Re: How many different unit fractions are lessorequal than all unit fractions? (circle limits)

On 10/02/2024 12:45 PM, Chris M. Thomasson wrote:

On 10/1/2024 7:40 PM, Ross Finlayson wrote:

On 10/01/2024 06:37 PM, Chris M. Thomasson wrote:

On 10/1/2024 2:11 PM, FromTheRafters wrote:

Chris M. Thomasson wrote :

On 10/1/2024 6:28 AM, FromTheRafters wrote:

Chris M. Thomasson wrote :

On 9/30/2024 4:13 AM, Richard Damon wrote:

On 9/29/24 3:16 PM, WM wrote:

On 28.09.2024 14:58, Richard Damon wrote:

On 9/27/24 3:06 PM, WM wrote:

On 25.09.2024 19:12, Richard Damon wrote:
>

The problem is that it turns out the NUF(x) NEVER actually
"increments" by 0ne at any finite point, it jumps from 0 to
infinity (Aleph_0) in the unboundedly small gap between 0 and
all x

0,

>
How do you distinguish them?

>
They have different values, so why can't you?

>
Then distinguish the first one.
>
Regards, WM

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There isn't a first one.
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Show me a circle with 4 sides.

>
;^) Humm, an n-gon where n is taken to infinity is a circle?

>
As n goes to infinity, the angle of the vertices goes to 180 degrees
-- is a straight line a circle?

>
No. As n goes to infinity it makes a circle. Think of a finite view
of a "large" number for n:
_________________________
n = 696969
>
normal_base = 1.f / n;
>
for (i = 0; i < n; ++i)
{
     normal = normal_base * i;
     angle = pi2 * normal;
>
     p0 = { cos(angle), sin(angle) };
>
     plot(p0);
}
_________________________
>
I typed this in the newsreader, so sorry for any typos! This a finite
view of a unit circle. Not a line.
>
Take n to infinity, well, its a circle...
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Taking an n-gon to infinity is a circle.

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It is 'never' a circle.
>
https://www.craig-wood.com/nick/articles/pi-archimedes/

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an n-gon as n goes to infinity approaches a circle? Fair enough?

>
Is the sum of its interior angles infinity, or, 2pi?

[...]
>
a square would be 2pi / 4, a pentagon 2pi / 5, ect...
>
Fwiw, check this out:
>
https://www.shadertoy.com/view/4s3yWj
>

Actually the usual idea is that a regular triangle
is pi/3 * 3 = pi, a 4-gon pi/2 * 4 = 2pi, a 5-gon
3/5 pi * 5 = increasing, that both the angle widens
to pi and the count increases linearly.
Yet, there's another way I read about in Wertheimer's
"Productive Thinking" that makes sense in that the
angle it only the central wedge and always adds up
to 2pi, as I talk about in "Moment and Motion: points
and space inversion".
Then here there are some various ideas about things
like "if a regular polygon has infinitely many sides,
there's a proof that it's a circle", or not, then
also as with regards to if not: then howsoever it
would be different, such a beast of a finite or bounded
regular polygon with infinitely many sides, each of
which would have a distinct tangent, each of those
parallel to exactly one other.
What I have in mind about that is that what we get
from antiquity is the ancient means of exhaustion
that was the first sort of approach to compute pi
as from the ratio of the perimeter of a regular
polygon to a line across it, as the number of
sides increases, the perimeter being computed
from as radial segments, radial triangles,
the vanishing base of the radial triangle
as the count of those goes to infinity.
So, the idea is that in one sense, that's
_compressing_ a bunch of triangles, so
they are more pointed in the middle and
flex out at the end. Then, in the other
sense has those when _expanding_, see,
as when threading through the corners,
and threading through the bases, then
extending those same, how they make as
like an accordion, that the base and
the point are the same.
It's sort of a "second wind" approach to
exhaustion, then, in as regards to that
there's an infinitely-sided polygon, and
it contains any different in its perimeter
or area, as with regards, to otherwise
anything else going on in the circle as
with regards to methods of exhaustion,
for example here: when the curve keeps
its linear length yet the area under it
vanishes and as it goes to the diameter
in the limit, how it can be that in the limit:
it's not the same as inductively, yet it's
established that's its bound.
Or, you trimmed the actual interesting bits
about the curve through infinitely-many
equi-distant centers down the diameter
that both has length D and has length pi/2 D.