New possibilities opened up by the concept of four-dimensional space (and difficulties involved in trying to visualize it) helped inspire many modern artists in the first half of the twentieth century. Early Cubists, Surrealists, Futurists, and abstract artists took ideas from higher-dimensional mathematics and used them to radically advance their work.

The idea of the fourth dimension is conveyed at the beginning of the twentieth century by the most diverse channels. A field of work in the exact sciences, it has been distributed by Poincaré publications to a wide audience. The fourth dimension quickly became a favorite subject of popular mathematics, science fiction, esotericism and art.

It could be called the hyperspace philosophy. The representations of the mathematical vulgarizations of the higher dimensions give rise to a flowering of illustrations of geometric solids all more complicated than the others.

N – dimensional geometries and non – Euclidean geometries are two separate branches of geometry that can be combined, but not necessarily. A confusion has been established in popular literature about these two geometries. Because Euclidean geometry was three-dimensional, it was concluded that non-Euclidean geometries necessarily had larger dimensions. But it is especially the idea of fourth dimension, possible mode of theoretical apprehension of the new cubist painting, which will fascinate the artistic world.

What must be remembered is the highly relational side of Cubism. Everyone knows each other, ideas circulate and take on mathematical, literary and pictorial forms. However, it is important to focus these ideas around those of Poincaré. It is he who for the first time gives this distinction between a geometric space and a representational space. This may explain not only the birth of Cubism in France, but also a minimum of public close to receive it.

Early influence

French mathematician Maurice Princet was known as “le mathématicien du cubisme” (“the mathematician of cubism”). An associate of the School of Paris, a group of avant-gardists including Pablo Picasso, Guillaume Apollinaire, Max Jacob, Jean Metzinger, and Marcel Duchamp, Princet is credited with introducing the work of Henri Poincaré and the concept of the “fourth dimension” to the cubists at the Bateau-Lavoir during the first decade of the 20th century.

Princet introduced Picasso to Esprit Jouffret’s Traité élémentaire de géométrie à quatre dimensions (Elementary Treatise on the Geometry of Four Dimensions, 1903), a popularization of Poincaré’s Science and Hypothesis in which Jouffret described hypercubes and other complex polyhedra in four dimensions and projected them onto the two-dimensional page. Picasso’s Portrait of Daniel-Henry Kahnweiler in 1910 was an important work for the artist, who spent many months shaping it. The portrait bears similarities to Jouffret’s work and shows a distinct movement away from the Proto-Cubist fauvism displayed in Les Demoiselles d’Avignon, to a more considered analysis of space and form.

Early cubist Max Weber wrote an article entitled “In The Fourth Dimension from a Plastic Point of View”, for Alfred Stieglitz’s July 1910 issue of Camera Work. In the piece, Weber states, “In plastic art, I believe, there is a fourth dimension which may be described as the consciousness of a great and overwhelming sense of space-magnitude in all directions at one time, and is brought into existence through the three known measurements.”

Another influence on the School of Paris was that of Jean Metzinger and Albert Gleizes, both painters and theoreticians. The first major treatise written on the subject of Cubism was their 1912 collaboration Du “Cubisme”, which says that:

“If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidian mathematicians; we should have to study, at some length, certain of Riemann’s theorems.”

In a review of the 1913 Armory Show for The Philadelphia Inquirer, the influence of the fourth dimension on avante-garde painting was discussed; the paper’s art-critic describing how the artists’ employed “..harmonic use of what may arbitrarily be called volume”.

Maurice Boucher’s essay on Hyperspace in 1903, which Matisse had in his hands, quoted the regular figures in a n dimensional space of Stringham, which had published these plates in 1880 in the American journal of mathematics.

“As a non-Euclidean world, we can imagine a world in four dimensions,” wrote Henri Poincaré in 1902 in Science and Hypothesis.

Even if, as such, the non-Euclidean curved space rarely appears in cubist painting, the new geometries were at the heart of the intellectual preoccupations of the artists of the beginning of the 20th century in France and Russia.

Dimensionist manifesto

In 1936 in Paris, Charles Tamkó Sirató published his Manifeste Dimensioniste, which described how

the Dimensionist tendency has led to:

Literature leaving the line and entering the plane.

Painting leaving the plane and entering space.

Sculpture stepping out of closed, immobile forms.

…The artistic conquest of four-dimensional space, which to date has been completely art-free.

