“You could probably find jut as accurate encodings for pi, phi, and e in lots of random pyramids compared to the Great Pyramid.” Debunked with computations for 48k randomly generated pyramids!

The Great Pyramid of Giza encodes the values of pi, phi (golden ratio), and e (Euler’s number), which are some of the most fundamental constants in mathematics. Here is how pi, phi, and e can be obtained from the dimensions of the Great Pyramid, whose base length is 440 Egyptian cubits and height (original height) is 280 Egyptian cubits:

• 2 * base length / height = 3.14286. The relative error compared to pi or 3.14159 is 0.040%.
• slant length / half of the base length = 356.0899 / 220 = 1.61869. The relative error compared to phi or 1.61803 is 0.041%. Phi can also be obtained more accurately with the following: lateral face area / surface area = 78339.7728 / 506959.0911 = 1.61782. This has an improved relative error of 0.013%.
• 2 * west-east cross section corner angle / west-east cross section vertex angle = 2 * 51.84277 / 38.15722 = 2.71732. The relative error compared to e or 2.71828 is 0.035%.
  - Both of the angles in the above calculation come from the same triangle. “west-east cross section” refers a triangle formed by cutting the pyramid from west to east like a cake. The other triangle that can be formed via such cutting is a cut made from the pyramid’s southwest corner to it’s northeast corner. While a couple of other cuts can be made, cutting from south to north gives you the same triangle as from west to east, and cutting from southeast to northwest gives you the same triangle as from southwest to northeast. Aside from those triangles, there’s the four lateral triangles, i.e., the four faces of a pyramid that one can see from the outside. Because the base of square pyramids are square, all four faces or lateral triangles are identical, in terms of their dimensions and angles anyways.

The sum of the relative errors for pi, phi, and e found just earlier is 0.089%.

There is a claim that is possible to obtain accurate values for pi, phi, and e from basically any pyramid, and thus, obtaining highly accurate values for pi, phi, and e from the Great Pyramid is unfortunately just a coincidence lacking meaningfulness. To put this to the test, I wrote a program that generates pyramids with random base lengths and heights. The program then finds the closest approximations of pi, phi, and e that can be found for each pyramid by comparing and scaling the ratios of their various dimensions. Here are the rules and procedures that the program followed:

• All of the following dimensions of each pyramid are allowed in calculations: base length, height, base perimeter, base diagonal, slant length, lateral edge length, angles of the lateral faces, angles of the west-east cross sections, angles of the southwest-northeast cross section, base area, lateral face area, surface area including the base, visible surface area (does not include the base).
• Every possible ratio between two dimensions is calculated with the caveat that the dimensions must have the same units. That means the length of a base in cubits can’t be divided by an angle in degrees! It is important for the units to be the same because pi, phi, and e are dimensionless numbers, and ratios of numbers with the same units can have their units cancel and become dimensionless.
  - When the ratio of let’s say base length and height is calculated (base length / height), the inverse is calculated too (height / base length).
• When ratios of angles are calculated, only angles from the same triangle are considered. My opinion here is that comparing an angle from one triangle with any other is a bit convoluted because those angles lack the relationship of being from the same triangle.
• All the ratios that are calculated are each scaled by the following to see which scaled value may be closest to pi, phi, or e: 1/10, 1/8, 1/4, 1/2, 1 (no scaling), 2, 4, 8, 10.
• The generated pyramids are allowed to have base lengths ranging from 1 Egyptian cubit to 1000 Egyptian cubits, which far exceeds the ranges found for Egyptian pyramids that have actually been built. The heights for the generated pyramids also share the range of 1 to 1000 Egyptian cubits. The choice of Egyptian cubits is in a sense irrelevant, because the units get cancelled in the calculations, so they could be feet or meters or even miles 🤷‍♂️.
  - Only whole cubits are allowed. The Egyptians didn’t build pyramids that were 220.4 cubits high or anything.
• If the ratio of the height to the base length is more than 3 or less than 0.333, then the pyramid is considered too narrow or too wide and is passed over.
• A generated pyramid is also passed over if its volume doesn’t fall within 14 million and 22 million cubits³. Pyramids outside of that range are way smaller or bigger than the Great Pyramid (18 million cubed cubits) and would not be a realistic pyramid that the Egyptians would have built.

