Math & Biological Sciences

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Our bodies and things around us are three dimensional objects. Hence concepts of length, area &amp; volume are inherent in them. Understanding the mathematical relations between these parameters gives us insights into the physical &amp; biological world.

Length, Area &amp; Volume 

Length is a linear measure. Area varies as per the square of the linear measure. Volume varies as the cube of the linear measure. Because of the mathematical nature of these parameters, their rate of change varies differently. Let us take some examples.

Scaling a Plane Figure

If in any cuboid, the length, breadth &amp; height are doubled, then its

This can be checked out by taking a cuboid of any particular dimension. These changes would happen in any solid or plane figure regardless of its shape.
 * 1) Perimeter (of any of its sides) will become twice its original size
 * 2) Area (of any of its faces) will become 4 (2^2) times the original area and
 * 3) Volume will become 8 (2^3) times the original volume

Hence the volume (a cubic function) will increase or decrease much faster than area (a square function) which in turn would be faster than length or perimeter or circumference. This leads to interesting but not so obvious effects.


 * 1) The volume of an apple which is twice the diameter of a smaller apple would be 8 times more. While buying apples, we are paying for their volume and not diameter!
 * 2) Amoeba takes its food through its entire body and hence is related to its surface area. The food taken by the amoeba has to feed its entire weight or volume. Now as the amoeba grows, its volume grows much faster than its surface area! Hence as it grows a point would come when the surface area cannot absorb enough food to feed its volume. Hence the amoeba splits and becomes 2 smaller amoebas.
 * 3) The same logic of the amoeba also operate in the cells of all living beings.
 * 4) When a fruit ripens, its volume grows faster than the skin which covers it. Hence in many cases the skin splits.
 * 5) The human body is supported by 2 legs. The body weight has to be supported by the bones of the leg. This depends on the cross-section of the leg bones. The weight of the body depends on the volume which is a cubic function. The cross section of the bone is an area and hence a square function. Hence after a particular height, the bones cannot support the weight of the body. Hence the height of the human body is decided by math!
 * 6) All animals have a skin (surface area) and weight (related to volume). For small animals like rats, the ratio of surface area to weight is much larger than that for big animals like a horse. Hence when a rat falls from a height, the air drag due to its surface skin would considerably slow down its fall. This will not happen in case of a horse. Hence a rat can fall from a tree safely whereas a horse will fracture its legs even if it falls over a few feet.
 * 7) The small intestine allows absorption of nutrients from the food to the blood in the capillaries. It needs a structure which has a large surface area in comparison to the volume. This is why the small intestine is very long with a small diameter.
 * 8) The capillaries are also long with very small diameters so that their surface area is proportionately more than its volume.
 * 9) The alveoli in the lungs help in exchanging oxygen &amp; carbon dioxide in the capillaries. The volume of the lungs is limited. But the capillaries &amp; alveoli require a large surface area for the gas exchange to take place efficiently. This is the reason both need long tubes with very small diameters. The fractal structure helps in packing such long tubes efficiently in a small volume.

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