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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: Thu, 3 Oct 2024 16:11:31 -0700
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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,
>>>>>>>>>>>>
>>>>>>>>>>>> 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
>>
> 
> 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.