N-gonometry

[Ross A. Finlayson]

Trigonometry is regularly known as the development of the sine and
cosine as functions of side lengths of triangles/trilaterals in
Euclidean geometry. Sine and cosine as orthogonal functions see many
applications in differential analysis and the mathematics of infinite
series as approximations and representations. The sine and cosine can
also be diagrammed from the evolution of the points of a vertex of an
unhinged triangle. Generalizing this to other n-gons, there is to be
developed a general theory of an n-gonometry as rotograph or rotogram,
then the development of functional systems with corresponding
identities and relations.

Ideas....

Regards,

Ross Finlayson

[1treePetrifiedForestLane]

Schoute, "polygonometry," may be that.

when I phoned Coxeter, and asked him
about "tetrahedronometry," he said that
I really meant polygonometry, but he was wrong;
after I looked it up.

[Ross A. Finlayson]

On Apr 30, 6:19 pm, 1treePetrifiedForestLane <Space...@hotmail.com>
wrote:

Polygonometry is a different thing, into the pyramidal, which is why n-
gonometry was coined instead.

Still haven't worked much on it beyond the general notion that the
resulting structures may be of interest.

Thanks, regards,

Ross Finlayson

[William Elliot]

The generalization of planar trigonometry is spherical trigonometry.

> Ideas....

Yea, replace your random thought and pseudothink gibbering
generators with something appropriate for homosapiens.

[Ross A. Finlayson]

Are we not men?

_A_ generalization of trigonometry is that what is also called
polygonometry. Above is described a generalization of trigonometry,
from the geometrical evolutes that generate sine/cosine, generalized
to parameterized evolutes of n-gons or polygons.

Then not having bothered to create machines to chart them with plots,
or yet enough time on a computer program to visualize their
progression, or simply in geometry to derive symbolically their values
and relations, as sine and cosine of trigonometry have seen since
when, still it seems that with so much value in the humble triangle's
ratios, that as features of geometry there's as much to garner as from
other generalizations, as are mathematics.

Or: thanks, thank you very much.

Regards,

Ross Finlayson

[JT]

What is the units in a trigonometry where the unit circle is one, and
the angles fractions of one?

[Robin Chapman]

Each is three hundredth and sixty times the units where
the unit circle is three hundred and sixty :-)

[JT]

Yes that is called degree when a circle is divided into 360 subturns.
I found this on answers.com
->Two reasons. Divisibility was important to ancient mathematicians.
360 is divisible by all numbers to 10, other than 7; plus a lot more
larger numbers. The second is that the earth took just around 360 days
to orbit the sun.
From a high level of divisibility we have now moved to a situation
where, in advanced mathematics, the measure of one full turn is 2*pi,
not even a rational number

So let me ask this then why is 2*pi used, because that is just an
approximation while 1 turn is a clear cut, and you can slice it into
any you are going for. You can not do that with irrationals because
they are simply approximation.

And again what do you call the angles of a unit circle using
fractions? Subturn?
Using fractions of course you can see inner sums of angles in full
turns...
Full turns
3 triangle or trigon 1 turn
4 quadrilateral or tetragon 1 turn
5 pentagon 1,5 turn
6 hexagon 2 turn
7 heptagon 2,5 turn
8 octagon 3 turn
9 enneagon 3,5 turn
10 decagon 4 turn
11 hendecagon 4,5 turn
12 dodecagon 5 turn

Did i get it correct?

[JT]

Sorry you can slice it into any precision you are going to making
clean cuts with fractions, am i right? But while using 2*Pi you are
again into the water of approximation, and your angles may not end up
to be one full turn without some adjustment.

[Richard Tobin]

In article <ec9fe7fc-e377-434d...@e13g2000vbn.googlegroups.com>,

JT <jonas.t...@gmail.com> wrote:

>From a high level of divisibility we have now moved to a situation
>where, in advanced mathematics, the measure of one full turn is 2*pi,
>not even a rational number

Divisibility and rationality of angles may be important to engineers,
but they are not important in "advanced mathematics".

>So let me ask this then why is 2*pi used

It's what you get when you use the natural unit of the radius.

>You can not do that with irrationals because
>they are simply approximation.

Where did you get that idea from?

