Circles

In the regular course of life I probably would not have met Matt. And since he spends a lot of time on the river, I know that we would not have run into another in the midst of our hobbies....but because we enjoy and are open to the perspectives, outlook and life meanderings of others, here in cyberspace is where we got the opportunity to meet, share and visit. What a world!
--Sheila from Black Tennis Pros

We all have circles we revolve in, but unlike the nearly concentric orbits in which the planets revolve around the sun, they are far from orderly. Therein lies the interesting.

Sheila. Taylor Mali. Me. Gabriela Sabatini. Four individuals. It is a most improbable event indeed that the four of us would ever sit down at a table together and discuss the career of Former U.S. Open Champion Gabriela Sabatini. And yet, today, or tomorrow, at least three out of the four of us will have read these words because today, or tomorrow, at least three of our circles will intersect.

Sheila is mostly correct. Like asteroids in orbits that cross the Earth's as they orbit the sun, in the course of regular life, the Earth and the asteroid probably will not meet--at least I hope they won't because such a meeting would be cataclysmic for both us and the asteroid. More importantly, us. But our circles intersect all the time. And every now and then, one, two, or three or more of us will be in the exact same place at the exact same time. That event is a point of intersection.



Today, Sheila will visit this blog because I left a comment on her blog indicating that recently I've been thinking about the blogosphere and that my thoughts echo hers as quoted above, and she commented that she is interested in what I have to say. Today, or tomorrow, Taylor Mali will visit this blog because today I will send him an email letting him know I'm quoting his work in this blog post and linking to his website.

Now who is Taylor Mali, you ask?

Well that's an interesting story. In short, Taylor Mali is a teacher and a poet, wouldn't you know it? As well as an advocate for analytical and critical thinking skills, creativity, and unique and strong voices. Back in the early 1990s when I was coaching high school debate in suburban Detroit, Michigan; one of the other coaches in my circle, Steve Marsh, invited me to attend a Poetry Slam competition that he was hosting and competing in at the Heidleberg in Ann Arbor. Taylor Mali was one of the competitors, and if my memory serves me correctly, Taylor led his team to victory that night performing his poem Labeling Keys below. The video performance of the poem is 3:30 long, and very worth your time to watch--not the least of which for his creative way of including Gabriela Sabatini in the poem.

Taylor Mali performing Labeling Keys



At the event in Ann Arbor I purchased a couple of Taylor's books of poetry, he signed them, and I emailed him a couple times. But then I stopped being a debate coach and gradually stopped revolving in that circle as my love of whitewater rafting was born in 1996 and I started revolving faster and faster in whitewater boating circles. The funny thing is, about a year ago on a whitewater forum called BoaterTalk, a kayaker from Washington D.C. had recently discovered poetry slams and posted a question asking if any other boater had heard of poetry slams. Of course, I had, so I looked up poetry slams on the internet. There's now a national organization and guess who its Executive Director is? Steve Marsh. Then I Googled Taylor Mali and found quite a lot of info and quickly discovered that Mr. Mali was doing quite well for himself. So I posted on BoaterTalk with how to contact the
National Poetry Slam organization and I emailed Taylor with a quick note inquiring about his recent published works--which I still need to order--and just casually mentioned how the circle had come around again in another intersection.

I hope you enjoyed Labeling Keys, but if you didn't have time to watch Taylor performing it, I hope you'll read the following poem that should ring true for every blogger circling around in the blogosphere by Teacher and Poet Taylor Mali:


Totally like whatever, you know?
By Taylor Mali
www.taylormali.com

In case you hadn't noticed,
it has somehow become uncool
to sound like you know what you're talking about?
Or believe strongly in what you're saying?
Invisible question marks and parenthetical (you know?)'s
have been attaching themselves to the ends of our sentences?
Even when those sentences aren't, like, questions? You know?

Declarative sentences - so-called
because they used to, like, DECLARE things to be true
as opposed to other things which were, like, not -
have been infected by a totally hip
and tragically cool interrogative tone? You know?
Like, don't think I'm uncool just because I've noticed this;
this is just like the word on the street, you know?
It's like what I've heard?
I have nothing personally invested in my own opinions, okay?
I'm just inviting you to join me in my uncertainty?

What has happened to our conviction?
Where are the limbs out on which we once walked?
Have they been, like, chopped down
with the rest of the rain forest?
Or do we have, like, nothing to say?
Has society become so, like, totally . . .
I mean absolutely . . . You know?
That we've just gotten to the point where it's just, like . . .
whatever!

And so actually our disarticulation . . . ness
is just a clever sort of . . . thing
to disguise the fact that we've become
the most aggressively inarticulate generation
to come along since . . .
you know, a long, long time ago!

I entreat you, I implore you, I exhort you,
I challenge you: To speak with conviction.
To say what you believe in a manner that bespeaks
the determination with which you believe it.
Because contrary to the wisdom of the bumper sticker,
it is not enough these days to simply QUESTION AUTHORITY.
You have to speak with it, too.




Each of us is at the center of our own world. But our worlds are revolving in circles that intersect with other circles, and in those intersections is where we meet other people, other bloggers, and yes, even a poet such as Taylor Mali and perhaps a champion tennis player such as Gabriela Sabatini--whom Sheila, Taylor, and I have seen play tennis and discussed her game from time to time in our own circles which today, or tomorrow, will intersect here on MTMD.


Love Is a Circle (Circles) by The Captain and Tennille



If you have time, please scroll through the following fascinating information about circles from Wikipedia. Or if you don't please scroll through to the bottom anyway.
____________________________________

What is a circle?
CIRCLES are simple shapes of Euclidean geometry consisting of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. A circle with center A is sometimes denoted by the symbol A.

