Monday, April 7, 2014

The surface properties of the human body.

What I would like to talk about today is the human body. Specifically the surfaces of the human body.  As a mathematics enthusiast, it is always our goal to explain nature through a set of equations.  The human body, a product of nature can be analyzed using mathematical techniques derived from Newton.  Most students who have studied second year calculus will be able to understand this analysis.

A surface or a volume can be described by its curvature.  For example the curvature of a circle is the inverse of its radius.  A curve in the space domain is characterized by its spacial derivative or the gradient of the curve.  Our team at SFU has devised a way to analyze the curves of the female body.  Here are a few characteristics of the female body, translated into the mathematical jargon, so commonplace in differential calculus.

Zero or near zero derivative.

This is not a appealing characteristic when we are speaking of a specimens thorax or gluteus.  However is a desired characteristic when referring to the abdomen.  In mathematics, taking a spatial derivative which results in zero or near zero implies there is zero rate of change with respect to space.  On the surface of the body, this means flatness.  We consider a typical Cartesian coordinate system, with z being in the upwards direction and perhaps x being normal to the surface of the body and y being orthogonal to those two vectors. We are describing.
x= Cz
This means as we travel down the z direction of the body, there is a constant value. As you can assume this is a favorable quality in some positions of z, and not favorable at all in other parts of z.

Non-zero derivative
At some points we can expect to see linear functions and quadratic functions of z.

x=(Cu(z+z0)z+Du(z+z1)z^2)z

Where u(z+z0) represents the unit step function. For example if we are going in the negative x direction from the top of the neck to the stomach. The initial curve likely shall be positive linear, however at some point it will be negative quadratic in order to return to the original height of x.  Clearly we are interested mostly in the values of C and D.  These constants will determine how extreme these curves are.  It is desirable to see high values of C or D in certain regions, however it is extremely undesirable for nonzero values of D (or to even imagine C) on other regions.  Unfortunately, due to the obesity epidemic of the western world, we have seen many instances of these non-zero derivatives in the for-mentioned regions.

Piece-wise-differentiable

This is a very rare situation to see.  In mathematics this means there is a cusp or a corner, in which the derivative is non-continuous.  The electric field in a space charge region, the dirac impulse, a square wave, etc are all examples of piece wise derivatives which have sudden changes in rate of change.  In our field this is seen on mostly african american woman, and to some recent researchers  with two copies of a recessive gene on chromosome 16, also known as redheads.  We see this and the immediate response is "Daaamn" or a collection of unintelligible words.  This occurs when the sacrum or lumbar region is characterized by a near zero derivative, and the adjacent gluteus has a non-zero derivative with a zero value for C the linear term, and a larger value for D, immediately where the gluteus starts and the lumbar region ends.  Many great writers like Biggie Smalls and Tupac Shakur have wrote full length sonets on this topic.  In "Can i get Witchu" Mr. Smalls summarizes his desire to gain carnal knowledge of a women with a piece wise-differentiable lumbar/gluteus.  In the song the woman asks "Why you want to get with me" upon Biggie replies "cause you got a big B-U-T-T."
Our group at SFU is most interested in the latter of situations.  If you know someone with a piecewise differentiable lumbar/gluteus we encourage you to have them contact us at the "Irmacs" centre for Mathematical Study.