The manifesto was signed by many prominent modern artists worldwide. Hans Arp, Francis Picabia, Kandinsky, Robert Delaunay and Marcel Duchamp amongst others added their names in Paris, then a short while later it was endorsed by artists abroad including László Moholy-Nagy, Joan Miró, David Kakabadze, Alexander Calder, and Ben Nicholson.

Crucifixion

In 1953, the surrealist Salvador Dalí proclaimed his intention to paint “an explosive, nuclear and hypercubic” crucifixion scene. He said that, “This picture will be the great metaphysical work of my summer”. Completed the next year, Crucifixion (Corpus Hypercubus) depicts Jesus Christ upon the net of a hypercube, also known as a tesseract. The unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The Metropolitan Museum of Art describes the painting as a “new interpretation of an oft-depicted subject…[showing] Christ’s spiritual triumph over corporeal harm.”

Some of Piet Mondrian’s (1872–1944) abstractions and his practice of Neoplasticism are said to be rooted in his view of a utopian universe, with perpendiculars visually extending into another dimension.

The fourth dimension has been the subject of numerous fictional stories.

Experiments on the use of new visual aids were taken by many famous artists. The role of prospects has decreased; for example, Cubists in their paintings often depicted people and objects simultaneously in different angles, thereby adding dimensions to them. In the visual arts appeared such modernist (sometimes called avant-garde) trends, like surrealism, futurism, abstractionism and others.

Today, scientists no longer confine themselves to the three dimensions of Euclid. And the artists, which is quite natural, attracted new opportunities for spatial measurements, which in the language of modern studios came to be called the fourth dimension. Having in mind a way of plasticizing an object, the fourth dimension is born from three known dimensions: it is the immensity of space in all directions at every given moment. This is the very space, the very dimension of infinity; The fourth dimension gives objects plasticity.

The popular mathematics

It could be called the hyperspace philosophy. The representations of the mathematical vulgarizations of the higher dimensions give rise to a flowering of illustrations of geometric solids all more complicated than the others.

The trial of Hyperspace of Maurice Boucher in 1903, and Matisse had in hand, quoting the regular figures in a space of n dimensions of Stringham, who had published these boards in 1880 in the American journal of mathematics.

This essay by Stringham inspired Hinton’s (A New Era for Thought, 1888). They pave the way for a series of publications in the early years of the xx th century as the isometric perspective of the 16 fundamental Octahédrons a Icosatétrahédroide of Jouffret and elementary Treaty of 4-dimensional geometry published in Paris in 1903, Le Cube’s Fourth Dimension of Hinton, London and New York in 1904, The Hypercubefrom Manning to New York in 1914.

Science Fiction

E. Abbott publishes Flatland in 1884, quoted by Jouffret in 1903 in his Basic Treatise on Four-Dimensional Geometry (A Square meets a Sphere) When the square asks the sphere to be carried into the higher dimensions, it sends it back into its flat Earth). Machine The Time of HG Wells is published in the Mercure de France in 1898-1899. Dodgson, aka Lewis Carroll, published in 1865 Dynamics of a particle, following a lecture at Oxford introduced by the history of Euclidean lovers, and in 1873 Defense of Euclidean geometry. Through the lens. Criticism of the craze for higher dimensions. Alfred Jarry, admirer of Lord Kelvin, is illustrated in 1911 with Gestures and opinions of Dr. Faustroll, pataphysician. Gaston de Pawlowski is the author of a science fiction novel: Journey to the Land of the Fourth Dimension, published in 1912. A great admirer of Poincaré, he is also editor of Comœdia magazine. His thinking has strongly influenced Gleizes and Metzinger for the writing of Cubism.

Marcel Duchamp was probably influenced by all these works, but it seems that he had some knowledge in mathematics quite advanced. In any case, he has thoroughly read Poincaré, whose hypotheses he often discusses very seriously in his notes.

Lovecraft will be the first writer to truly relate a journey in other dimensions other than time. There are traces in particular in “The house of the witch” and ” Cthulhu “.

