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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (circle limits)
Date: Wed, 2 Oct 2024 12:45:51 -0700
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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
>>>>>>>
>>>>>>> There isn't a first one.
>>>>>>>
>>>>>>> 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...
>>>>
>>>> Taking an n-gon to infinity is a circle.
>>>
>>> It is 'never' a circle.
>>>
>>> https://www.craig-wood.com/nick/articles/pi-archimedes/
>>
>> 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