardot,  Mateo Lopez, Mona Hatoum, Nicolás Robbio, Paulo Nazareth, Reginaldo Pereira, Rivane Neuenschwander, Robert Winter, Rodrigo Matheus, Rosângela Rennó, Tomás Saraceno