Euclidean geometry is flat- it is the space we are familiar with- the kind one learns in school. Non-Euclidean geometry is more like curved space, it seems non-intuitive and has different properties. It has found uses in Science such as in describing space-time. It has also been used in art, to lend a more other-wordly, non-conformist feel to the work, especially during the Surrealist movement. I shall delve into the use of such non-Euclidean geometry in art. For the unfamiliar eye, what is non-Euclidean geometry? How would one define and identify such geometry? What are examples of it being used in art? How was art influenced by developments in theoretical physics? What is the artist trying to convey and how does non-Euclidean space help him? Do these ideas align themselves with a larger purpose?

*Non-Euclidean Geometry *by i2ebis on *deviantart.com* depicting the fictional city ofR’lyeh from H.P. Lovecraft’s novel *The Call of Cthulhu ^{[1]}*

The first question that comes to one’s mind is what does non-Euclidean space look like? To provide a familiar example, one can think of the surface of a sphere such as the earth. To find the distances between two places, we would not draw a straight line between them, instead we would have to traverse the surface of the earth. If we follow the same principle and draw a triangle on the surface , the sum of the angles would not necessarily be 180° as is the rule in Euclidean geometry that we are familiar with. This raises the need for a new-Geometry. This new geometry- Euclidean space has gone on to find application in Mathematics and Physics. Kinematic properties and concepts such as the world line and proper time have been explained in Minkowski space using hyperbolic geometry, the space-time membrane in Relativity uses such non-Euclidean geometry as well. It is found in nature, the Folded Coral Flynn Reef is an example.

Euclidean geometry was formulated around 300B.C. based on Euclid’s postulate as stated in his first book of the *Elements.* His postulates state that:

1.A straight line can be drawn joining any two points

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.

4.All right angles are congruent.

5. If two lines are drawn that intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, the two lines inevitably must intersect each other on that side if extended far enough.

The space created in which this parallel postulate does not hold is termed non-Euclidean space. Its surface is not flat and it does not have null curvature. Examples of this are Hyperbolic space and Elliptical space.

Mathematicians were troubled by the fact that they could not prove Euclid’s fifth postulate. From Ibn al-Haytham in the 11^{th} century to Giovanni Saccheri in the 18^{th} century, they all failed. It was only in the 19^{th} century that we begin to see steps in the creation of non-Euclidean geometry by Carl Gauss and Ferdinand Schweikart independently around 1818. Around 1830, Janos Bolyai and Nikolai Lobachevsky separately published treaties on non-Euclidean geometry. Bernhard Riemann in 1858 formulated the geometry in terms of a tensor, allowing it to be used in higher dimensions as well. Einstein’s Relativity shot the hyperbolic space to fame. Once such space had caught the attention of the human mind, art could not have evaded its clutches. ^{[2]}

An easy way of identifying non-Euclidean geometry is by studying two lines perpendicular to a third line.In Euclidean space, the lines remain at a constant distance from each other even when extended to infinity. This geometry is characterised by zero curvatureIn Hyperbolic space, the lines curve away, increasing in distance as we move away from the third line. This is characterised by a constant negative curvature. In Elliptical space, the lines curve towards each other, with a positive constant curvature.

Another way of identifying this is observing the sum of internal angles of closed figures such as triangles. In Euclidean space, the sum of the angles is equal to 180°, in Hyperbolic space it is less that 180° and in Elliptical space, it is greater than 180°

To elucidate this matter, one can imagine a stretchable membrane with a square grid drawn on its surface. When this fabric is placed on a flat surface, the squares will remain intact. If we were to distort the fabric by pushing our finger upwards at the centre from underneath the fabric, the grid will distort and the squares will be transformed to other shapes due to the elongation along certain axes. If we were to distort the fabric to a much greater extent by pushing our finger further, the shapes at the centre will face more elongation appearing larger due to the strain. For a more accurate description, one can imagine a three-dimensional hyperbolic curve.

A clearer physical manifestation of this idea is in explaining the curving of space time fabric. Without the influence of mass, Einstein’s space time fabric would be flat. Once we introduce a massive body, due to gravitational effects, as explained by the Theory of Special Relativity, the fabric will bend.

It was earlier thought that Euclidean geometry was the only true depiction of the universe. With the progress of Maths and Science, we now know that the universe can also be described using non-Euclidean geometry. It has been used in art. The artist, who attempts to describe the universe, a feeling or an idea also began using this newly conceived geometry around the time when Einstein’s Relativity came to the fore-front. Whether the artists always knew and/or understood the kind of geometry they were using is debatable.

