Plain English Physics 101 part 1 long version

[thejohnlreed]

The Universe and the Mathematics:
Why They Are So Well Matched
Take 1A - Modified August 8, 2006, Saturday, 14 April, 2012
John Lawrence Reed, Jr.
Part 1

When I was a boy, I suspected that there was a common thread that ran
through all physical systems, and connected all physical laws. The
more I learned, the closer I came to identify it. A recurring thought
of a short lived image. A focused but momentary insight. A sudden and
clear panoramic view, but again and again, it disintegrated and was
gone. Defining this thread, putting my finger on it precisely, was for
a long time, just outside the range of my consciousness.

The most difficult physics problem for me, at that time, was the
conceptual understanding of atomic structure. A mathematics had been
conceived and refined by Bohr, Heisenberg, Schrodinger, Born, Dirac,
Feynman, and others, developed expressly for the operational, or
scientific analysis of atomic phenomena. My view of atomic structure
remained unclear for a long time [1], with or without the mathematics.

Today the mathematical descriptions of the universe on the blackboard
and in the published papers are abstract and (to me), devoid of any
conceptual connection to physical reality [2]. The American physicist,
Steven Weinberg, wrote, "... it is always hard to realize that these
numbers and equations we play with at our desks have something to do
with the real world." With the phrase,"...something to do with the
real world.", Weinberg reveals that the physicist mathematician has an
unformed idea as to what many of his or her, quantitative abstractions
represent conceptually.

Consider the words of the late Hungarian mathematician and physicist,
Eugene P. Wigner, "...the enormous usefulness of mathematics in the
natural sciences is something bordering on the mysterious... there is
no rational explanation for it." Eugene Wigner wrote this in a 1960
essay and continued by noting that, the ease by which the mathematics
applies to the universe is, "a... gift which we neither understand nor
deserve..."

While I did not concern myself, at the time, with our intellectual
qualifications, as the beneficiaries of the gift, I did seek to
understand why it was so effective. Wigner's essay was a major
influence on my early thinking, so it was with interest that I read
the recent words of Lawrence M. Krauss in his 2005 book titled,
"Hiding in the Mirror". Krauss addresses the ideas presented by Wigner
in the 1960 [3] essay. Krauss writes, "... are our physical theories
unique...do they represent some fundamental underlying reality about
nature... or have we just chosen one of many different, possibly
equally viable mathematical frameworks within which to pose our
questions... in this... case would the physical picture corresponding
to... other mathematical descriptions each be totally different"?

Krauss colors Wigner's concept in a shade perhaps, more reflective of
his own. My coloring of Wigner's concern is slightly different.
Although Wigner questioned the uniqueness of our physical theories,
Wigner did not doubt that the mathematics reflects a fundamental
aspect of the universe. Rather, Wigner pointed out the "uncanny"
usefulness of mathematics, and expressed some uncertainty with respect
to our reliance on the significance of the experimentally supported
predictions of mathematics, to serve as a sole and solid basis on
which to verbally formulate our "unique" conceptual physical theories.

Wigner approaches the idea that the selection of a mathematical model
determines the questions that we ask. He suggests that once we select
a mathematical model, both, our questions, and the answer to our
questions, are preordained. In other words, because the mathematics
adapts to the real world so well, our mathematical model may be easily
colored by the possibly erroneous "a priori" subjective assumptions
that we attach to the quantities that we perceive.

Where Wigner noted the "uncanny" usefulness of mathematics, I noted
that the usefulness remains, regardless of the veracity of our a
priori assumptions. As an example, first consider the Ptolemaic, earth
centered model of the solar system. The sole quantitative connection
to the real universe, in this "still useful" model, is the efficient,
least action, time-space property, attendant to each of the otherwise
contrived, circular, cyclic and epi-cyclic orbits. A circle is an
efficient enclosure of area. Equal arc lengths will radially enclose
equal areas of the circle. This is an efficient area enclosing
property of the circle itself. It is consistent with Kepler's law of
areas which law would be redundant in the case of perfectly circular
orbits. With the circle it is the circumference arc and its radially
enclosed area. In the orbit it is the time interval of the orbit
trajectory and its radially enclosed area. The law of areas is a
function involving time and space. It is a least action function. So
Ptolemy constructed several imaginary mathematical circles upon
circles to match the time space function of the real orbits. The least
action aspect of the mathematics in describing the least action
aspects of stable universe systems, assured his success. Imagine it
otherwise.

