Some notes, towards political action.

https://en.wikipedia.org/wiki/Galaxy_rotation_curve

The galaxy rotation problem is the discrepancy between observed galaxy rotation curves

and the theoretical prediction, assuming a centrally dominated mass associated

with the observed luminous material.

 . . . . while pi, is a result of a convergence series, and a transcendental number.

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Hello.

Generally, the baryonic matter, comes in groups of three,

and this could be related to nilkantha_convergence_series,

for localities, where proton decay, has not been observed.

However, the series is an approximation, with some further work, included below.

And then, there are vast spaces in solar system, and between galaxies.

and there the Wallis method, that has been proven to result in pi; could be applicable;

however, it converges, relatively, slowly.

It is my conjecture and opinion, that this is related to "locality",  {{disclaimer}}

while lepton_universality, goes about being true.

https://en.wikipedia.org/wiki/Four_color_theorem#Simplification_and_verification

https://en.wikipedia.org/wiki/Percolation

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Including some source code

import math

## has been proven to result in pi

def finding_convergence_wallis_method(n):

    result = 2.0

    for j in range(1, n):

        i = 2 * j

        numerator_next = i * i

        denominator_next = (i - 1) * (i + 1)

        result = (result * numerator_next) / denominator_next

    print("convergence_result_wallis_mehod_for_pi  = ", "{:.40f}".format(result))

    print("python mathpi = ", "{:.40f}".format(math.pi))

## converges quickly, not exactly;;

def finding_convergence_nilkantha_series(n):

    result = 3;

    three_d = result;

    curvature_toggle = 1

    for i in range(1, n):

        denominator = (three_d - 1) * (three_d) * (three_d + 1) 

        g_additive = (4.0 / denominator)

        result = result + (curvature_toggle * g_additive)

        curvature_toggle = 0 - curvature_toggle

        three_d = three_d + 2

    print("convergence_result_for_nilkantha_series = ", "{:.40f}".format(result))

    print("python mathpi = ", "{:.40f}".format(math.pi))

#start

finding_convergence_wallis_method(99999)

finding_convergence_nilkantha_series(99999)

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results =>

convergence_result_wallis_mehod_for_pi  =  3.1415847995002321724200555763673037290573

                                       python mathpi =  3.1415926535897931159979634685441851615906

convergence_result_for_nilkantha_series =  3.1415926535897864546598157176049426198006

                                      python mathpi =  3.1415926535897931159979634685441851615906

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Also additional notes about nilkantha convergence series

 

 

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##  also Euler series, converges slowly

def finding_convergence_euler_series(n):

    result = 0;

    for i in range(1, n):

        g_additive =  1.0 /  (i *i);    # in an isosceles triangle, two sides, have the same length

        result = result + g_additive;

    approx_of_pi  =  math.sqrt (result * 6.0);    # six isosceles triangles, can form a planar hexagon

    print("convergence_result_for_euler_series = ", "{:.40f}".format(approx_of_pi))

    print("python mathpi = ", "{:.40f}".format(math.pi ))

finding_convergence_euler_series(99999)

finding_convergence_euler_series(999999)

results  =>

convergence_result_for_euler_series =  3.1415831041354680408517197065521031618118

                                python mathpi =  3.1415926535897931159979634685441851615906

convergence_result_for_euler_series =  3.1415916986585985526403419498819857835770

                                python mathpi =  3.1415926535897931159979634685441851615906

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In reality, pi_square, participates in the context of Poincaré conjecture,

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## this one uses Decimal precision in Python, and takes time;

from decimal import *
import math

def finding_convergence_nilkantha_series(n):
    getcontext().prec = 50
    base = Decimal(2.0)
    curvature_toggle = 1
    result = base + curvature_toggle
    three_d = result;
    numerator = base + base
    for i in range(1, n):
        denominator = (three_d - 1) * (three_d) * (three_d + 1)
        g_additive = (numerator / denominator)
        result = result + (curvature_toggle * g_additive)
        curvature_toggle = 0 - curvature_toggle
        three_d = three_d + base
    print("convergence_result_for_nilkantha_series = ", "{:.40f}".format(result))

#start

finding_convergence_nilkantha_series(99999)
finding_convergence_nilkantha_series(999999)

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Hello

New Pi Formula (the extra physics bit) - Numberphile

https://www.youtube.com/watch?v=AZxoENTRKxg

New Recipe for Pi - Numberphile

https://www.youtube.com/watch?v=nXexsSWrc1Q

OK, bye