Re: Defining a circle


"hetware" <massless@xxxxxxxxxxxx> wrote in message
news:SpKdnXOCOKRGRRnZnZ2dnUVZ_u-dnZ2d@xxxxxxxxxxxxxxxx
| The Sorcerer wrote:
|
| >
| > "hetware" <massless@xxxxxxxxxxxx> wrote in message
| > news:YKGdnR_G6c2FtR7ZnZ2dnUVZ_rmdnZ2d@xxxxxxxxxxxxxxxx
| > | The Sorcerer wrote:
| > | > All mathematics starts with primitive axioms which are to be
accepted
| > | > without proof.
| > |
| > | So you say.
| >
| > Yep, and so do other mathematicians.
| >
|
http://www.google.co.uk/search?hl=en&q=primitive+axioms&btnG=Google+Search&meta=
|
| Proof by concensus is not persuasive.

Agreed.
hetware's theorem of inverted mathematics:
Mathematics starts with complicated theorems created by hetware and then
the rest of us try to figure out what the *** he's babbling about.
Proof follows.

|
| > | I accept that as a definition for the circumference of a circle.
| >
| > Good, because it is.
|
| As long as it is accepted.

You did.
=================================================
Some ideas are defined. Some some are assumed
| without recognition. Some are stated as axioms (which I hold to be
| arbitrary in general). Some are stated as postulates (which I hold to be
| those axioms which are "self-evident"). The remainder of ideas in
| geometric reasoning are derived from the foregoing. When attempting to
| establish a model on the basis of first principles the questions become,
| what are the first principles? Which of them are definitions? which of
| them are axioms? And which are derived? When attempting to expose
existing
| ideas in terms of first principles, one of the most difficult aspects of
| the exercise is to separate those ideas which we hold to be true by habit
| from those which we have established as or from first principles.
=================================================

Preceding moot paragraph ignored. See hetware's theorem.
|
|
| > But you
| > | assumed more in saying 2pi radians is the "arc length" of a complete
| > | circle, etc.
| >
| > Did I? Oh well, I didn't give the definition of 'radian'. I'm not going
| > to write a text book, there are some things I expect an intelligent
| > student to understand.
|
| Understand in what sense? I have used the concept of a radian from a time
| so long ago that I cannot even recall when I encountered it, or a time
when
| I was not familiar with it.

Mathematicians are not responsible for Alzheimer's disease. <shrug>
See hetware's theorem.

| For practical purposes, I take it as clearly
| defined. What I am not sure of is how it can be defined in terms of
| Euclid's Axioms.

Ok... Three points.
2) I am not going to write a text book,
3) You are writing to a physics newsgroup, take it up in sci.math or
alt.math
1) When *I* give a definition of something, that means it is what it is
because
I say it is.

A radian is defined as "one". One what? One radian, of course.
It is equal to the radius laid alongside part of the circumference, and
there are
2pi radii to a circumference because I say so and because I can
roll a beer can (hopefully now empty) across a bar (which is a plane) and it
will travel pi diameters, and if you can make it roll further with "one"
revolution
then you cheated, you argumentative dingbat.
(Proof of hetware's theorem now given).


| > | In plane geometry, if there is a line segment drawn passing
| > | through a circle and sharing a point with the center I say the circle
| > | has been bisected.
| >
| > Ok...so do I. We agree.
| >
| >
| >
| > I believe that the arclength of one side of the bisection
| > | is equal to that of the other. But I cannot say that follows
| > | immediately from Euclid's axioms.
| >
| > In other words you cannot prove it to you.
| > Does this help?
| > http://en.wikipedia.org/wiki/Megalithic_yard
|
| No.

See hetware's theorem.

|
| > I read Thom's book about 40 years ago... interesting material.
| >
| > I like this clock, don't you?
| >
|
http://tinyurl.com/create.php?url=http://images.google.co.uk/images?q=Stonehenge&hl=en
|
| I guess you aren't aware of the prior art, eh?

