Robert Cordiak, Emily Horvath, Sarah Poe
The Elegance of Mathematics in Research, Art, and Literature

AwardBlue Ribbon Award

SchoolKenston High School

CityChagrin Falls, Ohio

TeacherGreg Koltas

Selected Research

Alternative Pain Therapies and Their Effectiveness in
Previously Medicated Surgical Patients, Marissa Rose

Selected ArtMusic Monitor, Dale Banas

Selected LanguagePain Scales, Carl Buchwald

Our project is an essay in three parts. Each of the three members in our group focused on one portion of the eXpressions™ Math project: literature, art, and research. --Robert Cordiak, Emily Horvath, Sarah Poe

RATIONALE: In creating a mathematical analysis of Marissa Rose's research, we first reviewed the summer research of each student intern. We settled on Marissa’s research because, as several of us are musical, we were intrigued by the relationship between music as alternative therapy and pain relief. Next, we reviewed the art and language pieces that accompanied the research and chose to focus our studies on Dale Banas' Music Monitor and Carl Buchwald's "Pain Scales." Our group also reviewed the statistics that were found on the eXpressions™ Math Program website and created a statistical analysis of the data. The final product of our efforts is an essay detailing the mathematical principles found in statistics, art, and language. Our main objective in this project was to expose the elegance of the underlying mathematics hidden in the student submissions. We feel that by researching the presence of the Fibonacci sequence, isometries, and frieze groups within the pieces, we have successfully completed our initial objective.

MATH IN SCIENCE: Marissa's data used the process of experimental design. An experiment is intended to demonstrate how someone reacts to one particular variable while being as random as possible. Without the random factor, results cannot be deemed valid for the population. Because we do not know how Marissa collected her patients, the random allocation cannot be confirmed. An experiment was chosen because the other choices did not apply to this instance. An experiment takes the patients and changes their natural habits in order to test a variable. A survey would not be conclusive because the patients would not know how they would react to the music. An observational study wouldn't be correct because it couldn't put the patients through any controlled variables to examine what would happen.

To begin, you have to identify your observational unit, or who/what is being tested. In Marissa's case, these were previously medicated surgical patients. A small sample of six people out of the entire population was chosen to represent the whole. After the patients took their medication, they were asked to share their pain rating from zero to ten, with ten being the worst pain. If the pain was reduced, we would put them in the first group. If this rating was the same as the initial, we would put them into the second group. Then the groups were divided again (see Appendix Figure A). They were asked if they would like music therapy to reduce the pain. This may not seem random; however, in reality, patients would have the option of music therapy, and some people may not enjoy the accompaniment of music. Everyone in the reduced-pain group accepted it, and one in the equal-pain group did as well. Afterwards, patients rated their pain again. The average pain went from an initial 8.75 to 6.75 after medication to 6.00 after music. For non-music, the ratings never changed from the initial average of three.

Marissa attempted to keep the principles of experimental design in place. She controlled everything that would be in a normal environment. There were two variables in this experiment. Whether or not the patient chose music is a binary categorical variable because there are only two options to choose from. The pain rating is a quantitative variable because it is a measurement. We must assume that she randomized her patients and put them in several groups. Lastly, she replicated her experiment. This was done to reduce chance in her results, although this is hard to accomplish with her amount of observational units.

Further experimentation would be needed with guaranteed random allocation to confirm these results.

References: Rossman, Allan J. Workshop statistics discovery with data and the graphing calculator. Emeryville, CA: Key College Pub., 2001.

MATH IN ART: What is a frieze? Picture the architecture of the ancient Greeks. Visualize the towering, stone pillars and the intricate, elegant carvings. The broad middle section that lies on top of the pillars, filled with pictures and geometrical patterns, is the frieze.

In the world of mathematics, the term "frieze" takes on a whole new—but absolutely related—meaning. Over time, mathematicians have determined through the study of these ancient friezes that there are four possible types of symmetry, and, using these, that there are seven possible types of linear patterns called "frieze groups." It is possible to take this principle even further by adding yet another dimension and forming seventeen planar patterns: "wallpaper groups."

But first, back to the basics. The four fundamental types of symmetry are horizontal reflections, vertical reflections, rotations, and translations. Glide reflections, essentially a translation-reflection combination, are also included with these basic symmetry types. The following frieze groups are then formed by combining the four components seen in Math in Art Figure 1.

In Dale Banas' painting, the frieze pattern, or translations only, can be identified in the heart monitor, or EKG. The sequence of crests and troughs in the heart beat is translated about three times (see red lines).

