The Circle of Geometry

Today is the birthday of William Metzler born 1863 in Odessa, Ontario, Canada.  Metzler was a Canadian Mathematician who taught at Syracuse University and Albany Teachers Training College.  He published papers in the Proceedings of the European Mathematical Society.  His focus was on the theory of matrices and determinants.

A long time friend, Dick, and I were enjoying each other's company, reminiscing about our high school educational experiences.  After a brief lapse in the conversation, Dick stated "I should send an apology to my high school geometry teacher."  I asked why.  Dick sighed and replied, "I hated that class.  I thought it was so pointless.  Who would of thought that many of my conversations now, are with architects and revolve around geometric relationships and buildings."  I smiled, "The circle of geometry."

The circle of geometry.  I am fascinated by the circle.  The circle has been symbolic of the sun, life, boundary, completion, returning cycles, and unification of two lives as represented by wedding rings.  As I consider the circle, I think a more appropriate symbol for me is the helix.  This geometric shape combines the returning cycles of life and the experiences I gather each year.  As I garner each experience and the circle attempts to complete itself, it rises slightly, never competing the task but starting a new revolution above itself.

In mathematics, a helix is defined as a curve in three-dimensional space for which the tangent at any point makes a fixed angle with an axis.  Springs, screws and hand rails on stairwells are examples of helices.  Helices can be right-handed or left-handed.  At my dad's welding shop, there was a machine that put threads on metal rods.  Most of those threads were right-handed.  What I view as the growth of the helix is called the pitch and is measured parallel to the axis of the helix.  The equation for a helix is a parametric equation.  An example would be: x(t) = cost t; y(t) = sin t; and z(t) = t.  x(t), y(t), and z(t) are functions in three-dimensional space such that at t increases the point (x(t), y(t), z(t)) traces a right-handed helix with a pitch of 2pi radians and a radius of 1 unit about the z-axis.

In music, pitch space is often single or double helices.  These helices often extend out of a circle of fifths to represent an octave equivalency.  The DNA, double helix is probably the most famous geometric shapes known to mankind.

A molecule that encodes the genetic instructions used in the developing and function of all living organisms.  The circle is complete.  The circle of geometry.  The circle of life.

Proof and The Problem

Today is the birthday of Sir James Jeans born 1877 in Lancashire, England.  Jeans worked on astronomy, thermodynamics, heat and other aspects of radiation.  He was knighted in 1928.  Jean said, "We may as well cut out the group theory.  That is a subject that will never be of any use in physics."

I have been working on The Problem (see previous blog) for some time.  The problem has to do with a point moving about a circle and the construction of two lines and a line segment.  The first line passes through (0, 0) and the point (P) on the circle.  The line segment also has P and (1, 0) as its endpoints.  The second line is the perpendicular bisector (a line that forms a 90° angle and passes through a midpoint) of the line segment. The resulting intersection (T) between the first line and the perpendicular bisector forms an ellipse as P traverses the circle. A diagram with these conditions is imbedded in the proof below.  A colleague of mine asked me if the tracings of T truly formed an ellipse.  "Truly an ellipse" ... a simple question that requires proof from the recipient.

I was talking with my son recently and he was giving some feedback about my blogs.  I told him I wanted to blog about my proof and he asked why.  Why was proof important?  Why should he care?  Why should anyone care?

I remember when I first learned about proofs.  That learning occurred in geometry.  I struggled at first.  I was in a study hall, frustrated, and convinced I had reached the limit of my capacity to understand mathematics.  It was at this moment that the notation of a career as a lobster fisherman first emerged.  Almost simultaneously, the linkage of the proof registered.  I find a fact through the deduction process and use that fact to deduce another, linking those facts to my final conclusion ... a proof.  I justified each linkage with either a definition, an axiom, or a previous known fact (theorem).  I had an argument that verified truth without exemption.  A truth that wasn't based on opinion, emotion, tradition, or limited experience.

Recently, I experienced a conversation between two of my colleagues.  The first colleague was stating what he believed were facts based on his experience and "research".  He was attempting to link his supposed facts to prove a conclusion.  The intended conclusion was not only unknown by the listeners but also unknown by him.  The second colleague was challenging some of the facts and logic of the first colleague in a tactful manner.   The following is an excerpt of the conversation.  First colleague, "What do you think the median annual income of our area is?"  Second colleague, "Oh, I would say about $50,000."  First colleague, "No, you are way off, $46,000 at best".  After a brief search on my laptop, I chirped in "The median annual income of our area is $51,000."  First colleague, "No that can't be right.  What figures are you looking at?"  My response, "2013 figures stated in the local paper dated last week".   The frustration I have with proofs that are formed by many people to demonstrate a particular point of view is that those proofs often are unstructured and use "facts" that have not gone through verification.  Mathematics offers structure and verified facts.  I can determine truth in mathematics given a set of agreed upon definitions and conditions.  Euclid demonstrated that in his book, Elements.  Mathematical proof can serve as a model for a more logical and coherent argument.

I think we all look for absolute truths.  I believe they exist.  Kurt Godel proved that there exist absolute truths that cannot be proven mathematically.   "Is it truly an ellipse?"  "If it walks like a duck and quacks like a duck, is it a duck?"  The mathematician would say prove it.

Proof for me breaks the concept down in workable parts.  I am somewhat confident that the individual that determined the construction of the ellipse that I stated above used the proof that I am to submit, to foster its creation.  My proof is not unique.  I am sure of that but the proof verifies that what I believe to be true, actually is.  As I examine what I am trying to prove, I am forced to look at each individual part, determine both its linkage and its justification.  I feel this process allows me to understand the workings of the concept, its connections to other concepts, and its applications.

Why should my son care about proof?   As I have aged I have become more reluctant to believe in any statement that cannot be verified.  My colleague was indignant that his belief was not a fact and its lack of credibility weakened his argument, his proof.  Why should anyone believe what I have constructed is really an ellipse?  No one should.  My son should examine statements, arguments, and heated discussions from a mathematical proof viewpoint.  If anyone's proof or argument meets that criteria, they can be satisfied that the proof is true.

I cannot predict for any individual what will happen when they examine any mathematical proof but trying to understand something that is difficult, something that is a struggle creates a path of intellectual growth.

Yes, the construction is an ellipse and this is why.

90

Today is the birthday Luigi Federico Menabrea born 1809 in Chambery, Savoy, France.  Menabrea was a French-born soldier and engineer who made contributions to elasticity theory and became prime-minister of Italy.

Today was also the first day of my 34th year of teaching.  At the end of this year, I will be 56 and have completed 34 years of teaching.  56 + 34 = 90.  The prime factorization of 90 is 3x3x2x5.  Ever since I started teaching older teachers talked about the rule of 90.  The magical number when a teacher's age and years of teaching add to 90.  Once this magical number is reached, a teacher whose career started prior to 1990 can retire with full benefits.  At the end of this year, I could choose to retire.  I have been playing with that thought like I played with Silly Putty when I received it as a gift as a child.  In my mind, I have stretched, bounced, and placed that thought on a newspaper comic.  I really love what I do and this past summer was one of the most carefree that I have experienced in quite so time.  The intersection of two wants.  I assume my obsession with 90 will more than likely will be tossed aside as my Silly Putty was but there are those days when I see that toy in the store.