Time for Mathematics

AwardBest in Show

SchoolKenston High School

TeacherGreg Koltas

Selected Research

"Effect of Chronic Heart Failure Clinic on Hospital Admission Rates," Jacqueline Graham

Selected Art*Ticking Away,* Monica Neff

Selected Language

"Cuando el corazón deja de latir/When the Heart Stops Beating," Sydney Davis

*The mathematics induced from a single heartbeat can be expanded upon and divided into infinitely many areas. We took Monica Neff's Ticking Away and explored the qualities of time and its relation to angles. Upon examining the poem "When the Heart Stops Beating," we determined a relationship between the earth's motion and a person's location using coordinates. Jacqueline Graham's research showed connections between the normal rates of blood flow compared to the participants in the study.*

-Jeffrey Chudakoff, Robert Cordiak, Nick Gratto

- MATH IN ART
*Ticking Away*by Monica Neff has several mathematical qualities to its design and layout. First, the shadows of the clock hands have a mathematical relationship with their physical counterparts. The hands of the clocks all have shadows behind them indicating that they are outside and in sunlight of some sort. If we assume that the clock is lying face up and the 12 is facing the positive x-axis (north), we can get an idea of the location of the light source by examining the displacement of the shadows. A displacement in the x direction represents a corresponding north/south displacement of the sun, while a displacement along the y-axis represents a corresponding east/west displacement of the sun.We can determine the direction to the sun in terms of a three-dimensional vector. Using the hour and minute hands as the main reference points, you can measure the distances of the shadows from their actual counterparts. To begin, set the clock at the origin of a three-dimensional graph. For every positive 1 unit on the x-axis that the sun would be, the sun would also be 5 units up the z-axis, thus a ratio of 1:5. Also, for every 1 unit west (back) the sun is on the y-axis, it is 6 units up the z-axis. If you combine this —1:6 with 1:5, you get an ending ratio of 6:-5:30. So, the sun is 30 units up every 6 units north and 5 units west. A displacement vector ofAnother mathematical concept found in this piece of artwork is the angle between the two hands on the clocks. At any time of the day, you can take the time that the hands read and figure out the angle between them. In order to do this, you need an equation into which you can insert the time in hours and minutes and it will return the angle in degrees. One way of doing this is through computer programming. By using the computer language JAVA, a language commonly used in computer science and mathematics, we can write a program that asks for the time and gives us an answer in degrees. If we assign the variable m to minutes and h to hours, you can use equation: angle of hands = |30h – 5.5m| to solve this problem. What a JAVA program can do is make this more user-friendly and understandable. Also, you can create your program to catch certain scenarios that the equation cannot; for example, if the resultant angle is greater than 180 degrees, you can program your equation to automatically subtract your answer from 360 to get an angle between 0 and 180 degrees. To see the JAVA code and how it is used, see the Appendix.

If you wanted to make a mathematical function for the same data without programming, you would need to convert the original equation into a 1 variable function. If given a total time in minutes (m), the MOD function can help us determine the angle between the two hands. MOD(.5m,360) determines the angle between the hour hand and 0 degrees (midnight). MOD(6m, 360) determines the angle of the minute hand. If we subtract the two values:|(MOD(.5m, – 360)–MOD(6m, 360)| this will give you a degree measurement between the two hands between 0 and 360. To get the smaller angle between 0 and 180 degrees, we modify by adding 180 – |(180–answer)| to the function. This results in a final equation of: y = 180 – |180– | MOD (.5m, 360) – MOD(6m, 360)||

This chart represents the angle of the hands over a period of 4.5 hours. The angles peek every 32.7 minutes at 180 degrees and decrease afterwards until every hour and five minutes. This linear function has a minimum of 0 because the two hands cross over each other. This happens just over once an hour. Between noon and midnight, including noon and midnight, the hands will cross twelve times.

- MATH IN SCIENCE
Jacqueline's study, "Effect of Heart Failure Clinic on Readmission Rates," studies the effect of how a specialized clinic can affect the hospitalization of congestive heart failure patients after the initial diagnosis. The human heart is a delicate, complex organ which must function perfectly to ensure life within its host. The heart is divided into two unrelated halves. The left side is responsible for the systemic cycle, a periodic regulation of pumping which supplies the cells with oxygen and removes carbon dioxide. The other half also has a periodic cycle, known as the pulmonary cycle, where blood is enriched with oxygen. Both of these periodic cycles can vary from person to person, but researchers have been able to mathematically model these cycles.

