Read here about 4D Geometry and its influence on the development of tensegrity.

Summary

Four dimensional (4D) geometry was a theme of art, science and philosophy in the late 19th century. It influenced leading scientists and artists and was covered in popular literature and newspapers. This atmosphere had an indirect influence on the discovery of tensegrity and its acceptance as a genuine mode of comprehending structure, space, and reality in general.

For tensegrity researchers, the main influences are via Ouspensky, Hinton, Bragdon, Einstein, Constructivism (Ioganson), and Bauhaus (Black Mountain College faculty such as Albers), and the general acceptance of alternative ways of organizing space and its constituents, with particular non-perspective visual focus. For example, Fuller often spoke of how his structures originated from the feeling of constructing struts, and not the visible aspects of structure, which convention forced most architects to adopt variations of the cube.

The Fourth Dimension and Tensegrity

This introduction follows Henderson's definitive article (see reference [1]) with some occasional references added to tensegrity. Henderson's book, "The Fourth Dimension" is highly recommended as well.

Tensegrity as we know it was discovered at Black Mountain College in 1948. This discovery occurred partially due to the previous 70 years of intellectual activity regarding Euclid, Kant and geometry. By 1900 the fourth dimension as a non-Euclidean way of thinking was a concern common to intellectuals and artists in nearly every major modern movement: Analytical and Synthetic Cubists, Italian Futurists, Russian Futurists, Suprematists, and Constructivists, American modernists, Dadaists, and members of De Stijl. German Bauhaus members, and more. Today, we commonly overlook these ways of thinking, and it is largely overlooked in typical histories of tensegrity, since by the end of the 1920s the temporal fourth dimension of Einsteinian Relativity Theory had largely displaced the popular fourth dimension of space in the public mind.

Henderson says, "For all of these individuals non-Euclidean geometry signified a new freedom from the tyranny of established laws. Codified in Poincare's philosophy of conventionalism, which stated that Euclid's geometry was merely a "convention," the recognition of the relativity of knowledge was a powerful influence on early twentieth century thought. Thus, even artists who concentrated on the fourth dimension alone owed something to the non Euclidean geometries that had prepared the way for the acceptance of alternative kinds of space.

"Like non-Euclidean geometry, the fourth dimension was primarily a symbol of liberation for artists. However, the notion of a higher dimension lent itself to painterly applications far more easily than did the principles of non-Euclidean geometry. Specifically, belief in a fourth dimension encouraged artists to depart from visual reality and to reject completely the one-point perspective system that for centuries had portrayed the world as three-dimensional. For artists whose distrust of visual reality was most deep-seated, belief in a fourth dimension was an important impetus to create a totally abstract art."

Even Salvador Dali's famous limp watches in The Persistence of Memory of 1931 has non-Euclidean overtones. In his 1935 book The Conquest of the Irrational Dali discussed the watches in the context of his comments on non-Euclidean versus Euclidean geometry and the theories of Einstein. Noting their immediate visual source in a plate of Camembert cheese, Dali described the melted watches as "the extravagant and solitary Camembert of time and space." The Russians were particularly exemplary of this approach. Ioganson and the constructivists elevated pure space to a constructive material, and created the first three strut tensegrity prism in 1920.

Theosophical beliefs were also supported by fourth dimensional ways of thinking. In fact, the most utopian, idealist view of the fourth dimension aligned with calls for a new 'language' that were widespread in this era. Ranging from a geometric, purely spatial concept in the hands of Poincare to a mystical vision provisionally incorporating time in the hyperspace philosophy of Hinton, Bragdon, and Ouspensky, people read and cited these authors as well as Pawlowski, Jouffret, Poincare, Boucher, and A. S. Eddington's Space, Time and Gravitation.

By 1930 widespread public fascination with ultra-dimensions was over, but this notion, along with non Euclidean geometry, played a vital role in the development of art, architecture, physics, mathematics, and hence the birth of tensegrity. However, many influences lingered. Henderson cites Sirato's Mamfeste Dimensioniste published in 1936 which declared, "Sculpture [is] 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 Minkovsky.... Rigid material is abolished and replaced by gaseous materials." The manifesto was signed by Calder, Moholy-Nagy, and others. Black Mountain's plastic artists hewed to this vision of empty space or "vaporized sculpture" and this was the milieu that Fuller and Snelson joined in 1948.

Fuller and Michael Burt are two examples of tensegrity-aware individuals who continued to think about four and more dimensions. Artists did as well. Henderson cites the American painter Pereira, who matured in the 1920s and was influenced by Hinton's writings. Sensitive to the mystical and intuitive aspects of the fourth dimension, Pereira also studied the physics of Einsteinian Relativity during the 1930s. The result was what she termed a "pure scientific or geometric system of esthetics'', which sought "to find plastic equivalents for the revolutionary discoveries in mathematics, physics, biochemistry and radioactivity." As late as 1966, in The Transcendental Formal Logic of the Infinite: The Evolution of Cultural Forms, Pereira still sounded much like one of the devotees of evolving consciousness and dimensional awareness in the Stieglitz circle in 1913. Man Ray's photographs of polyhedral geometrical models at the lnstitut Henri Poincare, taken for the 1936 exhibition of Surrealist objects, also continued to inspire artists. These types of echoes remain strong through the 1970's and 1980's.

Fuller himself forever retained echoes of this liberating, non-Euclidean way of thinking. He explicitly rejected the Einsteinian simplification of the temporal fourth dimension, and until his death in the 1980's posited physical 4th, 5th and 6th dimensions realizable in physical tensegrity models.

