User:2^67-1/New Hemipyth Nonsense: Difference between revisions
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==Not coincidences== | ==Not coincidences== | ||
* The 24-cell with side length 1: it only has lengths of sqrt(1), sqrt(2), sqrt(3), and sqrt(4) between any two distinct vertices. | * The 24-cell with side length 1: it only has lengths of sqrt(1), sqrt(2), sqrt(3), and sqrt(4) between any two distinct vertices. | ||
** 96 sqrt(1) lengths, 72 sqrt(2) lengths, 96 sqrt(3) lengths, 12 sqrt(4) lengths | ** 96 sqrt(1) lengths, 72 sqrt(2) lengths, 96 sqrt(3) lengths, 12 sqrt(4) lengths (all 3-smooth numbers!) | ||
** | ** Total edge lengths are 96, total area is 24*sqrt(3), total cell volume is 8*sqrt(2), total volume is 2. | ||
** Radius of sphere touching the middles of all the cells is sqrt(2)/2 or sqrt(1/2) | |||
** Radius of sphere touching the middles of all the faces is sqrt(6)/3 or sqrt(2/3) | |||
** Radius of sphere touching the middles of all the edges is sqrt(3)/2 or sqrt(3/4) | |||
** Radius of sphere touching the vertices is 1 or sqrt(1) | |||
* [https://www.youtube.com/watch?v=ZMkIiFs35HQ Matt Parker calling 2 and 3 subprimes] and 2 and 3 literally forming the bedrock of almost every 'natural' JI system (I agree with him) | * [https://www.youtube.com/watch?v=ZMkIiFs35HQ Matt Parker calling 2 and 3 subprimes] and 2 and 3 literally forming the bedrock of almost every 'natural' JI system (I agree with him) | ||
* The truncated Spiral of Theodorus | * The truncated Spiral of Theodorus up to the 1-sqrt(3)-2 triangle | ||
* Trigonometric ratios (Note the denominators of the sine and cosine fractions are all 3-smooth!): | |||
** sin(pi/6) = cos(pi/3) = sqrt(1)/2 = 1/2 | |||
** sin(pi/4) = cos(pi/4) = sqrt(2)/2 | |||
** sin(pi/3) = cos(pi/6) = sqrt(3)/2 | |||
==Coincidences== | ==Coincidences== | ||
* EulerGamma ≈ 1/sqrt(3) | * EulerGamma ≈ 1/sqrt(3) | ||
* QPochhammer[1/2] ≈ 1/sqrt(12) | * QPochhammer[1/2] ≈ 1/sqrt(12) | ||
Latest revision as of 01:45, 26 April 2025
NO I AM NOT STARTING A CULT
Not coincidences
- The 24-cell with side length 1: it only has lengths of sqrt(1), sqrt(2), sqrt(3), and sqrt(4) between any two distinct vertices.
- 96 sqrt(1) lengths, 72 sqrt(2) lengths, 96 sqrt(3) lengths, 12 sqrt(4) lengths (all 3-smooth numbers!)
- Total edge lengths are 96, total area is 24*sqrt(3), total cell volume is 8*sqrt(2), total volume is 2.
- Radius of sphere touching the middles of all the cells is sqrt(2)/2 or sqrt(1/2)
- Radius of sphere touching the middles of all the faces is sqrt(6)/3 or sqrt(2/3)
- Radius of sphere touching the middles of all the edges is sqrt(3)/2 or sqrt(3/4)
- Radius of sphere touching the vertices is 1 or sqrt(1)
- Matt Parker calling 2 and 3 subprimes and 2 and 3 literally forming the bedrock of almost every 'natural' JI system (I agree with him)
- The truncated Spiral of Theodorus up to the 1-sqrt(3)-2 triangle
- Trigonometric ratios (Note the denominators of the sine and cosine fractions are all 3-smooth!):
- sin(pi/6) = cos(pi/3) = sqrt(1)/2 = 1/2
- sin(pi/4) = cos(pi/4) = sqrt(2)/2
- sin(pi/3) = cos(pi/6) = sqrt(3)/2
Coincidences
- EulerGamma ≈ 1/sqrt(3)
- QPochhammer[1/2] ≈ 1/sqrt(12)