See hetware's theorem.
|
|
|
3.1415926535897932384626433832795028841971693993751058209749445923078164062862\
| > |
| >
|
089986280348253421170679821480865132823066470938446095505822317253594081284811\
| > |
| >
|
174502841027019385211055596446229489549303819644288109756659334461284756482337\
| > |
| >
|
867831652712019091456485669234603486104543266482133936072602491412737245870066\
| > |
| >
|
063155881748815209209628292540917153643678925903600113305305488204665213841469\
| > |
| >
|
519415116094330572703657595919530921861173819326117931051185480744623799627495\
| > | 673518857527248912279381830119491
| >
| > Ok, very good, what is the difference between that and pi?
|
| If the number I posted is designated as phi, then the difference between
| that and pi would be phi - pi.

You made an arbitrary definition of 'designate'. The number you posted is
defined as (pi + epsilon) because I say so and I wanted to know the value
of
epsilon, phi is too arbitrary. See hetware's theorem. Only I am allowed to
define or designate because I unreasonably demand so, just to be
argumentative.
What the *** this has to do with mathematics is irrelevant, but it sure is
nice to trade arse kicks, which is the real purpose of these newsgroups
(proven by consensus).

| > |
| > | No. Perhaps I did not express myself well, but I meant to accept the
| > | definition of pi as regards the circumference of a circle.
| >
| > Okay... do you accept an arc is longer than a chord, no matter
| > how small the angle subtended?
|
| Working in plane Euclidian geometry, given two lines which are not
parallel,
| and a circle centered at the point of their intersection, I can and shall
| *define* arcs as those portions of the perimeter of the circle delimited
by
| the points shared by these lines and the perimeter of the circle[*].


You are not allowed to define. You must begin with a finished theorem which
is
to be broken down into lesser theorems and finally to axioms. See hetware's
theorem of inverted mathematics.



| Therefore, four arcs are thus determined. There are some clear
| relationships between these four arcs. For example, through obvious
| argument based on Euclid's axioms I can show opposing chords to be equal,
| and adjacent angles to be supplementary (Mr. Hoag is probably stunned that
| I actually learned anything in his class[**]). For the sake of this
| discussion, I shall assume we have chosen one of these four arcs as the
arc
| under consideration[***].
|
| I can agree that,

That's a tautology. Of course you agree with yourself.

| given any arc, I can select a point on that arc which is
| distinct from either of its endpoints. It is obvious to the intuition

Arbitrary stuff, intuition. Straight sticks in water are bent, we can see
they are. It's intuitively obvious and also empirical data.
Proof by intuition is not persuasive.

| that
| the triangle

All triangles are isoceles, it is intuitively obvious.
http://www.jimloy.com/geometry/every.htm

I find it redundant to include the rest of your post.
Androcles.