Esotericism and fourth dimension

In 1895 Leadbeater compared the theosophical idea of “astral vision” with that of “four-dimensional view”. Mrs. Blavatsky wrote in 1888 in The Secret Doctrine: “The familiar expression can only be an abbreviation of the complete formulation” The fourth dimension of matter in space. “In France, Revel publishes in 1911 The spirit and space: the Fourth Dimension., where he is to develop his abilities to move to “a more subtle world” Revel also pays tribute to the work of Poincaré on the visual space, tactile and motor. Noircarme published in 1912 Fourth dimension, with a “bicarré”

Ouspensky (Moscow 1878- London 1947) Russian mathematician and disciple of the esotericist Gurdjieff developed “Tertium Organum” a number of concepts on space and time. Using eclectic references (the East, Christianity, scientific discoveries…), it is based on Kant’s postulate that space with its characteristics is a property of our consciousness and not of the outside world. there, he develops the idea that, the world dependent on the psychic apparatus, a work on this psyche would thus make it possible to modify the man into a super-human.Completion in 1934, to “A new model of the universe”, he pursues his idea towards a possibleimmortality of man (Man and his possible evolution, 1945) acquired through an awakening of consciousness, excluding any possibility of psychic work during sleep, let alone dream.

N – dimensional geometries and non – Euclidean geometries are two separate branches of geometry that can be combined, but not necessarily. A confusion has been established in popular literature about these two geometries. Because Euclidean geometry was three-dimensional, it was concluded that non-Euclidean geometries necessarily had larger dimensions.

Painting and new geometries

The painter “brings his body” says Valéry [cited by Merleau-Ponty in The Eye and the Spirit], it is by “lending his body that the painter changes the world in painting”. Friend of Alfred Jarry (The Pataphysics, Ubu), and reader of Herbert George Wells (the machine to go back the time), Valéry was a passionate admirer of Poincaré. He had even started studying mathematics in 1890 and his notebooks, between 1894 and 1900, were full of equations.

Concerning matter, for Poincaré, one of the most astonishing discoveries that physicists have announced in recent years is that matter does not exist.

This statement questioned the painter Matisse who wrote to Derain in 1916 about Science and Hypothesis: “Have you read this book? There are certain hypotheses of dizzying audacity, for example, the destruction of matter. Movement exists only through the destruction and reconstruction of matter. ”

But it is especially the idea of fourth dimension, possible mode of theoretical apprehension of the new cubist painting, which will fascinate the artistic world.

What must be remembered is the highly relational side of Cubism. Everyone knows each other, ideas circulate and take on mathematical, literary and pictorial forms. However, it is important to focus these ideas around those of Poincaré. It is he who for the first time gives this distinction between a geometric space and a representational space. This may explain not only the birth of Cubism in France, but also a minimum of public close to receive it.

On the other hand, Einstein ‘s theories are unlikely to have influenced Cubism because they are known relatively late in France. The elaboration of the theory of relativity itself is done over a fairly long period.

The fourth dimension and cubism

“It is said that Matisse was the first to use this expression before Picasso’s first Cubist research. ”

This is what the Italian futuristic painter Gino Severini wrote in 1917 about the Fourth Dimension in the Mercure de France. Matisse, who, reading a treatise entitled Essay on Hyperspace, exclaimed: “Oh! but it is only a book of popularization! “. (It is Metzinger who quotes this anecdote in Cubism was born and he ends like this: “In the end, it showed that for the great wild beast, the time when the ignorant painter was running, carried by the wind, looking for a beautiful motive was well finished. “).

In 1909, Charles Camoin wrote to Matisse about his art:

“What a shameful profession at a time of such great speculation and after the discovery of the 4 th dimension. ”

If Matisse may, in the early years of the century, estimated at fair value a book on the new geometries and discuss the 4 th dimension, indeed certainly because he knows the authoritative scientific publications in the pre-war Paris namely, those of Henri Poincaré.

Including Science and Hypothesis, published in 1902, whose two chapters Non-Euclidean Geometries and Space and Geometry describe in a simple and precise way some essential notions on non-Euclidean geometries, n-dimensional geometries, and the fourth dimension. But the strength of this book lies in its description of the difference between the geometric space which is a convention (geometry is not true, it is advantageous), and the representative space with visual, tactile and motor components.

It was the mathematician Maurice Princet, who frequented cubist circles, who first established a formal analogy between the effect of facets obtained in the cavalier perspectives of the sixteen octahedrons of an Icosatetrahedroide by Jouffret and the cubist portrait of Ambroise Vollard by Picasso (1910). But Picasso has always emphatically denied having ever discussed mathematics with Princet (interview to Alfred Barr, 1945). The collector and art dealer Daniel-Henry Kahnweiler, in his 1947 book about Juan Gris, says about Princet “that he never had the slightest influence on Picasso or Braque, nor on Gray., who had followed his own studies of mathematics. ”

But the idea of the fourth dimension in Cubism undoubtedly has a mathematical origin, that of Poincaré. How was it transmitted?

