The best-laid plans of mice and men often go awry.
Robert Burns
1 Introduction.
How long does scientific knowledge take to become generally accepted among society? How long did it take for people to accept that the earth is round? What would you answer to questions like, which is the smallest particle that constitutes matter? Or, do parallel straight lines ever touch each other? These last two issues advanced into new widely accepted answers in the 20th and 18th centuries respectively. The current scientific body of knowledge includes them, namely, the Standard Model and non-Euclidean geometries [1, 4]. However, even after three hundred years, most of us are only familiar with the earlier Euclidean postulates and their corrections and developments, i.e., classical geometry. There, the line (straight or curved) was defined over two thousand years ago: It extends infinitely, it has one dimension [2]. Later Descartes gave it a slope, a direction and steepness, the constant that straight lines carry throughout their existence. This means that they don’t change direction. They symbolize the deterministic nature of things. Something that is linear, is something that remains the same. But more often than not, everything around us changes. No matter the efforts we put on planning our endeavors, no matter our preparation or will, things are bound to shift. We can also find many implementations of this non-linearity in physics, architecture, design, paint, art, and so on [3]. So, let’s contrast the ever-changing nature of things with their immutability in the realm of design. Let’s construct a visual representation that answers the question: What does it mean for a line not to be straight? Let’s attempt to design indeterminism. Does it make sense? Isn’t something that is built on a structured foundation and with logical building blocks, bound to follow a pattern in the end? This work focuses on answering this line of questions, and approaches them from a scientific and artistic intersecting perspective, even though art and science seem to be themselves two infinite untouching parallels.
2 The Science-Meet-Art Curve.
In a Euclidean context a line can be a straight line or a curved line. Therefore, if a line is not straight it must be a curve. Within this reference, if a line is curved, it means that it changes its slope. There are many ways in which a line can change its slope, because slopes are ratios of change over time, and the change can be bigger or smaller in shorter or longer amounts of time. Thus, rendering abrupt changes of direction and steepness or more subtle changes in orthogonal fashion (see Figure 1). In the end, a circle is a type of line with infinite radius. The characteristic of extending infinitely should be brought to our attention when analyzing its plausibility in Figure 1 (a), but not in Figure 1 (b).
How about the non-Euclidean geometries, e.g., elliptical or hyperbolic? There are no straight lines in these examples. Therefore, asking what does it mean for a line not to be straight, also means that we are referring to non-Euclidean geometries. Geodesics are the straight lines in these frames of reference [4]. The shortest path between two points on a curved plane. They are used in the General Theory of Relativity where the space-time continuum curves. The meaning of a non-straight line guides us on a wide and deep journey trough the world around us.
Furthermore, how does this scientific knowledge and mathematically proven information lead science to meet art? Some may say that they are two separate things and that they conflict with each other. Some may say that they are the same thing, different methods to understand life and seeing the world in new ways [5]. I think that under the light of infinitely possible geometries, it is possible that the two are just different faces of the same coin. However, science has rules that define it, but even then it can lead to abstract results. Art, on the other hand, is built in the realm of the abstract, but it is only considered art when it manifests itself as such in a collective view.
My first approach towards a design of non-straight lines was to use particle systems of straight lines (Figure 2 (a)). The simplicity of this scheme relied on extrapolating the Euclidean geometry. The idea was to use circular paths as trajectories for the motion of straight lines. It implicitly conveyed the underlying concept, but it was too weak. It lacked a visual lead to a non-Euclidean abstraction. Figure 2 (b) makes a more obvious attempt. Previous frames don’t get erased, and the progression of the generative animation can be seen in the trails of four different linear particle systems, two triangular, two straight lines. The way in which these particle systems interacted with each other, made their pixels mix and create non-linear and linear, diffused and defined shapes. The color scheme also went through several changes and was further toned down by a washed-out palette.
Researching Euclidean and non-Euclidean references on the Internet brings up results of saddle like planes and warping spheres, i.e., the Voronoi diagram, the Poincar´e disk model, toroidal shapes, and the alike. I aimed to roughly portray those references while applying my own aesthetic dimension by means of code. The experience led me trough a valuable path of motivating creativity and learning. I aimed to convey a visual space in which two manifestations of the same idea coexist. They are not different from each other, nor are they the same. Like science and art, lines in geometry evidence the human understanding of nature, which can be very well defined and very abstract at the same time when viewed from different perspectives. Perspective is everything! The final sketch of the non-straight-line concept encompasses most of these ideas (Figure 3). What is it missing?
3 Oh meeting, where are you?
Let’s first close the question of, what does it mean for a line not to be straight, by associating the topics discussed in this document with the picture in its cover page and also with Figure 3 in order to form a subjective answer.
Then I’d like to add the following points for future work: Although plenty of historical and general background of geometry has been researched for the elaboration of this work, it is still missing accurate mathematical implementations to claim a scientific attribution. At the time of writing this essay, I was learning to code the anaglyph 3D implementation in order to give the sketch the additional visual perception shifting capabilities to convey a paradigm shift. Implementing this two points will greatly enrich this work by bringing it closer to a valid response for the arguments debated herein.
References
[1] The Rise of the Standard Model: A History of Particle Physics from 1964 to 1979. Cambridge University Press, 1997.[2] Euclides, J. L. Heiberg, and Richard Fitzpatrick. Euclids Elements ofgeometry: the greek text of J.L. Heiberg (1883-1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883-1885. Richard Fitzpatrick, 2008.
[3] L.D. Henderson. The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Leonardo (Cambridge). MIT Press, 2013.
[4] H.P. Manning. Non-Euclidean Geometry. Dover Books explaining science and mathematics. Dover Publication, 1963.
[5] Wikipedia contributors. The third culture — Wikipedia, the free encyclopedia,[Online; accessed 25-August-2018].