defined by these three members of the circle will not share
| any other points with the circle. Furthermore, it is obvious that the sum
| of the lengths of the chords forming the obtuse angle is greater than the
| chord joining the endpoints of the arc. This process can be repeated ad
| infinitum. Every iteration of this algorithm will produce a polygonal
| "approximation" to the arc.
|
| But Euclid didn't give me any means of discussing the length of a curve
| explicitly. Nor did he give me a means of transcending the distinction
| between the arc and the polygonal "approximation". It appears I need
| another axiom here.
|
| [*]From here on, in this discussion I will use the term circle as
synonymous
| with the perimeter of the circle unless otherwise indicated - implicitly
or
| explicitly. This is actually the definition I began with, but it is not
the
| definition commonly used in geometry for a circle.
|
| [**] Due to my virtual lack of participation I earned an F in geometry.
My
| teacher, Mr. Hoag had offered a challenge to the class by presenting an
| exceptionally difficult problem, the solution of which would raise the
| students grade by a letter. I was the only student to solve the problem.
| I therefore earned a D in geometry.
|
| [***] I will also recognize the case of a bisector as a means of
determining
| two arcs, but will limit my discussion to arc which are defined by two
| non-parallel lines.
|
| > (I have to admit I do not know the name for one side of a regular
| > polygon enclosing a circle if it has one, but its centre point and a
| > point on a circle coincide)
| >
| > | It is the
| > | concept of partitioning the circle which I find ill-defined on the
basis
| > of
| > | Euclid's axioms.
| > Okay... So what are you going to do about that?
|
| Well, my first approach was to seek prior art. In lieu of finding such,
| perhaps I will be force to introduce my own postulate. I suspect this
| matter was discussed when the calculus was introduced.
|
|
| > Well, that was nice of him. A circle is the set of all points
equidistant
| > from a given point known as the centre.
| > As I understand you, though, you were having difficulty with a rotation
| > matrix for which you do not need a circle.
|
| I was trying to derive the rotation matrix beginning with the assumption
| that dx^2 + dy^2 represents an invariant under a set of transformations I
| will call rotations.
|
| > Can we focus on just what it is you wish to achieve?
|
| I am trying to determine a minimal set of assumptions which is necessary
and
| sufficient to define Euclidian space. In the case of Lorentz geometry,
| this was done by making three assumptions. We call one inertial frame the
| rest frame, and the other inertial frame - which moves with the speed beta
| along the x axis of the rest frame - the moving frame. We require the
| components of the transformation matrix to be such that:
|
| 1) they are independent of the event (point in 4-space) being transformed,
|
| 2) an object at rest in the moving frame has a velocity beta in the rest
| frame
|
| 3) any interval +-abs(dt^2 -dx^2)^(1/2) has the same value in both frames.
|
| From that we can arrive at the Lorentz transformations. The derivation
used
| in Spacetime Physics first arrives at the algebraic form of the Lorentz
| transformations. Hyperbolic trigonometric functions are then introduced
by
| defining beta = tanh(theta). Now the corelation between theta and our
| traditional concept of velocity is a somewhat interesting matter.
|
|
| My objective was to follow an analogous development for Euclidian space.
| From my way of thinking, it seems more obvious to show the projections of
| the basis vectors constitute the components of the transformation matrix,
| and then to show that they can also be expressed in terms of slope.
That's
| where I got stuck.
|
| This diagram is intuitively obvious to me:
|
|
http://baldur.globalsymmetry.com/open-source/org/sth/mma/UnitCircle/basis.gif
|
| But what it implies is not something I know how to arrive at using
geometric
| reasoning and Euclid's axioms.
|
| > | > The construction of the rotation matrix is by straight-edge, compass
| > | > and thought. I am not going to define those, and it is thought that
| > | > you are engaged in. Watch out for self-contradictions.
| > |
| > | I was attempting to reproduce the development of Taylor an Wheeler
| >
| > Hmm... I know something about Taylor-Maclaurin series,
|
| That appears to be plausibly relevant to this discussion.
|
| > but not too much
| > about Taylor and Wheeler, I consider them shitheads who wrote about
| > bright green flying elephants or black holes or something. I'm the
world's
| > expert on bright green flying elephants, not them, *and* I can prove it
| > to my satisfaction:
| > http://www.androcles01.pwp.blueyonder.co.uk/Genesis/BGFE.JPG
|
| I object to your calling these men "shitheads", and I will defer to you if
| and when I become interested in bright green elephants.
|
|
| > Ok... x = cos, y = sin.
| > Or did you require a proof of Pythragoras? I can do that too.
|
| http://baldur.globalsymmetry.com/open-source/org/sth/mma/Pythagoras/
|
| > |
| > | > You snipped. *** you, I will no longer answer you if you do that
| > | > again, you argumentative ***, you'll go straight in the killfile.
| > | > Androcles.
| > |
| > | Real men don't advertise their killfile. Nonetheless, I am completely
| > | at
| > a
| > | loss to what has your panties in a wad.
| >
| > You have not been killfiled - yet. I get somewhat frustrated by
| > the obviously intelligent yet strangely stupid student who challenges
| > "That is a set of examples, not a definition.", driving me into an
| > explanation that should be unnecessary.
| > Nor can I exactly quote you without referring back, which
| > pisses me off because although I have done so I'm partly lazy.
| > Below are my replies, now making the thread less tidy.
| > Whereas I am prepared to help, I expect the student to do his part,
| > not get his knickers in a twist and behave himself. This is your thread,
| > what you achieve with it is up to you, I see no other correspondents
| > and I can drop you in an instant. Only cowards snip.
|
| If you are talking about my snipping this:
|
| > Ahhh... you need only cos^2 + sin^2 = 1 for that, which is Pythagoras
| > again.
|
| I found it redundant to include that and this:
|
| > This may help.
| > http://www.androcles01.pwp.blueyonder.co.uk/AC/AC.gif
|
|
| --
| http://www.vho.org/GB/c/DC/gcgvcole.html
| http://www.vho.org/GB/Books/dth/
| http://www.germarrudolf.com/
| http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm


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