Congestive heart failure is a condition where the heart’s function as a pump to deliver oxygenrich blood is impaired. This usually causes the blood flow rate to fluctuate. The blood flow rate can be measured by the mathematical equation: The heart goes through periods of contraction and relaxation, known as a systole, which are performed over seventy times per minute. Blood flow and pressure both rise and fall periodically, and these processes can be represented by basic sine and cosine functions. The most ideal blood flow is represented by the wave equation: W(x, t) = 2.5cos(Πt)cos(2Πx) + 2.5, where x is the axial position along the length of the vessel, in centimeters, and t represents time, in seconds (see Figure 3). This equation represents the intensity of blood flow as a function of time and incorporates an individual's blood thickness and heart rate. In congestive heart failure, these periodic functions are skewed, often because of weakened or stiffened heart muscles.

To study the "Effect of Heart Failure Clinic on Readmission Rates," Jacqueline Graham utilized an observational study. An observational study is a type of research which draws a conclusion by comparing subjects against a control group, in which the researcher has no control over the experiment. In this instance, Jacqueline chose to use an observational study because inducing congestive heart failure upon patients is both ethically wrong and unrealistic. The explanatory variable (or independent variable) is divided into two groups representing patients before the congestive heart failure clinic and patients after the congestive heart failure clinic. This division follows the same separation used in the original research. The sample size of the control group, those who were not enrolled in the Marymount Hospital Congestive Heart Failure Clinic, was 648 people. The sample size of the treatment group, the patients who were enrolled in the Congestive Heart Failure Clinic, was 91, 47 males and 44 females.

From the data Jacqueline compiled, she was able to measure two response variables: the total number of readmissions to the hospital and the average length of stay during hospital readmissions. Jacqueline found that the response variables both matched her hypotheses, and quantity and length of readmissions both decreased. She found that the overall quantity of readmissions fell 44.12 percent. Also, the length of stay decreased on average 66.46 percent among the population.

The problem with all observational studies is that the researcher has no control over the samples. The researchers cannot control the variables, allowing for the intrusion of confounding variables. In an actual experiment, the experimenter uses random assignment to control confounding variables. For this study, the researcher is at the mercy of the data, which often creates biased results and can mask cause and effect relationships. Although Jacqueline's sample size is fairly large, it still cannot show a legitimate cause and effect relationship, therefore rendering her conclusions somewhat suspect.

- MATH IN LANGUAGE
"When the Heart Stops Beating" by Sydney Davis contains several mathematical properties and aspects within its structure and syntax. Among these, first and foremost, is the appearance of the poem which illustrates bilateral symmetry. The heart shape resembles a polar function, such as r = 7 – 7 sin θ (see Figure 4).

The poem's first line presents a rather interesting mathematical concept. It says, "When someone's heart stops beating, the world stops spinning." This idea demonstrates the author's viewpoint that one's heart must be beating in order for the world to spin. Thus, derived from this idea, one should be able to determine the distance the earth travels in between each heart beat. By constructing a function for this scenario, one can easily determine this result. First, the circumference of the earth must be taken into account. Because the earth is accepted to be a sphere, the equation "2Πr" suffices, with an accepted radius of the earth being 3,959 miles. Earth rotates once in 24 hours (equal to 1,440 minutes) and the average heartbeat is 72 beats per minute. So, on the equator, an individual travels about 17.27 miles per minute and, hence, about .24 miles per heartbeat. One must then account for the fact that mathematically, the further one moves from the equator, the slower they are moving, until one reaches the north or south pole, at which point they are not moving at all but rather spinning in place. This means that we need to adjust this equation for latitude. If we think of the earth compared to a stack of disks of different sizes, the radius of each disk, as we move up or down, can be represented by rcosθ (see Figure 5). To calculate the actual distance traveled per heartbeat (assuming 72HB/min), the equation would then be:

Using this equation, one is able to calculate that at the equator, the earth rotates .24 miles (rounded) between each heartbeat. Furthermore, at both the Tropic of Capricorn and the Tropic of Cancer, the earth would be rotating .22 miles (rounded) between each heartbeat. As for Cleveland, the earth rotates .18 miles (rounded) between each heartbeat. If you were to plot these results, the subsequent graph would be that of a cosine wave, with the crest centered on the y-axis being the relevant section (see Figure 6). The left half of this crest represents the southern hemisphere, while the right half represents the northern hemisphere.