-- Richard

[Robin Chapman]

On 01/05/2013 14:26, JT wrote:

> Yes that is called degree when a circle is divided into 360 subturns.
> I found this on answers.com

Bully for answers.com!

[JT]

On 1 Maj, 15:33, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <ec9fe7fc-e377-434d-8ff6-e2b051f8b...@e13g2000vbn.googlegroups.com>,

>
> JT  <jonas.thornv...@gmail.com> wrote:
> >From a high level of divisibility we have now moved to a situation
> >where, in advanced mathematics, the measure of one full turn is 2*pi,
> >not even a rational number
>
> Divisibility and rationality of angles may be important to engineers,
> but they are not important in "advanced mathematics".
>
> >So let me ask this then why is 2*pi used
>
> It's what you get when you use the natural unit of the radius.
>
> >You can not do that with irrationals because
> >they are simply approximation.
>
> Where did you get that idea from?
>
> -- Richard

So you say that Pi is not irrational, or not an approximation?
An irrational number is any real number that cannot be expressed as a
fraction a/b, where a is an integer and b is a non-zero integer.
Maybe i missunderstand your objection?

[JT]

Well it was the below part that were frin answers com, no i was not
sure about the random slicing of the unit circles origin. But i could
have guessed...

[Robin Chapman]

On 01/05/2013 14:48, JT wrote:
> On 1 Maj, 15:33, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
>> In article <ec9fe7fc-e377-434d-8ff6-e2b051f8b...@e13g2000vbn.googlegroups.com>,
>>
>> JT <jonas.thornv...@gmail.com> wrote:
>> >From a high level of divisibility we have now moved to a situation
>>> where, in advanced mathematics, the measure of one full turn is 2*pi,
>>> not even a rational number
>>
>> Divisibility and rationality of angles may be important to engineers,
>> but they are not important in "advanced mathematics".
>>
>>> So let me ask this then why is 2*pi used
>>
>> It's what you get when you use the natural unit of the radius.
>>
>>> You can not do that with irrationals because
>>> they are simply approximation.
>>
>> Where did you get that idea from?
>>
>> -- Richard
>
> So you say that Pi is not irrational,

He didn't.

Mathematicians successfully deal with irrational numbers every day.
Sci.math readers deal with irrational posters every day.

[Richard Tobin]

In article <833805a1-e087-4d63...@12g2000vba.googlegroups.com>,

JT <jonas.t...@gmail.com> wrote:

>So you say that Pi is not irrational, or not an approximation?

It is not an approximation.

-- Richard

[JT]

On 1 Maj, 15:55, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <833805a1-e087-4d63-b9c8-619e027fc...@12g2000vba.googlegroups.com>,

>
> JT  <jonas.thornv...@gmail.com> wrote:
> >So you say that Pi is not irrational, or not an approximation?
>
> It is not an approximation.
>
> -- Richard

?
I thought that any number that can not be expressed as a fraction or
sum of fractions was an approximation, isn't it so?

From wikipedia
Approximation usually occurs when an exact form or an exact numerical
number is unknown or difficult to obtain. However some known form may
exist and may be able to represent the real form so that no
significant deviation can be found. It also is used when a number is
not rational, such as the number π, which often is shortened to
3.14159, or √2 to 1.414

[JT]

I think you missed the or Robin, you will have a hard time deal with
bolean logic if you always cut the expression halfway before the
logical conjunction appears.

[Richard Tobin]

In article <18cf7805-17ad-4c8d...@f18g2000vbs.googlegroups.com>,

JT <jonas.t...@gmail.com> wrote:

>I thought that any number that can not be expressed as a fraction or
>sum of fractions was an approximation, isn't it so?

No.

>It also is used when a number is

>not rational, such as the number pi, which often is shortened to
>3.14159

The rational 3.14159 is an approximation to pi. Pi itself is
not an approximation.

To return to your statement above:

>I thought that any number that can not be expressed as a fraction or
>sum of fractions was an approximation, isn't it so?

If a number can't be expressed as a fraction, than any attempt to
express it as a fraction will have to be an approximation. But
that doesn't mean the number itself is an approximation; it's just
a limitation of fractions.

