Wesley Greiner, Keva Mickey, Nichelle Ruffin
Ultrasonic Numbers

AwardHonorable Mention Award

SchoolCleveland School of Science and Medicine

CityCleveland, Ohio

TeacherMary Simpson

Selected ResearchHigh Resolution Ultrasound, Brendan Mulcahy

Selected ArtHigh vs. Low, Chad Uehlein

Selected LanguageSeeing Via Sound, Elizabeth Beam

Under the influence of St. Ignatius student Brendan Mulcahy's high resolution ultrasound research project, we discovered that math can be hidden almost anywhere. Profoundly in art, literature and research. However, this was done with creative mathematical outlooks. Math in this particular assignment was, in our opinion, beautifully mysterious. The reason being is with just the slightest mathematical interpretation, we were able to vividly bring out the math in everything. --Wesley Greiner, Keva Mickey, Nichelle Ruffin

MATH IN SCIENCE: This research focuses on ultrasonic imaging. The high frequency ultrasonic transducers used in this research provide a much more detailed picture than the low frequency transducers. This is because the machines send out more sound waves. These waves can be measured using the trigonometric functions of sine and/or cosine to measure the frequency and the amplitude of the waves produced. These waves travel to the tissue and back to the transducer (D) in order to find the actual distance only from the transducer to the tissue (d). To actually measure the waves, the researcher would have to use the formula 1⁄2D = d.

When using the ultrasonic imaging, first there is a gel applied to the surface and it is used to smooth out ridges on the surface. Doctors/technicians must be sure to use a specified amount of the gel; the gel is used to keep those bumps from affecting the transducer’s image. Also, for the transducer to easily detect movement, it employs an x-y-z axis, then it plots points and the machine displays how the subject moves according to the points.

One of the ways a good transducer focuses is by having a spherical depression. A sphere is a shape, which is related to math. Measurements are needed to create a shape. In order to fill a 3-D shape (with air), you need to calculate the volume of the shape, which is what the technician does to focus the transducer. The shape is to be in the center of the film; to achieve this, precise calculations must be done to determine the distance the spherical shape is from each side of the transducer.

The axial resolution represents how thin the layer can be seen; this is a proportionate measure, because in many cases the image does not represent actual size. It is a comparison of the actual size of the part to the image on the screen, like a proportion, which is a big part of math. Both the axial and the lateral resolution can be found using mathematical formulas. Harmonic imaging sharpens images and pays close attention to detail. This process is very close to perfect, similar to applying rounding techniques in math.

MATH IN ART: "The most beautiful thing we can experience is the mysterious." -- Albert Einstein

In this particular picture, there is somewhat of a yin-yang effect to it. The yin-yang effect represents versatility, and in respect to mathematics, it takes on many different personalities.

For instance, the symmetry of the picture resembles a fraction: two numbers on top of each other, whereas in reality, two different colors are on top of each other. In fact, the line influencing the fraction resemblance can also be interpreted as a positive slope. Graphed linear inequalities could, as well, have relation, for their solutions are to be shaded on only one side of the graph. Coincidentallly, this picture is structured in that form.

In addition to the appliance of these algebraic concepts to the picture, there are the geometric concepts. This picture includes a few circular figures, in which each has a unique circumference, diameter, and radius, and the line that “slices” the picture in half forms two right triangles. The bottom left corner of the picture displays lines that intersect, and which come to be perpendicular, forming two right angles. This very detail can also become an angle bisector, per se. Overall, in this picture, there are a total of ten shapely figures. This amount can be translated in scientific notation as 10 to the power of 1, equaling 10. However, what truly concludes the observation of this picture is the fact that it’s so rich in mathematical reasoning — and this makes it all the more beautiful!

MATH IN LANGUAGE: The poem “Seeing via Sound” tells of how animals use a similar process as that of an ultrasound. The poem talks of how whales and bats use a sonar system, which they use to see and track down food to eat. The sonar has a certain frequency. The formula for the frequency of a wave is wave speed divided by wavelength; if numbers are put in, it can be an easily solved mathematical equation.

Frequencies are used commonly in math to find patterns and cycles. Echoes can also be converted to math since they are just waves. Waves are used all the time in math to depict sine or cosine curves which tell frequency and amplitude. The poem also mentions a narrow artery. In a narrow artery, the diameter of the gateway is smaller than the diameter of a regular artery.

Another mathematical aspect of the poem is the cadence of the words in each of the lines. Sometimes the syllables of a word can be stretched or bunched to go with the flow of the poem. Even the word flow is mathematical: it means to keep a set speed or constant rate. So even though it may not seem like it, all around us is math. Whether you see it in an ultrasound or in a work of literature, math is everywhere!