AwardRed Ribbon Award
SchoolKenston High School
TeacherGreg Koltas
Selected Research“Analysis of Cardiac Arrest Patients: Pre-Therapeutic Hypothermia,” Kalee Bilinovic
Selected ArtWire Study of a Heart and Brain, Hallie Rasner
Selected Language“Ice in My Veins,” Richard Matonis
Math is everywhere. However, in cases such as these, it requires a comprehensive search with multiple minds in order to seek it out and unravel the details. In choosing our project's subject, it was the art that caught our eye. The mathematical possibilities of interpretations of the artwork were most intriguing, but the ironic idea of inducing hypothermia in order to save lives also sparked our interest. It was an enlightening experience examining and embellishing the mathematical aspects of the art and literature reflections and the medical research from which they came.
- Cara Fagerholm, Grant Levy, Emma Rastatter
The mechanics of the human body can be overlooked or taken for granted in everyday life. This conundrum presents vast areas of research yet to be done. In the poem entitled "Ice in My Veins" by Richard Matonis, there are mentions of blood flow and heartbeat; both involve numerous scientific and mathematical concepts, laws, and theories.
Cardiac myocytes (heart cells) communicate through electrical signals and fire by their electrical charge. All the individual myocytes fire in unison to cause the physical pumping of the heart. The activity of excitable cells can be displayed using the Theta Model where. This equation relates the rate at which θ changes as a function of time in which A is a determination of the cell's predisposition to spike (A > 0) and I is the input into the cell which can be excitory or inhibitory. Voltage can be represented by y = sinθ. If I = 0, there is no input and the cell has the ability to fire only once. In the case of excitory input (energy that "excites" the cell to make it spike), I > 0, θ will continually increase because the cell will fire periodically, going around the unit circle [Figure 1] each time. Lastly, there is inhibitory input, I < 0, at which point the equation is dependent on θ as to whether or not the cell fires. There is a threshold level of activation energy that the cell must reach in order to fire, if the myocyte is unable to breach this point, there will be no spike at all.
The law of conservation of mass states that matter can not be created or destroyed. By the implications of this law, when blood is flowing through one's veins and comes to a more constricted area of decreased diameter, the velocity must increase, causing an acceleration of the fluid. Contrary to how it may seem, according to the Bernoulli Principle, the pressure of a liquid or gas fluid decreases as the speed of the fluid increases. Thus, a high-speed flow results in lower pressure, while a low-speed flow results in higher pressure [Figure 2]. (This is known as the Venturi Effect after Italian physicist, Giovanni Venturi). Another interesting interpretation of the heartbeat is the symmetry of its pattern. Such patterns are categorized as frieze groups.
There are seven of these groups named for their descriptions of footsteps. The first group, Hop, characterizes the pattern of the heartbeat's spikes as depicted by an electrocardiogram (ECG), simply translation. The frieze groups incorporate every possible pattern of symmetry, translation, horizontal and vertical reflection, and rotation. The other six are named the step, sidle, spinning hop, spinning sidle, jump, and spinning jump. Frieze groups are found in a wide variety of places beside the symmetry of the heartbeat's spikes, including some other unexpected places. Another area of mathematics that exhibits such versatility is fractals, which can be seen in one of the art submissions in response to Kalee Bilinovic's work.
The concept of fractals is a special area of mathematics that has been observed to occur both in nature and art-as well as many other man-made creations. Fractals are an infinite, geometric pattern created by repeating a simple process-the repetition produces an occurrence called self-similarity. When understanding fractals and their application in real-world scenarios, two aspects of fractals stand out as being important: fractal dimension and the different types of fractals. Both characteristics, once understood, have a wide variety of uses, such as solving problems (e.g., the famous example of finding the length of Britain's coastline), and explaining observable patterns (e.g., the leaves of a fern plant, tree branches or a design in a particular work of art). Both of these concepts can be seen in Hallie Rasner's jewelry piece, Wire Study of a Heart and Brain; to see how they have been applied in the creation of Rasner's piece, it is necessary to have a basic understanding of both fractal dimension and the various types of fractals.