In 1918, Louis Vauxcelles made fun of Princet and the way in which the idea of the fourth dimension in art had spread:

“It is well known in the workshops of Montparnasse, and everywhere else, that the inventor of Cubism was Max Jacob. We believed it ourselves. But it is necessary to render his honor to Caesar, and Caesar, in this particular case, is called M. Princet. It is, we think, the first time that this name is printed in the annals of Cubism. Mr. Princet is an “insurance agent” and very strong in mathematics. Mr. Princet calculates as Inaudi. Mr. Poincet (sic) reads Henri Poincaré in the text. M. Princet has thoroughly studied the non-Euclidean geometry and Rieman’s theorems, of which Gleizes and Metzinger have spoken so carelessly. So, one day, Mr. Princet met Mr. Max Jacob and gave him one or two of his discoveries on the fourth dimension. M. Jacob informed the ingenious M. Picasso, and M. Picasso saw in it the possibility of new ornamental schemes. M. Picasso explained his intentions to M. Apollinaire, who hastened to put them in formulas and to codify them. The thing proliferated and spread. Cubism, M. Princet’s child was born. ”

In fact Max Jacob makes mention in an article late 1915 for an American periodical 291 of his meeting with Galani whose attempt to explain the fourth dimension was converted by Max Jacob the religious into an explanation of the apparitions and disappearances of the risen Christ.

Matisse, in a letter to Derain in 1916, speaks of Galani who has just read Science and Hypothesis, a book in which he found the origin of Cubism (Matisse adds three exclamation points in parentheses).

Gleizes and Metzinger had probably studied Poincaré’s work quite closely. But, on the one hand, the discussions with Princet, and on the other hand possible reading of Theosophists quoting Poincaré as Revel in The Spirit and Space: The Fourth Dimension could bring some confusion in the way the link between the new geometries and cubism was established.

For Gleizes and Metzinger there are two kinds of geometrical spaces, Euclidean space and non-Euclidean space. Euclidean space poses the indeformability of moving figures. The non-Euclidean space is the one to which the space of painters must be attached. Gleizes and Metzinger recommend in this respect to study the new geometries:

“If we were to relate pictorial space to a particular geometry, we should refer to non-Euclidean scholars; we would have to study, in the end, some of Rieman’s (sic) theorems. ”

In fact, despite the intellectual concerns of artists about new geometries, non-Euclidean curved space rarely appears in cubist painting except in works like L’Estaque, by Braque and Dufy (1908), Delaunay’s Eiffel Towers (1910-1911), or the Cubist Landscape of Metzinger (1911), where it seems that it has been applied to conscientiously implement the principles of deformation of non-Euclidean curved space.

Influence in Russia

The Russian painter Mikhail Matiushin, who had written in 1911 an article on “The meaning of the fourth dimension”, published in March 1913 a translation into Russian of “Cubism” of Gleizes and Metzinger enamelled with quotes from the “Tertium Organum” of the Russian esotericist Ouspensky. This comparison highlights certain conceptions common to both authors:

pre-existence of a form that is visualized on the object of experience (by turning around, increasing its “power” of visualization)

dependence of form and color

It is this qualitative leap from the non-Euclidean geometries to the N-dimensional geometries and hence to the physics that Matiouchine operates when he puts this passage “From Cubism” by Gleizes and Metzinger in the face of a passage from “Tertium Organum” from Ouspensky:

“For the famous mathematician Riemann, when higher dimensions of space come into play, time, in a way, transposes itself into space, and it identifies the material atom as the input of the fourth dimension in the three-dimensional space. ”

The fourth dimension was an intellectual and artistic challenge in the new Russian painting, strongly influenced by Cubism. Between Cubist painters, the debate was raging and Matiouchine, in particular, strongly criticized Malevitch for his misunderstanding of the new physics, as he mentions in a 1916 review:

“There is a negative point… in that Malevich understands insufficiently the modalities of the new dimension. ”

Poincaré’s writings did not go directly into Russian artistic literature, but through two filters: those of Poincaré’s interpretation by Gleizes and Metzinger and that of Ouspensky’s esoteric reading. It seems that it is particularly Ouspensky’s theories that Malevich was reluctant:

“You can see a current looking for a new man, new ways. They even go to India and Africa and search the catacombs, they think they will find something (what do you think?). They publish a mass of books (Ouspensky)… ”

Malevich’s remark in “God is not dethroned” directly targeted Matiushin insofar as he openly appealed in his work to the conceptions of space developed by Ouspensky, thus maintaining an ambiguity about the relations of art. modern Russian with science, art and spirituality.

Source from Wikipedia