## Week8_Space by Dennis Yeh

Through the use of Linear Algebra, mathematicians are able to manipulate space, and play with the universe.  For example, a 3 x N matrix could be used to describe N objects and their position in the 3-dimensional universe.  By adding another row, a 4 x N matrix can add information about the objects in the fourth dimension: time.  Any number of linear transformations and other operations can then be used to map or trace the objects as they travel through space.  Additional algorithms and theory can then be applied to the matrix, in order to manipulate it in a ton of different ways.
Like how we can only see our three dimensions, there are many “invisible” aspects in linear algebra.  Abstract ideas such as vector spaces, basis, kernel, rank, and nullity are taught in Math 33A, linear algebra, and involve imagining 3D spaces while performing operations to manipulate them.

A concept known as the “tensor” is used in multi-linear algebra has many practical applications.  A tensor is a geometrical quantity that can be expressed as a multi-dimensional array that is different depending on the basis chosen to define it.  In other words, a tensor is independent of any frame of reference.  Tensors and tensor fields are used extensively by engineers and physicists to map relationships between force and acceleration, electromagnetic fields, linear elasticity of materials, and many other concepts.  Specifically, Einstein’s theory of general relativity was devised almost exclusively through the use of tensors.

European physicists recently used such ideas in order to prove Einstein’s E=mc^2 formula, 103 years after its creation.  The computations involve “envisioning space and time as part of a four-dimensional crystal lattice, with discrete points spaced along columns and rows.”
SOURCE: http://current.com/items/89549729/einstein_s_e_mc2_theory_was_right_103_years_later.htm

So mathematics such as linear algebra is crucial in order to envision higher dimensions, including time and space.  But how does it relate to art?  There are many artists who use math as a backbone for their art.

“Who would have thought that matrix multiplication could be beautiful?” The very short description:  inspiration from an example of homogeneous coordinates in my linear algebra book and lectures about linear transforms”

I found this image by searching for examples of linear algebra in art through Google.  It reminds me of the art by Casey Reas that we’ve seen twice in class:

SOURCES: http://rotand.dk/blog/2008/06/14/matrix-multiplication/
http://reas.com/category.php?section=works

When I envision space, it helps me to consider the math behind it.  Mathematics is extremely versatile, and can explain a lot more than people give it credit for.  The universe follows a distinct set of rules, and only by determining what these rules are can we truly learn how to manipulate it.  These rules, such as F=Gmm/r^2, F=m*a, F=kqq/r^2, E=mc^2, etc. are always true, and thus used by astrologists to map the universe.  We may never be able to travel to the stars that we see (or can’t see), but through the intelligent use of mathematics, our universe seems a lot smaller.  For example, the IBM Powers of Ten video (http://www.youtube.com/watch?v=AUUkjWsNC9k) which we saw in class takes us on a trip out of galaxy, then back and into the cells and atoms that make up a man’s hand in less than 10 minutes, a feat made possible only through the imaginative use of mathematics in order to interpret space

-IBM Video: Relation between the infinitely small and the infinitely large.

-Dennis Yeh