Fractal dimension is a complex concept created to relate fractals to the simple (whole number) dimensions, as fractal dimension occurs between two whole number dimensions. Since the further one tries to "zoom in" on a fractal, the more the pattern repeats, fractals can be difficult to define. Using the equation: , where D is the dimension, e is the number of magnifications, and N is the number of identical shapes seen within the magnification, the specific dimension of the fractal can be defined. There are a number of methods to determine the e and N values of a given fractal, but the simplest and most widely used is similarity method.
This method involves literally counting the number of identical shapes found at a given magnification and plugging the numbers into the equation; this works well for simple fractals. Another method for more complex fractals is the geometric method - calculating the slope using and adding 1 to get the dimension. Fractals can be a quite diverse pattern, with the many different types taking distinctly different forms. One of the most commonly known types of fractals is the Base and Motif fractal. This type consists of an original form, which is then modified using a repeated process. Fractal canopies are another type. An initial line is created, and other lines branch off of it. The lines that branch off of the second-level branches must match the angles and length ratios of the second-level branches.
Other types include IFS fractals (using geometric transformations to transform it into smaller versions of the same image and repeat the original image into a pattern) and Julia Sets (which are formed using the output information of a specified algorithm).
Other more specific and complex types of fractals can be derived from these four basic types.
In Rasner's piece, one particular part appears to contain fractals. The section is the brain section, the yellow portion on the top of the jewelry piece. This piece immediately stands out as one type of fractal: Base and Motif. The original base was the two separate straight lines of yellow wire; both of these were folded in half, and then each new segment was folded again. A semisolid form was created by repeating this process for the rest of the wire. Because this art piece was most likely not created specifically to be a fractal, the repeated modification may be hard to observe initially-the wire bends around itself and not always in the same direction. Though it may not appear to look like either a naturally occurring of specifically produced fractal, the jewelry piece does in fact show the known characteristics of a Base and Motif fractal.
Based on the information known about fractal dimension and the data available to be interpreted in Rasner's piece, the jewelry can also be analyzed in terms of fractal dimension. At the center of this section, it can be seen that there are two main wires coming together. Each wire is then bent mostly in half; each section is then bent again. The new, bent segments of the wire are all bent in half again. This process seems to have been repeated 15 times on the top wire and 10 times on the bottom; on the top wire there are approximately 42 new segments created, and on the bottom wire there are approximately 25. Using the fractal dimension equation for the top half of the brain with e = 15 and N = 42, the dimension of the top half of the brain can be approximated. In the bottom half of the brain, with e = 10 and N = 25, the dimension of the bottom half of the brain can also be approximated. Though this method provides a fairly accurate representation of the fractal patterns seen in the jewelry piece, it is important to understand that when this piece was created. It was not created specifically as a fractal, and though it has many of the properties that fractals are observed to have, it may not be a fractal in the strictest sense of the term. Because of this, both the values plugged into the equation and the values calculated by the equation can only be considered approximations.
As explained above, the fractal dimension equation is used to define fractal patterns in comparison to simple whole number equations. The jewelry piece is made up of lines, which are 1-dimensional, and resembles a solid area, which is 2-dimensional. The dimensions of both of the two halves of the brain -1.380 and 1.398 - prove this concept, as both patterns have dimensions greater than 1 but less than 2.
Fractals can be seen in many aspects of daily life.While some of these fractals have come about naturally, others are produced and manufactured. Hallie Rasner's jewelry art piece proves that fractals can be found in unlikely places and situations, including those where the occurrence of fractals was unintended. Most likely, Rasner did not plan for the brain portion of her piece,Wire Study of a Heart and Brain, to include a fractal pattern, but after analyzing the piece with methods used to analyze fractals, the resulting data matched up with accepted definitions of fractals. Seeing fractals such as this in such unexpected places show that fractals truly are everywhere.
While fractals may appear in everyday life, math appears everywhere and anywhere, especially where one would not expect it-like a medical institution. Ms. Bilinovic's research was done on a clinical method that helps patients who have suffered from cardiac arrest. The method induces hypothermia, which reduces damage done to the brain. Ms. Bilinovic then presented the data she collected in various charts and graphs. She displayed data on all of the variables she recorded, from which patients lived and died, how each patient received their oxygen supply, to how the temperature of each patient was taken.
