The first two parts dealt with more important mathematical constants, and or otherwise interesting alignments. This part has “the rest”, which are either not so important mathematically (well, in terms of what we expect the pyramid builders to know) or less-accurate alignments, but are posted here “for the record’. There is minimal exposition.
We might as well start with something unexpected, and also the third most accurate on this page: Apéry’s constant (1.2020569…), which is basically the limit as n tends to infinity, of the sum of inverse integer cubes (1/1³ + 1/2³ + 1/3³ +1/4³ + …. + 1/n³)
The angle is 299.229, we want 299.487, out by 0.257 degrees, accuracy is 99.914%
Which leads us of course to wonder where they left the solution to the Basel Problem which is similar to Apéry above, but with squares instead of cubes (1/1²+1/2²+1/3² + … + 1/n²) which Euler showed to be π²/6 (1.64493…). Also known as the Riemann Function ζ(2). It’s right here, and the most accurate result on this page:
The angle is 218.830, we want 218.854, out by 0.024 degrees, accuracy is 99.989%
Root 5 (2.236…):
The angle is 158.957, we want 160.997, out by 2.040 degrees, accuracy is 98.733%
We can also find cube roots of the first three primes. Here’s ∛2:
The angle is 285.7082, we want 285.7322, out by 0.02396 degrees, accuracy is 99.9916%
Note that this is measured on the diagonal, the actual angle of which is 45.0931° not 45°.
Similarly with ∛3:
The angle is 250.6411, want 249.61, out by -1.0311 degrees, accuracy is 100.4130%
This is also measured on the diagonal, the actual angle of P3 diagonal is 44.7829° not 45°.
The angle is 209.1986, want 210.52928, out by 1.3306 degrees, accuracy is 99.368%
Note that this is measured between same two points as ∛2, but different base line.
I’m not really happy with most of the rest of the results, so I’ll just post a summary table instead of images for each one.
|Name||Ratio||Point 1||Point 2||Calculated||Desired||AbsDiff|
|ln5||360/ln5||P2 BL||P1 TR||226.066||223.681||2.385|
|log2||360(log2)||P2 TR||P1 TL||107.075||108.371||1.296|
|log3||360(log3)||P2 TL||P1 BR, P1 BL||171.370||171.764||0.394|
|log5||360(log5)||P2 TR||P1 TL||252.925||251.629||1.296|
|ln root2||360(ln root2)||P3 C||P2 TR||125.113||124.766||0.346|
|ln root3||360(ln root3)||P3 TR||P1 TL||197.075||197.750||0.675|
|ln root5||360(ln root5)||P2 BR||P1 BL||289.330||289.699||0.369|
|τ/ℯ||360(τ/ℯ)||P2 C||P1 TL||155.033||155.746||0.713|
|ln(τ)||360/ln(τ)||P3 C||P2 TL||194.951||195.878||0.927|
|ⅇ/φ||360/(ⅇ/φ)||P3 BL||P2 C||213.717||214.287||0.570|
|Sophomore's dream||360/1.291286||P3 BR||P2 TL||278.240||278.792||0.552|
Continues in Part 4
2018-10-02: added cube roots, updated Tau to slightly better angle, added error summary table.
2018-10-03: added ln(π), ln(√2), ln(√3), ln(√5), π/ⅇ and composites with ln(π), π and ⅇ.
2018-10-04: added message image, added 7-11.
2018-10-07: started replacing the pics with cleaner (and in some cases, more accurate) versions.
2018-10-08: decided to split this post into three parts. First part will focus on mathematically important numbers, and at the risk of revealing that it was the butler wot did it, that means φ,π,τ,ℯ and squares and probably square roots.
2018-10-09: added πⅇ. Realised that my atan angle calculations assumed that the sides of the pyramids are aligned with N-S and E-W (ie x and y on Cartesian plane) when they’re actually fractions of a degree off. Will attempt to correct for this once rest is done … it’s not going to make major difference as far as I can see. May improve some results and worsen others slightly.
2018-10-11: added √7 and √11. Replaced √2 with much more accurate angle, and updated √2 and √3 merged angles. Added 7×11 much more accurate version.
2018-10-12: Discovered much better location for ⅇ so replaced the diagram plus assorted composites in this post and part 2. Also the relevant tables.
2018-10-13: Discovered small typo in co-ordinates of P1 TR corner (500115.3 -> 500115.0), so fixed angles which used that point: Tau, √7, 11×11, ln(5). Changes are very tiny. Also moved ⅇ to Part 2 where it fits better.
2018-10-14: Added 13² and √13. Why it’s there I have no idea….
2018-10-14: Tau and Pi have been bothering me for their relative inaccuracy, so took another look. Found proper location for Tau and made necessary replacements.
2018-10-14: Busy moving “the rest” from part 1 to part 3. Also added Apéry constant ζ(3) and solution to the Basel Problem ζ(2), both in Part 3.
2018-10-15: Added π^ⅇ and ⅇ^π, both in Part 2. Added table with “the rest” to Part 3. Moved changelog to end of part 3.
2018-10-15: Add Reciprocal Fibonacci psi ψ to Part 1, fixed all the images where I had labelled the wrong part of the arc for phi.
2018-10-15: Added the Omega function to Part 2.
2018-10-16: Added √τⅇ to Part 2. Created Part 4 with summary of the best results.
2018-10-17: Found correct location for 11×11, updated image and tables. Added Silver and Bronze ratios and updated tables.
2018-11-04: Added √φ to part 1, updated tables.
2018-11-05: Added √🜛 and √♀♃ to part 1, updated tables.
2018-11-19: Added φ² and φ³, updated tables.
Done for now.