Imaging things in another dimension outside of our own can
be a difficult, some say even imposable idea to grasp. The problem is not within the idea in and of
itself, instead the problem presents itself in the mental hurtles encountered
in attempts to visually interpret how another dimension might look. This
problem arises because we live in a three dimensional world, in which we observe
objects in a such a way that we can only view them in the present perspective
we have, from the position we currently stand, at a single moment in time. Viewing a fourth dimension would imply that we
are viewing an object inhibiting characteristics that render it as if it were viewed
from multiple vantage points at a single
moment in time, suggesting what is physically impossible, being in more than
one place at the same time. While mathematicians
seem to have no problem interpreting such possibilities using corresponding coordinate
systems, the vast majority of individuals can’t even begin to fathom the idea
of a fourth dimension or what it might look like. It is so hard to interpret an object in such a
dimension because doing so would entail trying to view a three dimensional
surface that inhibits fourth dimensional characteristics. The conception of
such a task can be considered paradoxical even when considering three
dimensional objects, because drawling’s can only be expressed on a two
dimensional surface. To create an object that inhibits three dimensional
characteristics one must illustrate such an object as an illusion to what it
actually is in the two dimensional sense.
This illusion portrays the object being three dimensional while in
reality what has been drawn is nothing more than a two dimensional geometric
shape. Unbelievably Picasso showed that
it was possible to depict a fourth dimension in a painting when he and his
collaborators discovered cubism. The
amazing part about the development of cubism is that Picasso was such an unbelievable
artist that he was able to depict objects inhibiting fourth dimensional
characteristics, on objects inhibiting third dimensional characteristics, both
of which being painted on a flat two dimensional plain. Picasso's multi-dimensional way of thinking
regarding art lead to further works of art following his cubist framework and
would even inspire futurism in which, time is depicted as an additional fourth
dimension represented in time-laps sequences or visualizations.
What cubism and futurism have tried to do is to convey an
image that obscures our traditional two dimensional way of viewing paintings
and images, in a way that accounts for a third and fourth dimension in order to
provide us with, more or less, an inter-dimensional perception of the image
being portrayed. I believe The
Peacock’s Tails summed up what cubism tries to accomplish best when he says,
“Such unusual images, typical in cubist paintings, may be interpreted as
attempts to catch views of the subjects by hovering into an unimaginable 4D
space and then projecting this new perception back onto the 2D canvas”. This sentence, in essence describes perfectly
what Cubism is trying to project with its strange and sometimes scary depictions
of a different reality viewed through a fourth dimension. Until reading these articles I was unaware
that cubism and futurism can both be regarded as interpretations of how one
might perceive a fourth dimension and I found it very interesting to see how
such complicated ideas were articulated through their discovery by some of the world’s
greatest artiest of all time. The image I
have a link to is called Dora Maar with Cat by Pablo Picasso. This image is a great example of Picasso’s style
in a cubist depiction of objects in a fourth dimension.
The difficuty involved in attempting to picture what a 4th dimension is like is akin, in my opioion, to trying to picture the world of general relativity in regards to space-time. I think it is interesting that mathematics and, in corelation, human being's ways of thinking, seem to become more and more abstract as time goes on. Euclidean geometry was rather straight foreward, which was then made more abstract by the addition of non-Euclidean, and then relativity. It is an interesting trend; that is, taking steps foreward in mathematics requires one to think "outside the box" and almost create an entirely new world view. I can honestly say that, even after reading these articles and looking at pictures, I have come to no better understanding of what a 4th dimension would look like. Hopefully we will look at more examples in class tomorrow!
ReplyDeleteI agree with Brandon. Attempting to picture a 4th dimension is extremely difficult and I believe that this difficulty is a direct consequence of a world perspective based on experience. Although I understand the concept of attempting to project multiple vantage points at a single moment in time onto a 2D canvas, I do not understand how this would translate to a 4 dimensional world. For example, when I see a projection of a three dimensional image on a two dimensional piece of paper I can imagine what that image would look like in a three dimensional world. This does not hold true for a 4 dimensional image projection. Honestly, when I look at Picasso’s picture above I don’t think of a 4th dimension, even though it depicts a side and frontal view of the head and face simultaneously. Confined to a three dimensional world makes it difficult to imagine a 4th dimension since it cannot be directly observed or experienced. This outlines the importance and power of abstract though, which can greatly impact our understanding of things beyond worldly experience.