The Ptolemaic model shows that accurate mathematical predictions serve
us to a limited operational extent, but provide no absolute basis for
an accurate conceptual view. Viewed through the clearer lens of
hindsight here, we can see that our conceptual questions must be
framed correctly prior to selecting the mathematical model. Must we
frame our conceptual questions any less correctly today?

Following my analysis of the Ptolemaic model of the solar system, I
concluded that our limited perceptive ability, combined with the ease
of application of the mathematics to the universe, in terms of time
and space, reflects both, a weakness and a strength. We cannot allow
the easily applied mathematics, to lead us into otherwise (outside the
mathematical model) incomprehensible conceptual ideas, that we
validate intellectually, solely on the basis of our limited perceptive
abilities. We cannot include quantities within our mathematical models
that are loosely defined by the words of the language we think in
terms of, and expect the rigor of a mathematical model to compensate
for our laziness in conceptual thought. The mathematics will provide
no greater precision than the meaning we attach to its operational
components allow.

As evidenced by the Ptolemaic model of the solar system our reliance
on perceived events to build the mathematical model, requires that our
conceptual foundation for the mathematical model be error free. If we
carry any erroneous a priori assumptive baggage into the mathematical
model whatsoever, that mathematical model will eventually be shown to
be a new age Ptolemaic mathematical model. We require circumspect
conceptual reasoning [4] concurrent with our use of the mathematics.
As a place to begin, we must precisely answer the comparatively
simple, fairly straight forward question: "Why does the mathematics
work so well on the universe?", if we wish to obtain a non-mystical,
non-fantasy based (non-new age Ptolemaic), rationally comprehensible
understanding of natural phenomena.

Krauss continues: "... because we have made huge strides in our
understanding of the nature of scientific theories... since Wigner
penned his essay... I believe we can safely say that the question he
poses is no longer of any great concern to scientists."

During the course of my life, my wide ranging research has included
the study of every publication in English print, that I have found,
that seeks to present a popularized view of theoretical physics and
the attendant mathematics. In my many years at this endeavor, Krauss,
to his credit, is the only author I have read, that directly
entertains Wigner's essay. Further, the cutting edge of science is
focused on technological progress. Consequently, the focus of Wigner's
concern is not seen as a subject that qualifies for research grants.
Therefore, as near as I can determine, the question posed by Wigner
was never of any great concern to other scientists.

Although Wigner's concern is clearly restated as a question, and the
answer to that question resides within obtainable bounds, we have been
content to leave the question unanswered and use the mathematics as
though the mathematics is a crystal ball, enabling us a near mystical
means by which we decipher the universe. I am reminded of the quote,
perhaps by Dirac, "... my equations are smarter than
me." (paraphrased).

Wigner's concern, together with many other concerns [5], did represent
a significant problem to me. Even to the extent that my intent to
pursue a professional career in physics was eventually derailed. Now,
much to my surprise, Krauss indicates that the question has been
answered as the result of "huge strides we have made in our
understanding of scientific theories..." Krauss continues: "We
understand precisely how different mathematical theories can lead to
equivalent predictions of physical phenomena because some aspects of
the theory will be mathematically irrelevant at some physical scales
and not at others."

The word "precisely" as used with the scientifically represented
verbal stream above, is a loosely chosen, unclear and misleading,
application of the English language. Many physicist mathematicians
today, regard any spoken language as inadequate, when compared to the
more rigorous, and more intellectually forgiving mathematics. The
initial difficulty of learning the mathematics, combined with its
operational effectiveness when applied to physical processes, provide
to the academic humanist, the physicist mathematician and to educated
humanity at large the illusion that a "deep" intellectual connection
to physical reality exists, that is revealed through the mathematics,
and only accessible to the physicist mathematician. This mindset
provides the unquestioned and unchallenged world academic platform
that enables the physicist mathematician to put forward any sort of
fantasy so long as the fantasy retains a mathematical consistency with
respect to experimental prediction. To the physicist mathematician,
any notion that is not "outlawed" by say, quantum mechanics or general
relativity, is viable.

As a clear and representative example of the extent of this view
consider the following quote from Steven Hawking in response to a
question on the conceptual validity of an extra-dimensional universe.
The question, "Do extra dimensions really exist has no meaning. All
one can ask is whether mathematical models with extra dimensions
provide a good description of the universe." and, "...one cannot
determine what is real. All one can do is find which mathematical
models describe the universe we live in."