A chord of a circle is a line segment whose both endpoints lie on the circle. A diameter is a chord passing through the center. The length of a diameter is twice the radius. A diameter is the largest chord in a circle.

Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle.

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

A circle of infinite radius is considered to be a straight line.

Mrs. Miniver's Problem
Mrs. Miniver's problem is a geometry problem about circles. Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference of A and B (the sum of the area of A − B and the area of B − A).

The problem derives from "A Country House Visit", one of Jan Struther's newspaper articles featuring her character Mrs. Miniver. According to the story:

She saw every relationship as a pair of intersecting circles. It would seem at first glance that the more they overlapped the better the relationship; but this is not so. Beyond a certain point the law of diminishing returns sets in, and there are not enough private resources left on either side to enrich the life that is shared. Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this; in life, none.

Alan Wachtel writes of the problem:

It seems that certain mathematicians took this literary challenge literally, and Fadiman follows it with an excerpt from "Ingenious Mathematical Problems and Methods," by L. A. Graham, who had evidently posed the problem in a mathematics journal. Graham gives a solution by William W. Johnson of Cleveland for the general case of unequal circles. The analysis isn't difficult, but the resulting transcendental equation is messy and can't be solved exactly. When the circles are of equal size, the equation is much simpler, but it still can be solved only approximately.

In the case of two circles of equal size, the ratio of the distance between their centers and their radius is often quoted as approximately 0.807946. However, that actually describes the case when the three areas each are of equal size. The solution for the problem as stated in the story ("when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle") is approximately 0.529864.

History of the Circle
Early science, particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. Notice, even, the circular shape of the halo. The compass in this 13th century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circles. The circle has been known since before the beginning of recorded history. It is the basis for the wheel which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Some highlights in the history of the circle are:

1700BC - The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π.
300BC - Book 3 of Euclid's Elements deals with the properties of circles.
1880 - Lindemann proves that π is transcendental, effectively settling the millennia old problem of squaring the circle.

Properties of a Circle
The circle is the shape with the largest area for a given length of perimeter.
The circle is a highly symmetric shape: every line through the center forms a line of reflection symmetry and it has rotational symmetry around the center for every angle.
All circles are similar.
A circle's circumference and radius are proportional,
The area enclosed and the square of its radius are proportional.
The constants of proportionality are 2π and π, respectively.
The circle centered at the origin with radius 1 is called the unit circle.
Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the center of the circle and the radius in terms of the coordinates of the three given points.

Great Circle
A great circle divides the sphere in two equal hemispheres. A great circle is a circle on the surface of a sphere that has the same circumference as the sphere, dividing the sphere into two equal hemispheres. Equivalently, a great circle on a sphere is a circle on the sphere's surface whose center is the same as the center of the sphere. A great circle is the intersection of a sphere with a plane going through its center. A great circle is the largest circle that can be drawn on a given sphere.

Great circles serve as the analog of "straight lines" in spherical geometry.

The great circle on the spherical surface is the path with the smallest curvature, and, hence, an arc is the shortest path between two points on the surface. The distance between any two points on a sphere is known as the great-circle distance. The great-circle route is the shortest path between two points on a sphere; however, if one were to travel along such a route, it would be difficult to manually steer as the heading would constantly be changing (except in the case of due north, south, or along the equator). Thus, Great Circle routes are often broken into a series of shorter Rhumb lines which allow the use of constant headings between waypoints along the Great Circle.

When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would look like a straight line on the map would actually be longer.

On the Earth, the meridians are on great circles, and the equator is a great circle. Other lines of latitude are not great circles, because they are smaller than the equator; their centers are not at the center of the Earth -- they are small circles instead. Great circles on Earth are roughly 40,000 km in length, though the Earth is not a perfect sphere; for instance, the equator is 40,075 km.

Some examples of great circles on the celestial sphere include the horizon (in the astronomical sense), the celestial equator, and the ecliptic.

Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. For aircraft traveling westerly between continents in the northern hemisphere these paths will extend northward near or into the arctic region, while easterly flights will often fly a more southerly track to take advantage of the jet stream. The area of a great circle is a quarter of the surface area of the sphere it belongs to.

Pi
When a circle's diameter is 1, its circumference is π, or:
ζ(3) – √2 – √3 – √5 – φ – α – e – π – δ
Binary 11.00100100001111110110…
Decimal 3.14159265358979323846…
Hexadecimal 3.243F6A8885A308D31319…

Pi or π is a mathematical constant which represents the ratio of any circle's circumference to its diameter in Euclidean geometry, which is the same as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159. Pi is one of the most important mathematical constants: many formulae from mathematics, science, and engineering involve π.

Pi is an irrational number, which means that it cannot be expressed as a fraction m/n, where m and n are integers. Consequently its decimal representation never ends or repeats. Beyond being irrational, it is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", probably by William Jones in 1706, and popularized by Leonhard Euler some years later. The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number.

Circumference = π × diameter. In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter.

Note that the ratio c/d does not depend on the size of the circle. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference c, preserving the ratio c/d. This fact is a consequence of the similarity of all circles.

Area of the circle = π × area of the shaded square. Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius

The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus.

The numerical value of π truncated to 50 decimal places is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

While the value of pi has been computed to more than a trillion digits, elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.

Because π is an irrational number, its decimal expansion never ends and does not repeat. This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer.

Calculating π
π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter.

π can also be calculated using purely mathematical methods. Most formulas used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus.


Full Circle

Note: Taylor Mali said this morning: You are doing the type of work I whole heartedly support.

Note: If you like this post, please take a moment and vote for it on Yearblook

Note: On July 7, 2008, this post was honored with a PlotDog Press WOOF Contest Award.

Thanks for reading.

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