ReplyDeleteI agree that it is difficult to comprehend a 4D world, seeing as it is not something we can directly observe. We live in a 3D world, and while it is not hard to imagine a 2D world, trying to conceptualize the fourth dimension is not something that is easy to grasp. Imagine that we live in a 2D world, instead of a 3D world. In this two dimensional world, it is impossible for us to move up or down. We are restricted to existing on a single plane. If the 2D world was all we ever knew and we had always been restricted to existing on this single plane, it might be difficult for us to imagine making the transition to live in a 3D world. This 3D world implies that there are an infinite number of planes adjacent to the 2D world we know. These planes exist above and below the plane we have always lived in. If a 3D entity were to walk across our 2D world at any time, we would only see a small cross section of the 3D entity at any given time. These concepts may seem somewhat tricky to grasp at first, but because we have had such extensive experience with both the 2D and 3D worlds, that for the most part, this is not a very difficult concept to understand. However, switching from a 3D world to a 4D world seems daunting and very difficult to imagine. The 4D world is not something I have ever directly observed, and so it is difficult for me to truly grasp what it would be like. The artwork we look at gives us an idea of how the 4th dimension can be perceived, however, I still find myself struggling to fully comprehend how the 4th dimension really works. I can imagine, however, that switching from 3D to 4D would be very similar to that switch we make between 2D and 3D. A 4D world implies that there are an infinite number of 3D spaces adjacent to our 3D world. Just as we were restricted to a single plane in the 2D world, in the 3D world, we are restricted to a single 3D space. Just as we are able to look at multiple 2D planes at a single time, a 4D entity would be able to see across a multitude of 3D spaces at a single time. It is interesting how mathematics started as a means of describing the physical world around us. Euclidean geometry was developed to help explain how and why physical entities on earth corresponded to each other. Now much of the study of math is no longer based on studying the inherently practical, and instead there has been a shift towards studying the complexity of the abstract. It is amazing how we are able to use the tools we have developed in math study and quantify a world that we can hardly even comprehend.
DeleteI tend to align with the above two responses on the difficulty of imagining a 4 dimensional world. The best concept I hold in my mind is that of a 3D space that can be viewed as transforming with respect to time, both forward and backward. Or, for simplicities sake, a map of time that has 3D objects depicted on a 2D plane (painting). This is probably naive at best but it's the best I can do without trying to think of 4D space. Because of this time changing map I like the futurism movement much more than the cubist but can see their impact on people by trying to ascend to a higher plane of depiction and visualization. I also entertained the idea of having 4D space + 1 time component (total of 5D space) until I found that trying to apply the hypercube talked about in the texts to time was impossible for me. If anyone has any other views, I'd like to read about them!
ReplyDeleteIf this posts multiple times I'm sorry...
ReplyDeleteAfter reading these three readings I found it interesting how the progression of art and mathematics relate. That as time continues and we discover more about the world that we apply it to mathematics and art. It was in the second reading where the Pavlopoulos states about having to discover this 4th contradictory because “painting about reducing dimensions rather than expanding them”. I think that mathematics is the same way. We take the natural world, science, or whatever your heart desires to compute mathematics about and reduce it to explain something. We connect the world, time and space to mathematics so we can understand it. I also found it interesting that when Picasso was talking about how everything relates to cubism, he was reducing it all to one topic. This I think shows how as humans we need everything to relate to each other to make sense. Not just to defend arguments, prove something is right, but for us to comprehend the world around us we need things to connect. This I do think came on with time and the age of discovery. I think this relates Christian iconology, they believed that heaven was right above the earth, and that is why they painted these icons they used the conventional linear principle?
I think Plato's description of the cave (brought up in the reading) is a very useful idea in considering the existence of a fourth dimension, or at least in beginning to understand the root of the limitations we experience. Individuals chained in a cave see only the shadows (2D) of the objects in the 3D world behind them. Thus, assuming that these individuals can only see the wall and nothing else, their reality exists solely in two dimensions. As the text describes, shadows would be ambiguous, and the “code” which translates points from 3 dimensions to 2 dimensions would be degenerate, such that some 3D objects could project an infinite number of unique 2D shapes on the wall.
ReplyDeleteTranslating this idea one dimension further, it seems logical that 4-dimensional objects could project themselves onto 3 dimensional surfaces, where they could be understood by human beings, who have experience with no more than 3 dimensions. More generally, I think that creatures (i.e., humans, the worm crawling through the book, the people in Plato's fictional thought experiment) can understand or maybe even perceive the "shadows" or projections of one dimension further than which they have experience.