## Table of Contents

## Summary

Four dimensional (4D) geometry was a theme of art, science and philosophy in the late 19th century. It influenced leading scientists and artists and was covered in popular literature and newspapers. This atmosphere had an indirect influence on the discovery of tensegrity and its acceptance as a genuine mode of comprehending structure, space, and reality in general.For tensegrity researchers, the main influences are via Ouspensky, Hinton, Bragdon, Einstein, Constructivism (Ioganson), and Bauhaus (Black Mountain College faculty such as Albers), and the general acceptance of alternative ways of organizing space and its constituents, with particular non-perspective visual focus. For example, Fuller often spoke of how his structures originated from the feeling of constructing struts, and not the visible aspects of structure, which convention forced most architects to adopt variations of the cube.

## The Fourth Dimension and Tensegrity

This introduction follows Henderson's definitive article (see reference [1]) with some occasional references added to tensegrity. Henderson's book, "The Fourth Dimension" is highly recommended as well.Tensegrity as we know it was discovered at Black Mountain College in 1948. This discovery occurred partially due to the previous 70 years of intellectual activity regarding Euclid, Kant and geometry. By 1900 the fourth dimension as a non-Euclidean way of thinking was a concern common to intellectuals and artists in nearly every major modern movement: Analytical and Synthetic Cubists, Italian Futurists, Russian Futurists, Suprematists, and Constructivists, American modernists, Dadaists, and members of De Stijl. German Bauhaus members, and more. Today, we commonly overlook these ways of thinking, and it is largely overlooked in typical histories of tensegrity, since by the end of the 1920s the temporal fourth dimension of Einsteinian Relativity Theory had largely displaced the popular fourth dimension of space in the public mind.

Henderson says, "For all of these individuals non-Euclidean geometry signified a new freedom from the tyranny of established laws. Codified in Poincare's philosophy of conventionalism, which stated that Euclid's geometry was merely a "convention," the recognition of the relativity of knowledge was a powerful influence on early twentieth century thought. Thus, even artists who concentrated on the fourth dimension alone owed something to the non Euclidean geometries that had prepared the way for the acceptance of alternative kinds of space.

"Like non-Euclidean geometry, the fourth dimension was primarily a symbol of liberation for artists. However, the notion of a higher dimension lent itself to painterly applications far more easily than did the principles of non-Euclidean geometry. Specifically, belief in a fourth dimension encouraged artists to depart from visual reality and to reject completely the one-point perspective system that for centuries had portrayed the world as three-dimensional. For artists whose distrust of visual reality was most deep-seated, belief in a fourth dimension was an important impetus to create a totally abstract art."

Even Salvador Dali's famous limp watches in The Persistence of Memory of 1931 has non-Euclidean overtones. In his 1935 book The Conquest of the Irrational Dali discussed the watches in the context of his comments on non-Euclidean versus Euclidean geometry and the theories of Einstein. Noting their immediate visual source in a plate of Camembert cheese, Dali described the melted watches as "the extravagant and solitary Camembert of time and space." The Russians were particularly exemplary of this approach. Ioganson and the constructivists elevated pure space to a constructive material, and created the first three strut tensegrity prism in 1920.

Theosophical beliefs were also supported by fourth dimensional ways of thinking. In fact, the most utopian, idealist view of the fourth dimension aligned with calls for a new 'language' that were widespread in this era. Ranging from a geometric, purely spatial concept in the hands of Poincare to a mystical vision provisionally incorporating time in the hyperspace philosophy of Hinton, Bragdon, and Ouspensky, people read and cited these authors as well as Pawlowski, Jouffret, Poincare, Boucher, and A. S. Eddington's Space, Time and Gravitation.

By 1930 widespread public fascination with ultra-dimensions was over, but this notion, along with non Euclidean geometry, played a vital role in the development of art, architecture, physics, mathematics, and hence the birth of tensegrity. However, many influences lingered. Henderson cites Sirato's Mamfeste Dimensioniste published in 1936 which declared, "Sculpture [is] 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 Minkovsky.... Rigid material is abolished and replaced by gaseous materials." The manifesto was signed by Calder, Moholy-Nagy, and others. Black Mountain's plastic artists hewed to this vision of empty space or "vaporized sculpture" and this was the milieu that Fuller and Snelson joined in 1948.

Fuller and Michael Burt are two examples of tensegrity-aware individuals who continued to think about four and more dimensions. Artists did as well. Henderson cites the American painter Pereira, who matured in the 1920s and was influenced by Hinton's writings. Sensitive to the mystical and intuitive aspects of the fourth dimension, Pereira also studied the physics of Einsteinian Relativity during the 1930s. The result was what she termed a "pure scientific or geometric system of esthetics'', which sought "to find plastic equivalents for the revolutionary discoveries in mathematics, physics, biochemistry and radioactivity." As late as 1966, in The Transcendental Formal Logic of the Infinite: The Evolution of Cultural Forms, Pereira still sounded much like one of the devotees of evolving consciousness and dimensional awareness in the Stieglitz circle in 1913. Man Ray's photographs of polyhedral geometrical models at the lnstitut Henri Poincare, taken for the 1936 exhibition of Surrealist objects, also continued to inspire artists. These types of echoes remain strong through the 1970's and 1980's.

Fuller himself forever retained echoes of this liberating, non-Euclidean way of thinking. He explicitly rejected the Einsteinian simplification of the temporal fourth dimension, and until his death in the 1980's posited physical 4th, 5th and 6th dimensions realizable in physical tensegrity models.

Links and references

[1] The Fourth Dimension and Non-Euclidean Geometry in Modern Art: Conclusion by Linda Dalrymple Henderson, Leonardo, Vol. 17, No. 3. (1984), pp. 205-210. Stable URL: http://links.jstor.org/sici?sici=0024-094X%281984%2917%3A3%3C205%3ATFDANG%3E2.0.CO%3B2-1

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