Extra dimensions are an exponential artifact of the mathematics. The
least action consistent property attendant to stable systems in the
universe allows us to use the least action consistent mathematics in a
variety of combinatorial ways that will ultimately function to
correctly predict the least action consistent behavior of the universe
that we observe. Recall that with Ptolemy we used combinations of
theoretical circles to ultimately explain orbits where the circles and
the orbits obey the least action consistent function known as Kepler's
law of areas.

Extra dimensions are brought to the real universe by Hawking with a
proclamation that I find uncomfortably similar to, "Verily verily I
say unto you..." "All we can ask..." and "all we can do..." will be
revealed by out crystal ball. Hawking a high priest in the field
speaks for all high priests in the field.

God like pronouncement on the limitations of our capacity for
knowledge coupled with the inneffectual (see Brian Greene's PBS
offering: The Elegant Universe.) disclaimers as quoted above together
with the unbridled faith humanity at large places in the conceptual
views attendant to the mathematics is one factor that caused me to
engage in what has turned out to be a life long quest. One purpose of
which was to understand why the mathematics extends the decreed
limits so effectively without providing adequate conceptual clarity.
The mathematics is capable of opening many locks. The mathematical key
must be ground so all the locks open. To accomplish this we must
understand the focus and limitations of the key itself.

Krauss continues, "Moreover, we now tend to think in terms of
"symmetries" of nature... reflected in the underlying mathematics."

First briefly consider the phrase "underlying mathematics".

Krauss is not the first author I have encountered that sets great
importance to the mystical notion for a symmetry in nature. He is
however, the first to place the notion directly at Wigner's door. Nor
is he the only physicist mathematician that considers the mathematics
as an "underlying" and therefore controlling aspect of nature, however
contrived the mathematics may, or may not be. Krauss perhaps offers
that the symmetries in nature are the reason that the mathematics
applies so well to the universe. I can agree with this to the extent
of its conceptual clarity. However, the idea for a symmetry in nature
is not new. The idea was held by the Ancient Greeks some thousands of
years ago. The Greeks believed in a divine, therefore perfect symmetry
for the motion in the "heavens". The Greeks conjectured that perfect
circles represented the symmetry. Have we progressed, as Krauss
indicates, only to the point of recognizing that the symmetry need not
manifest as a perfect circle? Indeed with enough exponential numbers
we cannot avoid obtaining some type of symmetry.

Through hindsight we can see that Ptolemy based his contrived
mathematical model on a centrist view of our place in the universe, on
experimental observation, and on a divine notion for symmetry. The
Ptolemaic model makes it clear that the notion for symmetry and
experimental observation is not sufficient to serve as a sole guide on
which we base our present day conceptual models. Ptolemy built his
mathematical model to match the observational data. One can thus say
that it predicts events. Recently we built our particle physics model,
according to a notion for symmetry and to match the experimental data.
Note that each model is built on a notion for symmetry and on
perceived data. Today, all we apparently lack is a centrist view of
our place in the universe. However it turns out that we are still
bilding the universe in terms of our own image.

We are surface earth inertial objects. We are composed of atoms. Our
particle physics model rests on the idea that atoms are composed of
more fundamental surface earth particles. The particle notion began
with the Ancient Greeks and was applied to the internal structure of
the atom after J.J. Thompson separated the electron from an atom. We
assumed that the electron maintained a granular state inside the atom,
and patterned its structural existence, inside the atom, after our
solar system, following the results obtained from the gold foil,
particle impact and penetration experiments, carried out by Rutherford
and his students.

The problems this model presented, guided our investigation through
the 20th century. Where we required extra mass, we predicted that a
neutrally charged particle existed within the atomic nucleus. Such a
particle was located outside the atomic nucleus, by the use of a cloud
chamber to detect cosmic particles that passed, unaffected, through
the magnetic field within the cloud chamber. Finding the track of the
particle was regarded as a successful prediction for the internal
description of the atomic mathematical model.

With the Ptolemaic model we had some naked eye observational evidence
to support it. Today we predict a particle and on finding a transitory
trace of it somewhere outside the model, we conclude that our internal
structure of the atomic mathematical model is predictively sound. We
say that it predicts experimental results. One problem is that the
likelihood of finding (sooner or later), say, any particular
additional unstable particle is probable, with or without the
mathematical model that requires its existence. Another more subtle
problem is this: When an atom releases a packet of energy, either
spontaneously, or, as the result of experimental modifications, or as
the result of severe natural causes, we have no absolute basis on
which to conclude that the released or absorbed packet maintains a
granular stable state inside the atom.

