Let's try another round to talk you and other people over.
Wallis' product
:::16*16*14*14*12*12*10*10*8*8*6*6*4*4*2*2
:::17*15*15*13*13*11* 11* 9* 9*7*7*5*5*3*3*1
converges towards π/2, ie. τ. The numerical relation in question is much better described by symbol τ (Greek tau, 2τ =π), as suggested by mathematically inclined Youtube artist Vihart (
https://www.youtube.com/watch?v=5iUh_CSjaSw).
Let's rewrite the product in the following form, by multiplying consecutive pairs of fractions:
:::324/323*/256/255*196/195*144/143*100/99*64/63*36/35*16/15*8/7*4/3 : τ
Corollary: The product contains also interesting sequence of Egyptian fractions:
:::(1+1/323)*(1+1/255)*(1+1/195)*(1+1/143)*(1+1/99)*(1+1/63)*(1+1/35)*(1+1/15)*(1+1/7)*(1+1/3) : τ
4 8 16 36 64 100 144 196 256 324
3 7 15 35 63 99 143 195 255 323
4 8 16 36 64 100 144 196 256 324 minus
.. 4 8 16 36 64 100 144 196 256
...4 8 20 28 36 44 52 60 68 minus
......]4 8 20 28 36 44 52 60
.......4 12 8 8 8 8 8 8 divided by 4
........1 3 2 2 2 2 2 2
4 8 16 36 64 100 144 196 256 324 minus
...3 7 15 35 63 99 143 195 255
...5 9 21 29 37 45 53 61 69 minus
........5 9 21 29 37 45 53 61
........4 12 8 8 8 8 8 8 divided by 4
.........1 3 2 2 2 2 2 2
which is
https://oeis.org/A104435 (Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.)
In order to measure a circle, a natural way is to draw a circle between equilateral triangles so that it touches both internal external triangle thrice. Hence we can define the size of a circle in relation to triangles as the meet where size the of the external triangle Te halts decreasing and the size of internal triangle Ti halts increasing, ie. where Te is neither more nore less than Ti. In relop we can write this double measurement operation in the following way:
> > < <
four operators in measuring process, insert metric of natural numbers.
>1>2<3<
The chosen metric does not act well with the basic semantics of the operators, so rewrite
<1<2<3<
>2<
>2<
>2<
etc. inpanding for more and more measurement values of nested square roots of 2 in the chosen metric, ie. Viète's formula.
Wittgenstein:
”Mathematics as such is always a measure, not the thing measured”. What is most fascinating, the pattern (4) 1 3 2 2 2 contains here in itself the general structure of measuring the relation in question with metric of natular numbers. As numerical metric for a measurement is a matter of choice and definition, we can try out the same Relop measurement also with other metric, say integers instead of natural numbers. But as this time the measurement is centered, instead of chiral, we can use the centered Relop form > > < <:
>1>0<1<
giving positive integer value 1 for τ.
For radius with numerical value 1 it takes 1 measurement turn to draw the unit circle, relation of integers 1:1. What this demonstration shows is that irrationality of π is only a relative phenomenon, depending on the choice of metric and what actual relation is being measured with what metric. Relational approach can more easily reveal and remind that measuring with centered n><n form is not necessarily very conmeasurable with measuring chirally only with L or R side of a numerical metric, and can produce fascinating and wonderful phenomena such as appearance of a transcendental π.