As far as the Great Pyramid goes, the Great Pyramid fits the rules above and would be one of the pyramids generated. A lot of flexibility is provided for the generated pyramids to give them a chance to outperform the Great Pyramid. For example, the Great Pyramid only used scaling factors of 1/2, 1 (no scaling), or 2, while the generated pyramids were allowed those and an additional six scaling factors. Later in the article, I discuss what happens when the generated pyramids are provided even more scaling factors than the six I described, just to see how that affects the results! Part of what makes the Great Pyramid’s encoding though is the simplicity of the calculations: only ratios are involved (division) and only scaling factors of 1/2, 1 (no scaling), or 2 are involved.

Back to the program itself: once all the pyramids are generated along with all the calculations between their many many dimensions, the closest approximation each generated pyramid had for pi, phi, and e are converted into relative errors. Those relative errors are then summed for each pyramid and considered that pyramid’s score in the “approximate pi, phi, and e as best as possible” competition. Lower relative errors and their sums are more accurate and better, so like a track race where a 13.4s 100m is lower/faster/better than a 16.8s 100m, lower scores are better/preferable.

Results

A total of 47,753 pyramids were generated (excluding the Great Pyramid) that met all the criteria discussed earlier. Each of those pyramids had a sum calculated for their relative errors of pi, phi, and e, and the average of those sums was 3.4%. That means the average pyramid from the generated set of pyramids had a score of 3.4%. Since that 3.4% is the sum of three relative errors, that means the average relative error for pi, phi, and e for those ~48k pyramids was higher than 1%. Notice the Great Pyramid’s relative errors for pi, phi, and e were all less than 0.1%!

• The Great Pyramid’s exact score was 39x less than the average score, meaning the Great Pyramid was 39x more accurate with its encodings than the average generated pyramid.

Did any pyramids have better encodings found for pi, phi, and e than the Great Pyramid. Yep! There were 35 of them. Is that a lot? Nope…there were 47,718 pyramids with higher/worse scores than the Great Pyramid and hence worse encodings for pi, phi, and e. 35 vs 47,718 meant the Great Pyramid was more accurate with its encodings than 99.9% of the generated pyramids. The 0.1% of pyramids that were more accurate (< 0.1% to be exact) were those 35 generated winners.

Quick context: if you divide an axis of a graph that goes from 0 to 1 into ten pieces to mark 0.1, 0.2, 0.3, …, 0.9, then there would be eleven labels and nine ticks, because the 0.0 and 1.0 are marked by the sides of the graph while the 0.1–0.9 in between them require nine ticks. So anyways, the Great Pyramid had a combined relative error of 0.0089, which is 23 times smaller than 0.02. Thus, if you wanted to mark where the Great Pyramid’s score is along the y-axis, you’d have to divide the section that goes from 0 to 0.02 into about 23 pieces. Then the first of those 22 ticks would be where to mark the Great Pyramid’s score. And with a position/rank of 36 for the Great Pyramid (behind 35 pyramids score-wise while ahead of 47,718 pyramids), if you wanted to mark where the Great Pyramid finished its “race” along the axis, you’d have to divide the whole x-axis into 278 pieces and the Great Pyramid would be the first of those 277 ticks from the left side.

I may correct the above graph at some point, but the y-axis labels all need to be multiplied by 10 to become 0.01, 0.02, 0.03, …, 0.07. Anyways, the Great Pyramid had a combined relative error of 0.0089, which is located below 0.01, the first y-axis label. And with it’s rank of 36, if you divide the 0–200 section of the X-axis into four parts, the Great Pyramid would be in the first of those four parts.