Finally, the famed Golden Ratio can also be found in the painting. The Golden Ratio occurs when the ratio of a larger quantity to a smaller quantity is equal to the sum of the two quantities to the larger quantity (a : b :: (a + b) : a). It is an irrational number (represented by the Greek letter phi, j), but it can be approximated by dividing any number of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21…) by the number directly preceding it (i.e., 8/5 = 1.6). When larger numbers of the sequence are used, the approximation becomes more accurate. The significance of the Golden Ratio is it constantly appears in nature and art, planned or not, and is naturally aesthetically pleasing to humans. In Music Monitor, the ratio of the measurement from the crest to the trough (1.9) to the measurement from the crest to the center line (1.2) is 1.583*—very close to 1.6 (see green lines). The Golden Ratio strikes again!

References: http://en.wikipedia.org/wiki/Frieze_group, http://www.geom.uiuc.edu/java/Kali/, http://mathworld.wolfram.com/GoldenRatio.html.

MATH IN LANGUAGE: Math is embedded within the verses of Carl Buchwald’s poem "Pain Scales," if only one cares to find it. Visual symmetry, a continuous rhyme scheme and meter, and the number sequences all appear in the poem. These mathematical concepts, which are naturally appealing to the human mind, may or may not have occurred in Buchwald's work intentionally, proving just how strongly we as humans gravitate towards what is sequential, orderly, and patterned. The first aspect of the poem that jumps out to the reader is symmetry. There are four isometries used in transformational geometry. Rotation means to spin an image around a center point. To reflect an image means to produce its mirror image by essentially “folding” it over a center line. A translation is sliding an image without altering the direction. Finally, a glide reflection is a combination of a translation and reflection. Symmetry is found everywhere, from the graph of a parabola to the human body. "Pain Scales" contains reflective symmetry, with a vertical line of reflection down the middle of the poem.

Before a form of writing had been developed, people passed on stories to their friends and family by using only their voices and memories. One age-old memory tactic is to set whatever you are trying to memorize to a rhythm or, better yet, make it rhyme. "Broken Minor" follows an "ABAB" rhyme scheme, which means that the last syllable of the first and third lines rhyme, and the last syllable of the second and fourth lines rhyme. This pattern is broken in the last verse, whose rhyme scheme is "AAABBB." "Harmonic Major" follows an "ABA" rhyme scheme, meaning that only the first and third lines of each verse rhyme with each other. A love of patterns is deeply ingrained in the human mind, which explains why much of poetry follows some form of rhyme scheme.

The meter in which poetry is written is very important and, often, very mathematical. "Feet", which usually consist of two syllables each, are the building blocks of poetry. A line of poetry with three feet is referred to as trimeter, etcetera. There are two common types of feet: iambs and trochees. An iamb is a foot consisting of an unaccented syllable followed by an accented syllable. Conversely, a trochee is a foot consisting of an accented syllable followed by an unaccented syllable. "Broken Minor" is written in common meter, which means that lines alternate between iambic tetrameter and iambic trimeter. It is interesting to note that while there are eight syllables in each line of tetrameter, there are three feet in each line of trimeter. These numbers belong to a set, called the Fibonacci sequence, that has long-fascinated mathematicians.

There is a sequence of numbers known as Fibonacci numbers that occurs not only within "Pain Scales" but throughout nature itself. In the year 1202, a mathematician by the name of Leonardo of Pisa published a book of his discoveries in the field of mathematics after traveling through Asia. This book, entitled Liber Aci, was revolutionary in that it introduced the concept of using place holders while counting, instead of Roman numerals. However, this is not what Leonardo (more commonly referred to as Fibonacci) is remembered for. Also within this book, he introduced the recursive sequence known as the Fibonacci sequence.

This means that the next term of the sequence is calculated by taking the sum of the previous two terms. Thus, the sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

Fibonacci numbers are found in nature within everything from pinecones to sunflowers to leaves on stems of plants. In fact, the sequence originated with a hypothetical pair of rabbits. This pair of rabbits takes one month to mature, after which point it proceeds to have one pair of rabbits each month. The successive pairs of rabbits mature in the same manner. Assuming the rabbits never die and have only one pair of rabbits at a time, the result is the Fibonacci sequence. A sequence known as Lucasian numbers is also present in this piece of poetry. This is also a recursive sequence whose first two terms are two and one. The sequence is generated by adding the previous two terms to get the next term. Some of these Fibonacci and Lucasian numbers are found frequently within "Pain Scales". There are two sections to "Pain Scales": "Broken Minor" and “Harmonic Major.” "Broken Minor" has twenty-six lines. The number twenty-six is divisible by four numbers: one, two, thirteen, and itself. The majority (twenty-two lines) of the lines in "Broken Minor" (twenty-six lines) contain eight syllables. This ratio reduces to 11: 13. Within “Harmonic Major” there are seven verses with twenty-one lines total. As a whole, "Pain Scales" has twenty-one verses and forty-seven lines total.

The frequency of Fibonacci and Lucasian numbers intentionally or unintentionally included within "Pain Scales" is astonishingly high. Therefore, one must conclude that these patterns, and math in general, appear in everything we see, everything we do, and are certainly present in Carl Buchwald's "Pain Scales".

References: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html, http://mathforum.org/sum95/suzanne/symsusan.html.