Chapter thirteen of The Colossal Book
of Mathematics is, in my opinion, one of the more intriguing topics of math.
Hyper cubes and the different dimensions these cubes can exist in was the topic
of discussion throughout this chapter. Hyper cubes can range from the simplest
of objects to some of the most brain twisting structures your eyes have every
laid sight on.
Humans have a three dimensional view on the
world and can literally on see to the third dimension of an object or space.
Hyper cubes can extend in dimensions far beyond what the naked eye can perceive
and what our brains can manipulate. Even when we are claiming to see something
in a four-dimensional state, we are not. Because of our limited perception
humans have contemplated for one hundred plus years whether or not these
dimensions exist and if they do how can we stretch object to these dimensions
in real life? In order to achieve this we must first start with a point and
move it one unit in a straight line. Now we must take that line and move the
line one unit perpendicular to the line. We now have a square with four points
in a two-dimensional view. Now we must take the four points on our unit square
and shift them perpendicular to all three axes. This very shift is what brings
us into the visually boggling fourth dimension or 4-space. This shape is called
a tesseract, which has four perpendicular edges meeting at every corner. If you
try and draw this peculiar shape you may find yourself erasing quite a few
times trying to correct the shape. For us to understand why this shape has so
many different points, lines, squares, cubes and tesseracts we must first look
at a simple formula that can help us calculate these exact numbers for each nSpace. The formula of (2x+1)^n is the base formula to calculate all
the analogous of the cube in various dimensions. If you keep multiplying the
formula by itself you will see a pattern start to develop.
The most
interesting section of chapter thirteen is unfolding and cutting certain edges
of a hyper cube in order to form other shapes of see them shape in a different dimension.
Salvador Dali’s Corpus Hypercubus is
a great physical example of an unfolded tesseract in 4-space. The reason I like
this example the most out of all the examples is symbolizes the limited vision
of the human eye and mind.
It
came to my surprise that after reading the chapter and watching a couple of
videos of how hyper cubes and multi-dimensional shapes work that this topic is
as complicated as I first thought it would be. I was honestly frightened by the
title “Hypercubes” but after reading and understanding that it is a shape
simply shifted along axes in different dimensions my mind was able to
understand it and try and figure out how to stretch images into these
unexplored dimensions.
I
believe every who reads about hypercubes and tesseracts will be overwhelmed at
first but after reading about them and seeing how these shapes are formed
through the stretching and shifting process will definitely give you a better
grasp on the topic than you had before.
I would first like to say that you summarized the chapter well and your post really enlightened me. As i was reading the chapter i came across a section that contained numerous questions pertaining to unit squares, cubes and hypercubes. One of the questions was "What size hypercube has a hyper volume equal to its hyper surface?" I was wondering what you came up with or what you thought of this question?
ReplyDeleteI also found the topic very interesting, and as I researched more on it to better my understanding, I came across this video: http://www.youtube.com/watch?v=BVo2igbFSPE about unwrapping the tesseract to 3-D and back to 4-D. It's cool!! but how accurate is it in representing the 4 dimensions? I fail to comprehend the representation of the fourth dimension, so if anyone could enlighten me on that, I would have a much better view of the tesseract
ReplyDeleteSorry the link/address will not automatically redirect you, but if you want to see the video you can just copy the address above and paste it on the browser :)
ReplyDeleteThe topic was kind of hard to follow but by your summary made it a lot more clear to me. It helped me understand hypercubes a lot better. You did a great job explaining the chapter and making things more clear.
ReplyDeleteI thought this chapter was pretty compelling. It shows that even though we can project and simulate what the fourth dimension looks like, humans will never truly see what it actually is. This chapter poses many questions for the future like "Will eyes evolve enough to where we could visualize the 4th dimension?" Or even if "Can eyes be trained enough now that they can perceive 4D?" From what I learned in psychology, our eyes technically can only see in 2D, but our second eye is what allows us to perceive depth, and our brain blurs together the images it gets into what we see now, which is pretty cool. Hypercubes is definitely a complicated topic to get a grasp on visually.
ReplyDeleteThis chapter was very interesting in which our perception of three-dimension is as clear as day. The concept of 4-dimension is something we can only envision via simulation. The hypercube does show what it can represent in dimensions exceeding the third but can humans really see and understand the fourth dimension? Its something to think about from the hypercube.
ReplyDeleteI agree that this chapter was intriguing but my one issue with it is how can this be taught in a simpler way. When reading the chapter i had trouble understanding the chapter but with your summary it helped tremendously.
ReplyDeleteReading Chapter 13 was a bit confusing at first, but the way you phrased it and narrowed it down really helped me understand it better. Also, watching the movie that the Professor showed was useful. From the reading, I realized that every time a point generates a line, our mind is limited to visualize the shape of the hypercube. But by cutting open the corner of a square, its lines can be unfolded to form the one-dimensional figure. I found it interesting that it can be complicated to try to draw the irregular shape because of all its sides, points, and faces.
ReplyDeleteAfter reading the chapter I was a little lost on the whole subject. After reading your summary, I understood the topic a little better. I was already aware that we weren't able to see in 4-D but wasn't sure why until I read this. The whole chapter was very informative and interesting.
ReplyDelete