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

On 10/3/2024 4:11 PM, Chris M. Thomasson wrote:

On 10/2/2024 6:56 PM, Ross Finlayson wrote:

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,

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How do you distinguish them?

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

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Then distinguish the first one.
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Regards, WM

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

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

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

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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;
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for (i = 0; i < n; ++i)
{
     normal = normal_base * i;
     angle = pi2 * normal;
>
     p0 = { cos(angle), sin(angle) };
>
     plot(p0);
}
_________________________
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I typed this in the newsreader, so sorry for any typos! This a finite
view of a unit circle. Not a line.
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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.
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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?

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

[...]
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a square would be 2pi / 4, a pentagon 2pi / 5, ect...
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Fwiw, check this out:
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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.

 Points of an equilateral triangle inscribed in a unit circle is comprised of the following angles:
 angle_0 = pi2/3 * 0
angle_1 = pi2/3 * 1
angle_2 = pi2/3 * 2
 This works for any n-gon.
 

For an inscribed 5-gon wrt the unit circle:
angle_0 = pi2/5 * 0
angle_1 = pi2/5 * 1
angle_2 = pi2/5 * 2
angle_3 = pi2/5 * 3
angle_4 = pi2/5 * 4