Solution below the break.

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## Tuesday, January 31, 2017

### 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.

Solution below the break.

## 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 |

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 |

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. |

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) |

-->

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, January 13, 2017

### 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. |

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.

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