Problem of the Week 1-31-17

Here is this week's problem of the week! Let us know how you did in the comments below!

Solution below the break.

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. 

Any unwanted images in this article will be removed at the request of the owner.

Friday The 13th, Math and Music.

           Friday the 13^th is perhaps the single most superstition inducing date. Weather you are from China or Italy where 13 is considered a lucky number, or from America where pop-culture has developed Friday the 13^th into a paranoia and horror ridden day. Indeed, Friday the 13^th ranks up there with Halloween and Valentines Day (invisible fairies shooting you with magic arrows… no thanks!) as one of the spookiest days you and I will live through.

            In the spirit of horror and tingly sensations running down your spine, it is important to appreciate, or at lease inspect one of the most bone-chilling ballads of our time: John Carpenter’s Halloween Theme. 

            John Carpenter is one of the early experimenters of synthesizers and digital interfaces in music, and this horrifying song features both analogue and digital elements. Surprisingly enough, the reason both elements are so readily accessible to composers like John Carpenter is because of math!

            Let’s go way back in time, before computers and before the widespread use of mathematical techniques to calculate approximations of irrational numbers: the year is 1600. At about this time, music theorists are on the verge of normalizing the octave into a neatly partitioned scale; and in ten years Simon Stevin will draft a report postulating the 12^th root of two to be the frequency ratio between two semitones. It will be another 20 years after Stevin’s postulate until the French mathematician Marin Mersenne will calculate the 12^th root of two (even before logarithms were used for such calculations!), giving the octave a rigorous tuning standard. This development gave professional composers access to an easy system in which they could change keys freely (the Halloween Theme is in the spookiest key of them all: D), as well as allowing music to spread rapidly since tuning an instrument could now be done in a systematic way. All thanks to math!

The octave split into a nice geometric partition.

            Fast-forward to the 1970’s, electronic music is on the rise, but still very decentralized, it will be 1983 when Dave Smith and Chet Wood pioneer a musical instrument digital interface (MIDI) that will allow synthesizers to ‘talk’ to each other (MIDI uses an 8bit format, which increased the resolution of pitch and other sound synthesis parameters {like volume, panning, frequency oscillation} to 128, but I am getting ahead of myself here). Sadly, the year is 1978, and John Carpenter is composing the Halloween theme without the use of MIDI, relying on circuitry (and a fair amount of under-the-surface math) alone to produce chilling electronic sounds.

            Because of innovations in math, music broke free of the stigma that only professionals could create pieces; so next time you hear a song on the radio, thank mathematics for those sweet vibrations.