The research she did, and the data she compiled cannot really be counted as an experiment or an observational study. An observational study is a collection of data that shows a relation between two ideas. Ms. Bilinovic's research only has one idea, and doesn't show a relation.
An experiment takes that relation, neutralizes the confounding variables, and proves there is a "cause and effect" between the two ideas. A confounding variable is a second variable that prevents the "cause and effect" from being proven. An example would be the relation between ice cream sales and number of drownings. When ice cream sales go up, more people drown. Because of the presence of a confounding variable, in this case whether or not it is summer, it cannot be proven that ice cream causes people to drown.
What Ms. Bilinovic has done, is compiled data for half of an experiment. An experiment requires a population of subjects that are divided into two groups. Half the subjects are given a treatment, and the other half are not. All the other variables are neutralized ensuring treatment could be the only possible explanation for a different result. The data that Ms. Bilinovic compiled was on the control group, or the group that was not given hypothermic therapy. Her data could be used in an experiment to prove that the treatment causes cardiac arrest victims to have fewer problems. Although, from a strictly experimental view point, there are two improvements that could be made to eliminate the possibility of inaccurate results.
The subjects Ms. Bilinovic collected data on were not randomly selected, nor were they given a placebo. A placebo is a fake treatment, which prevents the power of suggestion from influencing the answer when the subjects are humans. Now, because the subjects were in a coma, not having a placebo probably will not affect the results.
As far as we know, the patients Ms. Bilinovic collected data on
were not chosen at random, they were chosen by who was in a coma
from a cardiac arrest in the hospital. When a population of
subjects is not chosen at random, various uncontrollable variables
can wreak havoc on the experiment. Things like gender, age, and
even eye color could take your work and make it biased towards one
group. You never know if having green eyes could cause you to be
less affected
by therapeutic hypothermia. A randomly selected population will
not eliminate these variables, but it will give the results a bit
more credence. Also, by increasing the amount of subjects tested,
the resulting precision can be increased.
One of the sets of data that Ms. Bilinovic collected was on which patients lived and died along with how many cardiac arrests they had. She displayed this data in a simple chart. I took her data and her chart, and converted it into an easier to read graph that displays the various aspects of her chart in a color coded form.
The various colors of the graph represent whether or not the patient died, and what the patient died from. The blue color means that patient died while having an arrest, and red color means that the patient has a history of arrests, and the green color means that patient has had a recent arrest.
Where there are measurements, there is mathematics. A patient's LOC, or Level of Consciousness, is measured on the Glasgow Coma Scale. A patient is rated on the Glasgow Coma Scale by rating their verbal response, motor response, and eye movement. Each section has attributes that are assigned a point value, and the patient is matched with the attribute that best describes that patient. The points are then added up, and the Glasgow Coma Scale tells what LOC the patient has.
The scale ranges from 1-15, with the lower numbers being the more severe brain injury. A person who scores between 13 and 15 is ranked mild, and their problem could be as simple as a concussion. Those that score between 9 and 12 are ranked with moderate disability. They may have lost consciousness for half an hour, and will benefit from rehabilitation. One who scores between a 3 and an 8 is ranked with a severe disability, namely a coma. Anyone who scores below a 3 is ranked as having a vegetative state; they have no interaction with the environment. Ms. Bilinovic's patients were ranked on this scale, but she assigned each patient to a numerical value on the scale, not in the category that they would fall into. I took Ms. Bilinovic's information and graphed it according to the Glasgow Coma Scale.
Bibliography
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"Frieze Groups." Learn Some Math. 1999. Web. 14 February 2010
Jalics, Jozsi Dr. "The Math in Your Heart." MathFest. Youngstown State University, Youngstown, OH. 4 November 2010. Lecture.
"Kenston Kindness." Kenston Local School District, Chagrin Falls, OH, USA.Web. 26 Feb. 2011.
Rossman, Allan J., Beth L. Chance, and Barr Von Oehsen. "Topic 3, Topic 4." Workshop Statistics. 3rd ed. New York: Key College, 2008. 45+. Print.
"What Are Fractals?" FractalFoundation.org. Web. 26 Feb. 2011.