The fact that we can view the atom in terms of imaginary "particles in
equilibrium", and conduct successful experimental operations with this
as a guide, does not mean that the "particles" of energy that exist
outside the atom, retain that form within the atom. The fact that high
energy particles pass through crystals must be studied in the context
of how low energy particles pass through crystals, etc The particle
wave duality demands such studies..

Even so, during the 20th century the notion for symmetry and our
unquestioned assumption that the particle maintains granularity inside
the atom, served to rescue us from the detritus covered field that
consisted of some 400+ so called, elementary particles [6]. Murray
Gell-Mann developed his new age Ptolemaic, symmetrical, mathematical
model, to account for what had become a sea of flotsam and jetsom as a
result of the high energy experimental research into particle
physics.

By picking and choosing from an array of already created transitory
particles, Gell-Mann put them together in a symmetrical order, that he
called "The Eight Fold Way". This model required some new, rather
bizarre properties, as well as the uncomfortable idea (then) for a
fractional charge. In desperation perhaps, and with some desire to
maintain credibility in the field, [*] and to secure the continuation
of research grants, the model was affirmed. Gell-Mann himself, had to
be cajoled into accepting it as real. As contrived as it is, it meets
our meager scientific requirements for some spare confirmation of
prediction. Who would challenge that? Clearly its name is a reference
to eastern mysticism. Our reliance on symmetry, while catering to a
very shallow requirement for successful prediction, together with the
inclusion of our erroneous a priori assumptive baggage, led us right
where we deserve to be. Perhaps Wigner saw further than I had first
considered.

Part 2

In any event, our problem did not begin with J.J. Thompson. Some 2000
years after the Ancient Greeks, Tycho Brahe's careful observations on
the behavior of celestial bodies and Kepler's subsequent careful
analysis of those observations, revealed that the symmetry was in time
and space. The predictable solar and celestial time-space symmetry was
subsequently co-opted by Isaac Newton, and used as the carrier for our
tactile sense of attraction to the earth, quantified in terms of our
locally isolated (surface planet) "inertial mass", and declared as the
controlling cause of the order we observe in the celestial, least
action consistent universe. This was heralded as Newton's great
synthesis [7] and is so considered even today.

Isaac Newton defined centripetal force in terms of his second law to
act at a distance by setting his first law planet surface object on an
imaginary circular path of motion at a constant orbital speed. Newton
allowed his moving (planet surface like object) to impact the internal
side of the circle circumference at equidistant points to inscribe a
regular polygon. He dropped a radius to the center of the circle from
each vertex (B) of the polygon to describe any number of equal area
triangles. "...but when the body is arrived at (B), suppose that a
centripetal force acts at once with a great impulse". Taking the
length of each triangle base to the limit (approaching zero) the force
vector [ma, mv/t, or dp/dt] at the vertex (B) is by definition
directed along the radius toward the center of the circle as [mv^2/r].
[**] Again, as with Ptolemy we have a perfect circle and perfect
motion where the law of areas falls out as an artifact of the circle
itself.

Newton generalized the equal areas in equal times artifact of the
perfect circle to any curved path directed radially around a point.
"Every body that moves in any curve line... described by a radius
drawn to a point... and describes about that point areas proportional
to the times is urged by a centripetal force... to that point"

Newton extended the property of his planet surface like orbiting
object to all celestial bodies. "Every body that by a radius drawn to
the center of another body.. and describes about that center areas
proportional to the times, is urged by a force.."

Newton then ties the force directly to the force he feels and calls
gravity... "For if a body by means of its gravity revolves in a circle
concentric to the earth, this gravity is the centripetal force of that
body." In short the force acted on any orbiting object as though that
object is identical to Newton's first law planet surface object where
the force [ma] would then be proportional to the areas and times.

We cannot overly generalize sensory quantities that operate solely
within least action parameters, beyond the specific frame within which
they directly apply. Where we quantify a force we feel, in terms of
our inertial mass, as isolated on the planet surface and applicable to
surface planet inertial mass objects within the planet field, we
cannot generalize that notion of force, to serve as the cause of the
least action consistent behavior of the celestial bodies that
apparently generate the field. We can, as inertial objects, use it to
predict our operational and navigational requirements through the
field.

Consider:
Either our tactile sense of attraction to the earth (gravity),
isolated quantitatively in terms of planet surface object inertial
mass, is the cause of the least action consistent planet orbits, or,
the least action consistent planet orbits are the reason we can
isolate the independent and emergent quantity inertial mass on the
balance scale, and our tactile sense of attraction to the earth is
caused by the planet attractor uniform action on planet surface object
non-uniforn atoms pulling us to the planet surface.