-- Richard

[JT]

On 1 Maj, 16:32, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <18cf7805-17ad-4c8d-87bb-5fb29f490...@f18g2000vbs.googlegroups.com>,

>
> JT  <jonas.thornv...@gmail.com> wrote:
> >I thought that any number that can not be expressed as a fraction or
> >sum of fractions was an approximation, isn't it so?
>
> No.
>
> >It also is used when a number is
> >not rational, such as the number pi, which often is shortened to
> >3.14159
>
> The rational 3.14159 is an approximation to pi.  Pi itself is
> not an approximation.

Well Pi is irrational so it do not have a finished decimal expansion
in any calculation.
There is no calculation using Pi without the use of approximations.

> To return to your statement above:
>
> >I thought that any number that can not be expressed as a fraction or
> >sum of fractions was an approximation, isn't it so?
>
> If a number can't be expressed as a fraction, than any attempt to
> express it as a fraction will have to be an approximation.

Well to my irrationals have a rather fairy like expression, everyone
claim them to be numbers but noone is able to write them down.
To me Pi is a function not a number.

>But that doesn't mean the number itself is an approximation; it's just
> a limitation of fractions.

No there is no limitation to fractions, fractions can express any real
number.
Fractions can not be used to express numbers that do not have a finite
representation though, that is because they are not numbers.

If Pi was a number there would be a numbersystem where it would have a
finite representation.
And there isn't so it is a function.
> -- Richard

[1treePetrifiedForestLane]

the spatial analog of trig,
is tetrahedronometry (or, dually,
tetra-asteronometry .-)

the main result, taht I know of, so far,
is the sine law on the dihedral angles.

[1treePetrifiedForestLane]

the direct application to tetrahedronometry follows
from the special angular configuration
of the right or left trigonum on the diameter,
the circumdiameter.

often ...

[1treePetrifiedForestLane]

hint: three of the dihedrals are "right" (or left).

[1treePetrifiedForestLane]

I generally use the surfer's value of pi,
for the back of the envelope stretchers.

I'd rather find a continued-fraction form for Brun's constant,
although it is a simple, algebric number, apparently.

[Ross A. Finlayson]

On May 1, 7:02 pm, 1treePetrifiedForestLane <Space...@hotmail.com>
wrote:

Basically the conservation of ratio and angle in the triangle or 3-gon
has then sine and cosine as orthogonal functions.

An idea is to generalize that to n-gons with families of n-1 many
"orthogonal" functions, then as to the development of extensions of
Fourier series for analysis, and other general simple, tractable
results. These constructs simply already have these features, they're
not well-explored where trigonometry is well-developed over time and
used throughout geometry and analysis. One might see this with
potential applications for spectrum analysis, n-D systems, and
approximation of n-many periodic components in extensions of the two
dimensions of planar analysis in Fourier series.

Another generalization then is as to each n-gon and variations in the
evolutions of points swept given isotropies and anisotropies of the
"(un-)rolling square" or here triangle. Then these would offer (more)
simply parameterized solutions, given the generalized framework, to
differential equations and etcetera.

Rolling squares and recycled triangles: Avenues for progress.

Regards,

Ross Finlayson

[1treePetrifiedForestLane]

MAY BE NECCESARY, BUT WAHT IS A SUFFICIENT

> Basically the conservation of ratio and angle in the triangle or 3-gon
> has then sine and cosine as orthogonal functions.

[1treePetrifiedForestLane]

I mean, the trigon only has two orthogonal functions;
the tetrahedron has only 3 that are orthogonal *to each-other*;
four dimensional is neccesarily homogenous, pointwise,
just like 3d in coordinates in the planar.

[Ross A. Finlayson]

The 3/4 are rolling around.

These are torsional components,
then also in those, all but one,
of the separate components of the
mutually orthogonal (n-orthogonal).

[abu.ku...@gmail.com]

I've got what I call a fundamental theorem of tetrahedronometrt,
if it is that

[Ross A. Finlayson]

I think you mentioned it,
but I am usually writing a
development in complete
sentences and as if proof-read.

This is a courtesy for
mathematics in English.

Please feel free to
write terse formulas.

[abu.ku...@gmail.com]

it is just the p.t, used spatailly