Re:
Arc! Arc! de Triangles design,
added to:
http://aitnaru.org/images/Tasty_Pi.pdf
Total segments length approaches Pi arc length as segments increase
relative to powers of 2 and decrease in individual length.
Wisdom: A million sqrt(2) mini bites of Pi do not comprise one Pi,
but you'll think you've eaten the whole Pi. Bon Appétit
How to calculate sqrt(2)-relative segments of Pi
(aka "what geometry nerds eat for breakfast")
Cartesian composition where large Diameter = 4
and 1/4 of Circumference = Pi (provides starting
values for the right triangle in the pattern):
a = (2(sqrt(2)) - 2) / sqrt(2)
a = 0.5857864376269049511983112757903..
^2 = 0.34314575050761980479324510316121..
b = sqrt(2)
b = 1.4142135623730950488016887242097..
^2 = 2
c = sqrt(a^2 + b^2) (Pythagorean Theorem)
= sqrt(2.3431457505076198047932451031612..)
c = 1.5307337294603590869138399361216..
The rest is a piece of Pi ! (they say)
About the "predictable geometry" ...
Tip 1: Half the hypotenuse in the beginning triangle
becomes the long side in the downstep triangle;
half of angle A becomes angle A in the downstep;
beginning angle A = 22.5 degrees.
Tip 2: Multiply downstep side c by 4 (next power of 2),
then by 4 to calculate total segments length relative to Pi.
Tip 3: A computer can be programmed to downstep
to tiny segments whose total length gets very close to Pi.
This is the Tasty Pi relative to sqrt(2)
Tip 4: This a new way of calculating Pi,
using only sqrt(2) values!
Decreasing sqrt(2) segments lead to Pi
(calculations using trigonometry calculator
where angle A = 45 /2 /2 ..
side b = previous side c /2 )
A = 22.5
b = 1.4142135623730950488016887242097..
c = 1.5307337268917396..
x 2 = 3.0614674537834792..
A = 11.25
b = 0.7653668634458698..
c = 0.7803612832416639..
x 4 = 3.1214451329666556..
A = 5.625
b = 0.39018064162083195..
c = 0.3920685572664755..
x 8 = 3.136548458131804..
A = 2.8125
b = 0.19603427863323775..
c = 0.196270686637696..
x 16 = 3.140330986203136..
A = 1.40625
b = 0.098135343318848..
c = 0.09816490543420567..
x 32 = 3.14127697389458144..
A = 0.703125
b = 0.049082452717102835..
c = 0.04908614609652455..
x 64 = 3.1415133501775712..
A = 0.3515625
b = 0.024543073048262275..
c = 0.024543535073374..
x 128 = 3.141572489391872..
A = 0.17578125
b = 0.012271767536687..
c = 0.012271825290234..
x 256 = 3.141587274299904..
A = 0.087890625
b = 0.006135912645117..
c = 0.0061359198643231..
x 512 = 3.1415909705334272..
A = 0.0439453125
b = 0.00306795993216155..
c = 0.0030679608345627..
x 1024 = 3.1415918945922048..
A = 0.02197265625
b = 0.00153398041728135..
c = 0.0015339805300815..
x 2048 = 3.141592125606912..
A = 0.010986328125
b = 0.00076699026504075..
c = 0.00076699027914077..
x 4096 = 3.14159218336059392..
A = 0.0054931640625
b = 0.000383495139570385..
c = 0.00038349514133289..
x 8192 = 3.14159219779903488..
A = 0.00274658203125
b = 0.000191747570666445..
c = 0.00038349514177352..
x 16384 = 3.14159220140867584..
Pi = 3.1415926535897932384626433..
This calculation emphasizes why so few decimal digits
of Pi are needed, especially for Cartesian exploration
where sqrt(2) is host of Cartesian space
Re:
https://www.jpl.nasa.gov/edu/news/2016/ ... ally-need/
"How Many Decimals of Pi Do We Really Need?"
"For JPL's highest accuracy calculations, for interplanetary navigation,
we use 3.141592653589793." (i.e., traditional Pi is abstract - not real)
For a proper slice of Pi (they say)
when Diameter = 4, Circumference = 4(Pi)
For sqrt(2) association to segments of Pi,
Pi digits must be evenly divisible by 4
(relates to circle's inscribed square)
3141592
/ 4 = 785398
31415926535897932
/ 4 = 7853981633974483
31415926535897932384
/ 4 = 7853981633974483096
314159265358979323846264
/ 4 = 78539816339744830961566
31415926535897932384626433832
/ 4 = 7853981633974483096156608458
31415926535897932384626433832795028
/ 4 = 7853981633974483096156608458198757
314159265358979323846264338327950288
/ 4 = 78539816339744830961566084581987572
3141592653589793238462643383279502884
/ 4 = 785398163397448309615660845819875721
314159265358979323846264338327950288419716
/ 4 = 78539816339744830961566084581987572104929
3141592653589793238462643383279502884197169399375105820
/ 4 = 785398163397448309615660845819875721049292349843776455
"What kind of genius math is that ?!"
The same where 1/4 of Circumference = one Pi.
sqrt(2) Circumference segments
for 16 decimal dgits of Pi where D = 4,
C = 12.5663706143591728 = 4(Pi)
3.1415926535897932 (divisible by 4)
/ 10 = 0.00000000031415926535897932
10 billion segments per 1/4 Circumference
with length = 0.00000000031415926535897932
total 3.1415926535897932
Long story short ...
Sides of beginning right triangle
(all three are sqrt(2) values)
1.4142135623730950488016887242097..
- 1 = .4142135623730950488016887242097..
x 2 = 0.8284271247461900976033774484194..
/ 1.4142135623730950488016887242097..
= 0.5857864376269049511983112757903.. = a
2(1.4142135623730950488016887242097..)
= 2.8284271247461900976033774484194..
/ 2 = 1.4142135623730950488016887242097.. = b
Pythagorean Theorem calculates side c
0.5857864376269049511983112757903.. = a
^2 = 0.34314575050761980479324510316121..
1.4142135623730950488016887242097.. = b
^2 = 2
sqrt(2.34314575050761980479324510316121..)
= 1.5307337294603590869138399361216.. = c
Ro ...
...