A mathematical look at cell size

Compound light microscopes are valuable tools in the lab. They magnify our ability to see in detail by up to 1, times, allowing us to study things as small as the nucleus of a cell. With them, we can determine the shape and structure of cells, observe the movements of microorganisms, and examine the smallest parts of plants, animals and fungi. To determine the FOV of your microscope, first examine the microscope itself. These numbers are the eyepiece magnification and the field number, respectively.

Also, take note of the magnification of your objective lens at the bottom of the microscope, if applicable — generally 4, 10, 40 or times. If your microscope only uses an eyepiece, this is all you need to do, but if your microscope uses both an eyepiece and an objective lens, multiply the eyepiece magnification by the objective magnification to find the total magnification before dividing the field number.

In the early 17 th century, two Frenchmen named Rene Descartes and Pierre Fermat co-developed what would become known as the Cartesian coordinate plane, more commonly known as the x,y -graphing plane. This invention was an extraordinary advance in the history of mathematics because it brought together, for the first time, the integration of the two great, but distinct branches of mathematics: geometry, the science of space and form, and algebra, the science of numbers.

The invention of the Cartesian coordinate system soon led to the graphing of many mathematical relations including the sine and cosine ratios. As it turns out, the trigonometric functions can also be defined in relation to the "unit circle," i. When we put the unit circle on the Cartesian plane , we can begin to see how this works if we draw a triangle within the circle, as seen in the diagram below.


The secret math of plants: Biologists uncover rules that govern leaf design

According to our earlier discussion, the sine of angle A in the diagram equals the ratio of the opposite side over the hypotenuse. However, remember that we are working with a unit circle and the length of the hypotenuse is equal to the radius of the circle, or 1. So the sine of A gives the length of the opposite side of the triangle, or the y-coordinate on our Cartesian plane.

Similarly, the cosine of angle A equals the ratio of the adjacent side over the hypotenuse. Since the length of the hypotenuse equals 1, the cosine of A gives the length of the adjacent side, or the x-coordinate on the Cartesian plane. If we redraw this triangle as we move counterclockwise on the circle, we can begin to see that the trigonometric functions, in this case sine and cosine, take on a periodic quality.

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This means that sine, for example, increases to a maximum at the top of the circle, decreases to zero as we sweep left, and begins to take on negative values as we continue around the circle. At the bottom of the circle the sine function reaches a minimum value and the process begins again as we reach the right side of the circle.

Cell Size and Scale

To better appreciate this idea, review the animation Sine, Cosine, and the Unit Circle linked below. Sine, Cosine, and the Unit Circle This animation illustrates how the values of the sine and cosine change as we sweep around the unit circle. As you saw in the animation above, as angle A increases, the values of the trigonometric functions of A undergo a periodic cycle from 0, to a maximum of 1, down to a minimum of -1, and back to 0.

There are several ways to express the measure of the angle A. One way is in degrees , where degrees defines a complete circle. If we now plot the sine of the angle measured in radians along the Cartesian coordinate system , we see that we again get the characteristic rise and fall.

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However, since the angle measure is plotted along the x-axis instead of the cosine of the angle , the graph that results is a continuous curve on the coordinate plane that resembles a physical wave, as seen below. If you look closely at this graph you will see that the wave crosses the x-axis at multiples of 3. One full wave is completed at the value 6. Understanding the origin of the sine function makes it easier to understand how it operates in relation to waves.

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  • As we saw earlier, the basic formula representing the sine function is:. In this formula , y is the value on the y-axis obtained when one carries out the function Sin x for points on the x-axis. This results in the graph of the basic sine wave. But how can we represent other forms of waves , especially ones that are larger or longer? To graph waves of different sizes we need to add other terms to our formula. The first we will look at is amplitude.

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    In this modification of the formula , A gives us the value of the amplitude of the wave — the distance it moves above and below the x-axis , or the height of the wave. In essence, what the modifier A does is increase or amplify the result of the function Sin x , thus leading to larger resulting y values.

    To modify the wavelength of a wave, or the distance from one point on a wave to an equal point on the following wave, the modifier k is used, as seen in the formula below. The multiplier k extends the length of the wave. Since waves always are moving, one more important term to describe a wave is the time it takes for one wavelength to pass a specific point in space. Understanding the mathematics behind wave functions allows us to better understand the natural world around us.

    For example, the differences between the colors you see on this page have to do with different wavelengths of light perceived by your eyes.