A Comparison Between Fourth Dimensional Geometry and Cubist Art Forms

The world around us exists with three different spatial dimensions.  However, from as early as the 1880’s, mathematicians have pondered the existence of a fourth spatial dimension, and even higher dimensions beyond that [1]. Yet, living in the third dimension, we are not able to perceive object in hyperspace.  In order to view such an object, we can use a slicing method.  For example, a sphere passing through a plane would appear, for something in the second dimension, to start as a point, grow larger as a circle, then eventually shrink back down to a point before it vanishes.  Using this same method, we can view a fourth dimension object as various changing three dimensional figures, as shown in Figure 1 [2].

Figure 1. (Top) A cube passing through a plane (Bottom) A hypercube passing through the third dimension [2].

This slicing method works only to partially describe a surface.  Figure 2 shows the various ways a square would pass through a plane, which is dependent on orientation.  Furthermore, viewing these slices separately does not give a clear picture of what the surface actually looks like.  Instead, we can use projection to see how a surface like this would exist.  Just as one can draw a cube onto a piece of paper by distorting the lengths of edges to give appearance of depth, we can project a hyper surface to the third dimension [1] Finally, perspective can also be used to render higher dimensions, just as Henri Poincaré suggested in 1902 [2].

Figure 2. Different orientations of a cube cast different images in the second dimension [1].

Figure 3. A projection of a hypercube, also known as a tesseract.

One art form that is heavily influenced by perspective, or the rejection of such, is cubism.  Cubist work relies on the use of the fourth dimension and its lack of perspective, as Guillaume Apollinaire states perspective is “that fourth dimension in reverse” [1].  Pablo Picasso is one of the prominent artists in the cubism movement, utilizing multiple perspectives, such as one technique in which Picasso layered two photo negatives, synthesizing four paintings in one photograph [3]. Because of the reliance of fourth dimensional geometry, parallels can be seen between renderings of hyper surfaces and cubist works, in some cases by sight alone as seen in Figure 4.

Figure 4. (Left) An example of cubist art by Juan Gris. (Right) Multiple perspectives of octahedra by E. Jouffret. Similarities can be seen between these two images.

Cubism is not the only art form that uses this fourth dimension.  Mathematician and artist Tony Robbin utilizes various properties of the fourth dimension to create his artwork [4].  One example of his use of higher dimensions in his work is in the braiding of sheets.  As he explains, braiding threads starts with one dimensional lines, which then go over an under one another, requiring the use of three dimensions.  Braiding sheets, therefore, requires jumping up two dimensions as well, utilizing this fourth dimension.  Figure 5 uses five of these sheets to create one painting.  By use of the mathematical fourth dimension, artist were, and continue to be, able to create new works of art, unbound from the world we live in, instead focusing on one we cannot see.

Figure 5. One of Tony Robbin’s paintings, 2006-6.

References:

[1]  Henderson, Linda Dalrymple. “The Image and Imagination of the Fourth Dimension in Twentieth-Century Art and Culture.” Configurations 17.1 (2009): 131-160.

[2]  Bodish, Elijah. “Cubism And The Fourth Dimension.” Montana Mathematics Enthusiast 6.3 (2009): 527-540. Academic Search Complete. Web. 3 Dec. 2016.

[3]  Ambrosio, Chiara. “Cubism And The Fourth Dimension.” Interdisciplinary Science Reviews 41.2/3 (2016): 202-221. Academic Search Complete. Web. 3 Dec. 2016.

[4]  Robbin, Tony. 2015. “Topology and the Visualization of Space.” Symmetry 7, no. 1: 32-39.

[5]  Henderson, Linda Dalrymple. “The Fourth Dimension and Non-Euclidean Geometry in Modern Art: Conclusion.” Leonardo, vol. 17, no. 3, 1984, pp. 205–210. www.jstor.org/stable/1575193.