Furthermore, when we look at a photograph, technically the paper is two-dimensional, but it is impossible to not see any various planes of depth—impossible to not understand the three-dimensional world it represents. I never thought I would say this, but geometry is like art, at least in this way: geometry sets out to understand reality through representation. In order to solve a problem we draw lines and points—or at least representations of lines and points. In order to be truly one-dimensional, a line cannot have width or height, and a point cannot have length, width, or height (zero-dimensional). The pencil mark alone has measurable length, width, and height—arguably because we see it that way. We have evolved to see in three-dimensions, and personally it hurts my head when I try to fathom space in a two-dimensional or 4+ dimensional world. Still, geometry strives to represent reality and analyzes many of the same concepts art does. Crazy stuff.
This chapter really tied up any loose ends I had. In the middle of realizing all of this (freaking out over the possibility that nothing is “real”), I suddenly connected this chapter to something we read at the beginning of the semester (Sachs perhaps?). It was about how blind people perceive space differently—constructed through time rather than visually. I believe space is real—in that it exists because we construct it. So this is to say that if we could perceive four or five dimensions, we would see things in a completely different way. Seeing, perhaps meaning finally understanding the many unanswered how’s and why’s of our world equally as it means physically seeing. As a seeing person, space is nearly inseparable from visual experience. I say that space exists for blind people (but that they experience it differently) because I see space—because the blind live in my visual world. Why do I see space and depth? Is it because I can see color, because I can see light, because I can see brightness and contrast which gives shape to objects, because I can see various forms of these shapes…
Two-dimensional drawings that depict three-dimensional reality usually shows scenes that are real/ spatially achievable relationships. Such art simulates a “photographic” representation of nature (only one viewpoint), and it does not allow us to see the scene from different vantage points or to view objects from various sides. However, pattern can be used to alter our judgment of a geometric shape in illusionary art. It can depict a three-dimensional scene, which could not occur in reality. This creative art is created and appreciated by the best mathematicians and artists.
M. C. Escher:
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