In *Manifeste Dimensioniste* written by Charles Sirato in 1936, the theories given by Eisnetin are cited as the impetus for “Dimensionisme”. The manifesto declares that ,

“Animated by a new conception of the world, the arts in a collective fermentation have begun to stir. And each of them has evolved with a new dimension. Each of them has found a form of expression inherent in the next higher dimension, objectifying the weighty spiritual consequences of this fundamental change, Thus, the constructivist tendency compels:

i.Literature to depart from the line and move to the plane.

ii.Painting to leave the plane and occupy space: Painting in space, Constructivism, Spatial Constructions, Multimedia Compositions

iii.Sculptre to abandon closed, immobile, and dead space, that is to say, the three-dimensional space of Euclid, in order to conquer for artistic expression the four –dimensional space of Minkowsky.” ^{[4]}

In order to understand the use of non-Euclidean space in art, we should attempt to look at works of art with a geometrical perspective in mind. The figures below depict a plane of regular hexagons rendered in Euclidean and hyperbolic space. The number of tiles in each subsequent layer sees a linear increase in Euclidean space whereas the same is not observed in hyperbolic space. Through our Euclidean glasses, the tiles grow smaller in the second figure, where as they are actually all the same size in hyperbolic space. Although the boundary of the second figure appears as a finite circle, in hyperbolic space it lies at infinity.

^{[5]}

M.C. Escher used this concept in his works of art. The first is titled “Regular Division of the Plane” and the second is “Cirlce Limit IV”. Once again, the angels and demons are the same size in both the figures, but because we view the hyperbolic space of the second with “Euclidean glasses” since it is rendered on a Euclidean plan, as we move away from the centre, the figures appear to grow smaller and smaller.

When we move into the depiction of such geometries in art, we can ask ourselves a couple of questions-, how did the use of such geometry come about in art, why has the artist used this geometry, how does it “add” to the image when compared to if Euclidean geometries were used, what idea does the artist try to bring forth with the use of such space, what effect does it have on the viewer’s mind, how did it influence thought in that age etc…

Linda Dalrymple Henderson throws light on these answers and much more in her book *The Fourth Dimension and Non-Euclidean Geometry in Modern Art *published in 1984. According to her, Andre Breton and various surrealists (particularly the French surrealists) acknowledged Einstein’s theories and retained implications from non-Euclidean geometry with its main advocates being Duchamp, Cubists like Metzinger and Gleizes, Russian poet Khlebnikov and painter El Lissitzky, Benjamin de Casseres, Dada founder Tristan Tzara and later the surrealists. For them, non-Euclidean geometry signified a ‘new freedom from the tyranny of the established’. “Codified in Poincare’s philosophy of conventionalism, this recognition of relativity of knowledge was a powerful influence on early twentieth century thought. In this way, artists who depicted the fourth dimension too owed something to non-Euclidean geometries that prepared the way for the acceptance of alternative kinds of spaces. By 1930, the concept of the fourth dimension and non-Euclidean geometry had played a vital role in the development of modern art and theory.” ^{[7]}

Henderson goes on to talk about how the new geometries suited their arguments^{ [7]}. Andre Breton associated it with his ideas of a new “surreality”. Freud’s analysis of the unconscious mind was a major source for this theory which was further accompanied with themes related to non-Euclidean geometry. It was a method of trying to escape the control of reason. He carried on the Dada attack on logic and reason. In 1936, Non-Euclidean geometry was officially incorporated into this Surrealist attack on reason and logic. Gaston Bachelard cited Lobachevsky’s non-Euclidean geometry as one of the sources for “surrationalism” and went on to argue that human reason must be restored to its function of turbulent aggression. Many surrealist painters shared this view and they used non-Euclidean geometry as another support for rejecting established laws. This is reflected in titles of works such as Yves Tanguy’s *The meeting of paralles* and Maz Ernst’s *Young Man Intrigued by the Flight of an on-Euclidean fly. *Salvador Dali’s* The Persistence of Memory* has non-Euclidean overtones, in his book* The Conquest of the Irrational *he discusses the watches in context of his comments on non-Euclidean versus Euclidean geometry and the theories of Einstein. He described the melted watches as “the extravagant and solitary Camember [cheese] of time and space”.