Is this a reasonable "either/or" proposition? Mass causes the least
action planet orbits, or, the least action planet orbits allow us to
isolate the quantity inertial mass on the balance scale? Or, can they
both be true as defined by Isaac Newton and postulated by Albert
Einstein?

Except perhaps for the attitude of the axis of rotation of the planets
and the spatial eccentricity of the orbits that may result from the
inertial mass of the planet in opposition to the super-electromagnetic
controlled orbit, I cannot show that inertial mass enters into the
earth attractor or celestial attraction mathematics. I can show, to an
experimental accuracy of twelve decimal places that inertial mass
"does not" enter into the earth attractor mathematics during freefall
and during planet surface object orbit and escape velocity experiments
[10].

However, I can also show that the least action consistent planet
surface object orbits are the reason we can isolate the quantity,
inertial mass, on the balance scale.

The planet surface object orbits function within the constraints of a
least action consistent, time-space principle. Freefall functions
within the same constraint. Whatever the cause (see Take 1D [8]) of
the shared principle, that principle allows us to isolate planet
surface object inertial mass on the balance scale. For: if all planet
surface objects did not fall at the same rate, when dropped at the
same time from the same height, we would be unable to separate the
planet attractor surface, accelerative action (g) from the mass of the
planet surface inertial object (m) with respect to the "tactile sense
of attraction" we feel as resistance and quantify as force (force =
weight = mg). In other words, if all objects did not fall at the same
rate when dropped at the same time from the same height, we would have
no emergent quantity called inertial mass to investigate. In such a
case, the "unencumbered" field with respect to mass, required for
Newton's first and second laws, would not exist. Consequently, I say
that planet surface object inertial mass is emergent in a field that
does not act on the property of matter we feel as resistance and
quantify in terms of our planet surface object inertial mass, as
weight [mg]. Therefore as experiments show the planet attractor acts
on planet surface object atoms and not on the mass of planet surface
object atoms. Matter not the mass of the matter. Atoms.

Einstein's idea that Newton's first law applies to planet orbits
because the planets follow a curved space-time geodesic, merely
extends Newton's definition of an inertial planet surface object mass
definition of force as the cause of the planet orbits by further co-
opting the least action planet orbits, within another new age
Ptolemaic, mathematical model. Here we gained a new and further
obfuscating label for the least action consistent planet orbits: the
geodesic.

This is not to say that the new age Ptolemaic mathematical model is
without value. On the contrary. As Krauss concludes: "Thus seemingly
different mathematical formulations can... be understood to reflect
identical underlying physical pictures." Here, the phrase "Thus
seemingly different...", when replaced by the phrase "New age
Ptolemaic..." states the case more precisely.

So much for the meaning of the word "precise" when used by the
physicist mathematician outside the rigor of her or his discipline. So
much for Wigner's question. So much for the conceptual significance we
attach to mathematical predictions based on loosely defined objects of
our perception. As Eugene Wigner noted, we have yet to acquire the
intellectual framework that is necessary to properly use our gifted
crystal ball.

Fortunately, many, many years ago, during one of my unrelenting
contemplative sessions on the mathematics and the operation of the
stable systems in the universe, I found and retained, the "precise"
rational intellectual framework for it. In one illuminating insight
that accompanied, what I remember as a spring like release of torqued
tension on my brain, I had the answer to the dilemma articulated by
Eugene Wigner, and I had the object of my long sought for "common
thread" that runs through all our physical laws. Galileo may have been
the first to formally assert that, "...the laws of nature are written
in the language of mathematics." Today we may elaborate: stability in
the field requires economy in cyclic motion. It is illuminating to
note that the action stable systems must follow to maintain perpetuity
in the field, is precisely an action that mathematics represents well.
The mathematics fits the stable universe because the mathematics
easily represents [9] the efficient, time-space, least action [10]
properties common to stable physical systems.

Least action lends itself readily to mathematical analysis. As a
consequence, and as Eugene Wigner alluded to, great care must be taken
to insure that in the study of our least action consistent universe,
we do not inadvertently allow our least action dependent, mathematical
models, to include our perceived, overly generalized, locally isolated
(surface planet), a priori assumptions, solely on the basis of a
quantified consistency within specific local (surface planet) cases of
kinematic least action events. And we must circumspectly guard against
including multi-dimensional exponential fantasies made possible by our
gifted crystal ball, especially in view of the open window that allows
for additional fantasy made probable by Heisenberg's uncertainty
principle, within the crystal ball constraints of Planck's constant.

