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Friday, January 27, 2017

Using Math to Create Something Beautiful

                   Think back to when you were first introduced to functions, thin lines depicting a single value output for each input in a domain.
A Function
For many people, the idea of what a function ‘looks’ like does not change much from this bland depiction of data. However, data can be crafted into something that carries much more information than just inputs and outputs, and in the right hands an enormous and messy set of data can be presented in a powerful way. Indeed, data representation is an important part of any scientific field.
            Compacting more data into inputs and outputs provides not only more information, but also a more stunning visualization of data. In a vector field, a single point can contain information about location, strength and direction of a force.  The more information a function tracks, the more stunning the display becomes, with 4 or even 5 dimensions represented on a graph of three-dimensional space and color.

Vectors depicting the strength and direction
of a magnetic field at discrete points.
A 3D graph, with a 4th color dimension
 Vector fields can even represent information that cannot be easily compiled into a simple function, which allows for out-of-the-ordinary occurrences in nature to be studied more carefully.
With developments in technologies that offer efficient data manipulation, the possibilities of what we can do with functions and data are more far-reaching than ever. Anne M. Burns of Long Island University Uses computers to create beautiful representations of functions.
Burns plots complex valued functions as a vector field, seen here.
The advantages of this technique transcend aesthetic purpose, and can be used to find roots of functions at a glance.
Attributing more dimensions to an occurrence is useful and can be beautiful, but what if the object or function in question is impossible to make sense of as it is? It is often handy to project or unfold an N-dimensional surface onto an (N-1)-dimensional surface. Most of the time, in calculus, a three-dimensional surface will be looked at as a two-dimensional projection on the xy, yz, or xz plane in order to set up an integral to find the volume of the object. In theoretical physics, this technique of reducing the dimension of mysterious happenings is used to speculate the nature of the universe. A common example, and perhaps the most accessible way to think of this process is the unfolding of a four-dimensional cube, the tesseract.
The nets of a 3D cube and a 4D hypercube above.
Dali's Corpus Hypercubus (1954)
This way of thinking about higher dimensions caught more than just the eyes of mathematicians and scientists. Salvador Dali, the great surrealist painter, was fascinated by the advances in science during the twentieth-century. In the 1950’s Dali was fascinated by nuclear physics and quantum mechanics, and found inspiration for many paintings in mathematics.

At its core, mathematics does not only seek knowledge, but also pursues beauty in the natural world.

Works Cited
  • The Function graphic was found on the page of Maret School's BC calculus page, and is spliced with Charlie Brown of Peanuts, created by Charles M. Schulz.
  • The Magnetic field graphic was found on Vassar College's Wordpress blog, under a lecture by Prof. Magnes.
  • 4D graph curtesy of user Blue7 on math.stackexchange. 
  • Find all of Anne M. Burn's Work here.
  • The Cube net image was found here

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