In my calculus class several years ago, the idea of triple integrals puzzled me in a way similar to this idea of 4-dimensions. When integrating "under" a line, one finds a 2-D area; integrating "under" a 2-D surface, a 3-D volume; and integrating over some 3-D "volume", one finds a quantity which would correspond to something that exists in 4-Dimensional space. Mathematically, this object seems to exist, and the 4-dimensional space it occupies can be quantified. But, from a human perspective, understanding is very limited since we have no experience with space or objects beyond 3 dimensions. But, allowing the consequences of mathematical principles to “guide” our understanding and search for truth (beyond the limitations of human perception) it would seem that dimensions beyond the third (4, 5, 6, . . . )do exist.
As most have said, I would have to agree with George on the paradoxical nature of dimensions outside our 3D image of the world. Until this reading, I never really thought of Cubism as an art that is a paradigm shift in how we interpret space in art. Rather than simplifying the dimensions, Cubism aims to add more.
ReplyDeleteUntil this class and especially over the last several readings, I have come to realize how our definition of the space we live in (Euclidean, non-Euclidean, relativistic, 4D, etc.) manipulates how we interpret the world. Through these interpretations, we act and hence our space defines how we act. With this said, I wonder if I had grown up with teachers/an atmosphere emphasizing other space interpretations besides Euclidean if these abstract theorems of space would be more natural and become a part of my rational in the way the world works?
Furthermore on the topic of removing the Euclidean bias, I wonder if when I dream my subconscious thinks past 3D/in non-Euclidean or relativistic or even Cubist space (or its own space-time relation)? If this is the case, could we argue that these other forms of space are more "innate" to us then Euclidean interpretations?
I find it nearly impossible to imagine a "4th dimension" as described in the readings. It doesn't seem like a reasonable idea, at least for me. I picture in my mind a pair of dice. I can see the front, I can rotate those dice in my mind to see the front and the back but you can only rotate it so much before you just get back to the beginning. It is a very interesting thing to think about and it really rattles my brain. I've been looking up some additional things about 4 dimensional geometry and what it would be like to live in a 4 dimensional world. Something that I came across was the idea of shadows. Light shone on a 3 dimensional object in a 3 dimensional world cast a 2 dimensional shadow. Similarly, light shown on a 2 dimensional object in a 2 dimensional world cast a 1 dimensional shadow. Light shown on a 4 dimensional object in a 4 dimensional world would cast a 3 dimensional shadow. That seems like the coolest thing ever even though I really can't imagine what that would be like, the idea of having a 3 dimensional shadow is really wild to me.
ReplyDeletePerhaps I've read too many popularizations of how string theory works, but I don't find it very difficult to imagine the world having more spatial dimensions than what is apparent. Brian Greene, in his book The Fabric of the Cosmos, uses a version following thought experiment to explain how there may be physical dimensions that are not apparent to our observations:
ReplyDeleteImagine you're outdoors and look at a power line at some significant distance away. To you, the power line appears flat or line-like in nature. For all intents and purposes, it looks as if the object is 1-dimensional. Now zoom in on the power line. Turns out, there is an ant crawling around the surface of the power line. To the and there appears to be two dimensions to the surface it is on. How do we reconcile these two apparent accounts of the same object? Well, one may postulate that the second dimension the ant perceives (the circumference of the power line) operates on a length scale that is so much smaller than the scale we observe/live in and as a result is effectively "curled" up from our view. This simple explanation is the conceptual apparatus for string theory which postulate the existence of many more (some as high as 26) spatial dimensions that operate on a length scale known as the planck length which is about 10^-35 meters. To give you an idea of how small that is, if I were to shrink the observable universe, roughly 10^26 meters across, by a factor of 10^35, it would be 10^-9 meters across, which is about 5 hydrogen atom diameters.
I leave it to you to ponder this: could this existence we observe merely be curled up dimensions in some larger object that is effectively unobservable to us because we operate on a totally different length scale?
I really like this idea of a fourth dimension as well. It is hard to fully conceptualize this idea, but I am starting to be able to formulate from these readings how it can be done. When I was looking online for some paintings or whatnot to share, I stumbled on this picture. When I looked at it it was like a “whoa!” moment because I felt like I could see what was going on over time for this cat as this picture was being produced.
ReplyDeletehttp://www.flickriver.com/photos/morgannn/3144839312/
With just the 3D perspective, this reminded me of a Zocchihedron and hallway perspective pictures like this one.
http://cdemoremack.files.wordpress.com/2010/09/zzz.png
The hallway has to be one of my favorite perspective theme. This picture of just 3D cubist objects also really intrigued me.
http://home.comcast.net/~artistrickclement/images/gphorsehead.jpg