Clearly some artists used non-Euclidean geometry to fight against conformism and convention and to question established notions of reality. They aligned this new geometry to their ideals and used it to express they sense of liberation from established ideas. It also added to a feeling of surreality in their works. Something beyond what we have seen, sometimes even something beyond what we can imagine. If painters used their art to convey such complex ideas, where they successful? When viewing such work, were such ideas interpreted and emotions felt as well? To find an answer to this, I picked *Le Diabolo *and asked around a dozen batch mates without any idea about such geometries what they thought about the painting. I asked them what they thought about the kind of world that is depicted in the painting and what they particularly thought of the building.

Le Diabolo by Jacques Resch ^{[8]}

For people who have no idea of non-Euclidean geometry, their views on the painting depict that the artists message was conveyed. Some of the answers I received were: unconventional way of drawing buildings, spiralling, disorder, weird, skewed,lack of dimensionality, stretched and turned, strange, no science, don’t know how to make sense of it, being churned and muddledcurves and twists, claustrophobic, architecture doesn’t follow traditional boundaries, how is it possible?, can’t be real world, twisted, no sense of alignmet, someone’s twisted dream, things go round and round, chaos, sucked in, no linear perspective, unusual zig-zag, no single shape, land of wonder, tearing apart different aspects of life, distorted vision, viewing through a convex/concave lens, dream-like, convoluted, despair, explosive, ,converging and diverging, unnatural proportions, curvature, being stretched, through a vacuum tunnel, space time is warping, being away from reality, etc…

It is quite clear that even though the viewers had no idea of non-Euclidean geometry, the idea of such space is successfully conveyed. A viewer may not understand the mathematics, he may not even have a clue of the presence of mathematical elements, yet the idea and essence of such elements is successfully translated.

If this is such a powerful medium, why don’t we see the use of such space in more recent works? Henderson explains the trend of moving away from the depiction of non-Euclidean geometry in art certain antipathy among young American artists in the 1950s and 1960s.^{[7]} According to her, Barnett Newman’s * The Death of Euclid* is a generalized rejection of all geometry and not a tribute to non –Euclidean principles. She says that the next generation of painters whose styles emerged in the 1960s were so far removed from the period dominated by bon-Euclidean ideas that they were totally unaware of the importance of the new geometries for early modern art. The withering away of such depictions are explained by her as the modernist preoccupation with flatness that continued to discourage purposeful evocations of space as a goal in painting. Finally, spatial illusion was almost banished from modern painting with the Minimalist movement of the 1960s.

One cannot really complain about disappearing geometries in art as the very nature of artistic movement lies in the phasing in and phasing out of certain concepts and ideas. What one can take away from a study of such geometries in art is the fact that art, maths and science and intrinsically interconnected. According to Tony Robbin, art goes beyond mathematics or physics per se. He says,” Artists who are interested in four dimensional space are not motivated be a desire to illustrate new physical theories, nor a desire to solve mathematical problems. We are motivated by a desire to complete our subjective experience by inventing new aesthetic and conceptual capabilities…Our reading of history of culture has shown us that in the development of new metaphors for space, artists, physicsts, and mathematicians are usually in step…”

Sources:

* ^{[1] }*i2ebis ,

*Non-Euclidean Geometry*on

*deviantart.com*: http://i2ebis.deviantart.com/art/Non-euclidean-geometry-170697104

^{[2]}wikipedia,

*Non-Euclidean Geometry: History :*http://en.wikipedia.org/wiki/Non-Euclidean_geometry

^{[3]}Alexis Karpuzos,

*The Fourth Dimension in Art & Science:A visual narrative of the relationship between Science and Art, during the 20*, Think Labs, Athens 2013 : http://www.slideshare.net/akarpuzos/the-4th-dimension-in-art-science

^{th}Century^{[4]}Charles Sirato ,

*Manifeste Dimensioniste*,Revue N+1 ,1936

^{[5]}Caroline Series andDavid Wright,

*Non-Euclidean geometry and Indra’s pearls,*Plus Magazine, 2007 : http://plus.maths.org/content/non-euclidean-geometry-and-indras-pearls

^{[6]}Maurits Cornelis Escher,

*Regular Division of the Plane and Cirlce Limit IV*

^{[7]}Linda Dalrymple Henderson,

*The Fourth Dimension and Non-Euclidean Geometry in Modern Art,*1984

^{[8] }Jacques Resch,

*Le Diabolo:*

*http://www.jacquesresch.com/a23.htm*

Note: This paper was submitted as an assignment for the Art Appreciation course at YIF

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reminds me of this http://upload.wikimedia.org/wikipedia/commons/e/e3/Habitat_panorama.jpg

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