Week 4: Dimension / by Valzorra

Representation of the Third Dimension Into the Second Dimension

The representation of three dimensional space on a two dimensional surface has been explored over hundreds of years in a variety of artistic attempts. Mathematically, the idea of perspective and vanishing points is what is exciting about the representation of these dimensions. The basic idea of perspective is that objects appear to be smaller the further they are away from the viewer, and there is an accurate mathematical way to represent perspective. The most basic way to do that is to use a single vanishing point, which used to traditionally positioned in the centre of the canvas. With this technique horizontal lines are perpendicular to the canvas, while all vertical lines lead up to the same point in the centre. To make the perspective more exciting, artists later included multiple vanishing points, sometimes positioning them outside of the canvas. However, with more than one vanishing point, the horizontal lines can no longer be perpendicular to the canvas as they would not appear horizontal in the world of the paining or image. When there is more than one vanishing point, we refer to the line connecting those points as the vanishing line.

There are other ways to give the illusion of perspective other than using traditional vanishing points. Introducing Desargues’ Theorem, which relates to how triangles can be in perspective of each other without the use of vanishing points. In order to explain the two main notions of the theorem, let’s introduce two triangles ABC and abc. Desargues’ Theorem states that if the points Aa, Bb, and Cc all converge to the same point, then the two triangles are in perspective from a point. Now, to explain the other notion of the theorem, let’s call the meeting point of AB and ab = D, the meeting point of BC and bc = E, and the meeting point of AC and ac = F. If D, E, and F all fall on the same line, then the triangles are considered to be in perspective from a line. Desargues’ Theorem is a keystone notion in the field of projective geometry , which deals with the representation and transformation of geometric objects. If you would like to explore the proof of Desargues’ Theorem, please refer to the video below as it explains the concept better than I could ever hope to.

One of the most notable artists who have tackled the idea of representing the change from second to third dimension is M.C. Escher, who’s work is absolutely fascinating mathematically. Specifically, his experiments with Tessellations (which are simply infinitely repeating mathematical patterns, usually closely fitted together) continuously merge and play with the idea of dimensions and the constant shift between them. In the example below, Escher represents the cyclic nature of life by depicting the perpetual existence of the crocodile, which manages to escape its position, reach unknown heights, only to swiftly return and crawl back into position, repeating the whole process once more. The reptile manages to become a higher form of existence by entering another dimension, however, that is incredibly short lived. Additionally, M.C. has placed another object in this Graphic which shares a similar transitional nature, and that would be the dodecahedron, one of the regular polyhedrons. As discussed in a previous post, this is one of the few objects in existence that retains its mathematical properties while shifting between dimensions, thus further reiterating the main idea behind the reptiles.

Reptiles, 1943, M.C. Escher

Reptiles, 1943, M.C. Escher

Representation of Higher Dimensions

Representation of the fourth dimension can be excessively challenging, however, there are a few notable ways of visualising it, specifically through the Hypercube. One of the most famous examples of a depiction of the Hypercube is Salvador Dali’s Corpus Hypercubus, where he used a net of eight cubic cells glued to each other. Another possible method, commonly known as projection, features one cube located in the centre of another, with their corners joined together by edges. However, all of these are mere representations of the Hypercube through one format or another, while true vision of the fourth dimension has yet to be achieved, if at all possible. What’s interesting to me about this specific section of geometry is how these different shapes interact with each other, what their core principles are, how they help shape our understanding of dimensions and the main pillars of their construction. The transfer of information through dimensions and the existence of extraordinary shapes we cannot even imagine fascinates me and motivates me to look into them even further.

Crucifixion, 1954, Salvador Dali

Crucifixion, 1954, Salvador Dali

Mathematically speaking, the fourth dimension (and other higher dimensions) could be represented through the use of matrices, filled with Cartesian Coordinates and data points. The four vertices of a square can be represented by (0, 0) (0, 1) (1, 0) (1, 1). Adding one dimension, we can represent the cube as (0, 0, 0) (0, 0, 1) (0, 1, 0) (1, 0, 0) (0, 1, 1) (1, 0, 1) (0, 0, 0) (1, 1, 0) and (1, 1, 1). Adding a dimension one more time, and we would have the mathematical representation of the fourth dimension given by (0, 0, 0, 1) (0, 0, 1, 0) (0, 1, 0, 0) (1, 0, 0, 0) (0, 0, 1, 1) (0, 1, 0, 1) (0, 1, 1, 0) (1, 0, 0, 1) (1, 0, 1, 0) (1, 1, 0, 0) (1, 1, 1, 0) (1, 1, 1, 1), a total of 16 vertices. In order to represent even higher dimensions than the fourth dimension, we would simply need to add an additional coordinate. What’s even more exciting is that through the properties of matrices, one could then upscale or downscale the n-dimensional object, transforming it into any desired size.

Moving on from one dimension to another results in the loss and gain of certain information about those objects.

However, working purely in numbers is not very visual and does not provide a very intuitive idea of how to think or work with four or higher dimensional objects. I have previously explored the geometric properties of certain Polychora in Week 3: Research on Geometry, so feel free to explore that section of the blog for further visualisation. Looking into a way to make higher dimensions more intuitive, I came across this fantastic video that combines analytical and geometric methods of thinking about the fourth dimension, specifically a 4D Sphere. The basic method detailed in the video is to use a series of sliders in order to represent the points in four dimensional space, rather than to use strictly coordinates or strictly geometric shapes. Furthermore, what one will discover through this method is that in higher dimensions, the geometric shapes seem more counter-intuitive, which would force mathematicians and enthusiasts to be very creative when working and explaining their properties. More detail on the subject can be found in the video itself, but for my research purposes, the methodology of visualisation and representation is most significant.

Thoughts and Reflection

What I find really interesting about the idea of dimensions and going in between dimensions is this idea of information. Moving on from one dimension to another results in the loss and gain of certain information about those objects, which is absolutely fascinating to me. For example, a 2D representation of a 3D cube could not possibly display all of the edges as having the same size, as then the object would not appear to be a cube anymore, due to the rules of perspective. Additionally, the idea of different perspectives showing different sides of the same object, revealing new data about that object along the way could be perfectly translated into a game mechanic. What’s more is that this is the primary way in which one can create art with optical illusions and potentially use those in an environment. Overall, I am rather excited to see if this will go any further, but for now, onward with the research.