Should’ve just marked where the Great Pyramid was located on those graphs with a red dot 😅. I’m editing this article two years later just cuz and don’t want to pull out the computer program this article is based on from the dust or hop back into Excel and make new graphs. Sorry about the inconvenience!

Conclusion

Because the Great Pyramid has more accurate values for pi, phi, and e than the majority of generated/hypothetical pyramids that could’ve been built, this is strong evidence that the Great Pyramid’s accurate encodings for pi, phi, and e were intentional and not just present due to coincidence. Furthermore, for the Great Pyramid’s approximations for pi, phi, and e, only scaling factors of 1 (no scaling) or 2 were used. The generated pyramids were allowed much more freedom and it’s possible a majority of the 35 pyramids that had lower scores than the Great Pyramid used at least one scaling factor other than 2. And let’s not forget, the Great Pyramid encodes far more than pi, phi, and e. If the Great Pyramid really does encode the speed of light for example, then the Egyptians definitely knew the values of pi, phi, and e and could’ve purposely designed the Great Pyramid to encode them.

The end 🥳. Well, here is the code for the program that let this article come to fruition, and some final words up ahead for the skeptic-skeptics. Fwiw I am sometimes a skeptic and have been skeptical about the pyramids and various claims about them from time to time 🤷‍♂️. I don’t agree with all the “magical” values the pyramids allegedly had encoded intentionally.

Now, if I were to myself in the shoes of a skeptic, I may consider the rules for the program to be too strict in at least these three regards:

• The were only nine scaling factors that went from 1/10 to 10. Why not allow a range such as 1/20 to 20 or twenty scaling factors within 1/10 to 10 itself (e.g. add 3 and 1/3 as scaling factors)?
• Why not allow pyramids that are really narrow or wide? And why not allow pyramids with volumes much lower or much higher than the Great Pyramid too?
• What happens if addition, subtraction, and multiplication are allowed in addition to division for comparing values of dimensions? I mean sure, the Great Pyramid’s encodings for pi, phi, and e only used division and addition, subtraction, and multiplication lead to values with units while pi, phi, and e have no units and are dimensionless, but who cares 😏.

To address the above concerns, I reran the program multiple times with more freedom in those areas for the generated pyramids. End result: the Great Pyramid always outperformed, i.e., had lower scores, than at least 99.8% of the generated pyramids. That was regardless of all the freedom I tried to give the generated pyramids.

Generated pyramids: 😭.

Skeptic: What about seked 🦇? Egyptians chose the ratio of half the base length and height of their pyramids and consequently the ratios you used to calculate pi, phi, e from a range way way more limited than 1 to 1000. In fact, the range they picked from was more limited than 1 to 28! The height for their ratios defaulted to 28, and then they’d pick a number, usually between 16 and 36, for half the base length. Thus, all they have to do is get lucky and choose 22 for half the base length, then they end up with 44 for the base length, 28 for the height, and then the “👻👻👻”-ratio of 44:28 for the Great Pyramid and it’s “magical” encodings for pi, phi, and e.

* skeptic drops the mic*

Me: The skeptic’s remark on seked is true 😅. First let me address how some readers could be a bit confused how the Great Pyramid ended up with a base length of 440 cubits and a height of 220 cubits from the 44:28 ratio. After Egyptians picked a ratio for their pyramids, they’d multiply both numbers in the ratio by a factor to obtain the actual base length and height they wanted for their pyramid, i.e., for construction. In the case of the Great Pyramid, they multiplied the 44:28 ratio by 10 to get 440 and 280 👾. more confusing if 44:28 was reduced to 22:14 first 🧐.

For the Skeptic’s point on seked, I address that in this section of this article. For those who are short on time, the conclusion on seked is that there are counter arguments to what the skeptic brings up, but the skeptic’s argument is still quite strong. Personally I don’t think the skeptic’s argument on seked debunks the Great Pyramid’s encoding of pi, phi, and e. At the same time, I can’t confidently claim the Great Pyramid proves the Egyptians knew the values of pi, phi, and e much more accurately than most historians believe because of seked.