Endnotes
1] Eleven years passed before the results I obtained from my study of
atomic structure, forced me to turn my focus toward gravity. A topic
that until then, represented a solid, unassailable pillar, in my
worldview. The wave nature of particles is a clue to the structure of
the atom. I have applied this clue in Take 6.
2] Except as noted herein.
3] Actually the Krauss books are informative and entertaining. The
subject complexity is daunting. My kudos to the author. However,
Eugene Wigner's 1960 essay is seldom seriously entertained by anyone
but me. I graduated from high school in 1961. Consequently, Wigner's
essay was a major and continued influence on my subsequent thinking.
4] In Take 1D, "Mass: The Emergent Quantity", I put forward a viable,
rationally consistent, conceptual alternative, to our theory for a
mass derived gravitational force. Through the "present-sight", more
finely ground conceptual lens, provided by Take 1D, we can, with some
unexpected amplification, again see the importance of succinctly
defining the quantities we use within our mathematical model, prior to
using the accurate time-space predictions provided by the mathematical
model, to point toward an investigative direction, and prior to
describing the universe in conceptual terms. In Take 1D, I define, and
so limit, the extent to which our perception applies within the
mathematical model, and a clarity falls out of the conceptual model.
Compare this to the many mathematical models today
that exploit our limited perception, in order to provide the
foundational basis for the veracity of the mathematical model, while
abandoning any requirement for conceptual contiguity.
5] As one example, consider Einstein's postulate that all inertial
observers measure the same speed of light, regardless the velocity of
the observer and the light source. Note that light comes in one speed.
It has no acceleration one way or another. It has many frequencies and
many corresponding wavelengths. The discrepancy of velocity with
respect to the observer and source is accounted for by the difference
in frequency and wavelength measured by each observer. Therefore, if
we require the Fitzgerald-Lorentz modification, originally proposed in
response to the missing (and not necessary) "aether" left undiscovered
by Michelson and Morley, it "may" have something to do with a time of
arrival, but it has nothing to do with the measure of lightspeed. As
another example: Take 6 together with Take 1D provides an alternative
view that eliminates the mathematically predicted "blackhole". The
blackhole eventually became another major concern in my thinking.
6] The particles that are created and released by the elements are
fundamental. Those particles found regularly in cosmic and solar
streams might also be regarded as fundamental. Those particles that we
have bludgeoned into existence are most certainly, primarily rubble.
7] My respect for Isaac Newton is as boundless as is my conception of
the universe. We must note that Newton justified the veracity of his
"system of mathematical points" by writing that "since it is true for
all the matter we can measure, it is true for all matter
whatsoever." (paraphrased).
8] One example of many in the math: When we differentiate the function
that describes the area of a Euclidean circle (pir^2), we get the
function that describes its circumference length (2pir). In other
words, we get a least action (efficient) "boundary condition" for a
given least action consistent closed area function factored by "pi".
This is the simplest example, but it holds true for the function that
describes the volume of a 3D sphere and every other least action
(efficient) closed area or volume function factored by "pi", that I
have investigated.
9] A simple example of an efficient or least action (when taken over
time) function, in terms of a static form, is a Euclidean circle. The
circumference is the shortest line length to contain the greatest
area.
A circle can be viewed as a polygon with an infinite number of
infinitely small sides. This idea can be used mathematically because
we can describe a side as small as the diameter of the smallest stable
atom. At each end of each side or atom we can drop a radius to the
center of the circle. This will result in a near infinite number of
small triangles ranged around the center of the circle filling its
area. This is similar to the thought process used by the ancient
greeks to deduce the formula to determine the area of a circle.
The area of a rectangle or square is length times width. If you
construct a diagonal in the rectangle or square you split it into two
identical triangles. The area of the triangles is half the area of
the square. So the area of each triangle is length times width divided
by two. This can be written as one half the length times the width or
in the case of our triangle 1/2 its altitude times its base.
Since the circle is filled with triangles with an altitude the length
of the circle radius and the base for all the triangles is the entire
circle circumference we have the circle area as 1/2 the radius times
2pir which provides [pir^2] for the circle area.

If the reader wishes to review the Takes referenced herein, type
"johnreed take" at the google.group screen and click the search
button. Then click on the sort by date option in the mid-upper right
of your screen to avoid my earlier attempts to succinctly articulate
these ideas.
johnreed