# Papers

The Beautiful Cubit System

An analysis of the Egyptian Royal cubit, presenting some  research and opinions flowing from that research, into what I believe was the original cubit, and how it was corrupted. I show various close arithmetic approximations and multiple ways of getting the divisions of the cubit, as well as some related measures. The cubit also encapsulates the basic components for the metric system.

Diskerfery and the Alignment of the Four Main Giza Pyramids

The curious alignment of the three main pyramids at Giza has puzzled many people over the ages. Various theories have been proposed, either based on site geometry or stellar alignments coupled with known ancient Egyptian religious beliefs. This paper shows that the precise alignment of the pyramids can be explained with mathematics alone, using diskerfery and other geometry. We then use the same techniques to identify the most- probable location of the now-dismantled fourth pyramid at Giza. The techniques and discoveries in this paper provide the basis for dating Giza, as discussed in the companion paper “55,500 BCE and the 23 Stars of Giza” .

55,500 BCE and the 23 Stars of Giza

This is a companion paper to and reliant on “Diskerfery and the Alignment of the Four Main Giza Pyramids” (Douglas, 2019 [1]). Following the geometric alignments shown in that paper, we now present the astronomical design plan with 23 stars. There is a perfect alignment with two stars, very close alignment with others, and close alignment with other prominent stars in the area. We propose that this was done to provide a date for the construction of Giza. The alignment occurs at circa 55,500BCE.

## The foot, cubit, metre, and φ, π and e

A short document highlighting the consistent relationship between the foot, cubit and metre, and combinations of them, and phi, pi and e, via e-1.

## Zep Tepi Mathematics 101 – How Giza was probably designed

A mathematics course from the Zep Tepi era, where we plan and analyse a large building site, showing how the design mirrors the stars.

A simple and elegant explanation of how Giza, with six main pyramids, was laid out, using √2, √3, √5, π and φ. The design incorporates the necessary elements for squaring the circle, area-wise. The design matches the heavens around 55.5k BCE. This could force a rethink of at least the history of mathematics, if not the broader human timeline, and effectively solves the puzzle of how Giza was laid out.

Khufu’s Coffer

An analysis of Khufu’s coffer dimensions and location, based on Petrie’s measurements. The analysis suggests that it can not be 4th Dynasty, due to the formulas involved.

Pi, phi and e in Khafre’s dimensions

Some formulas showing how approximations for pi, phi and e are in the dimensions of Khafre.

The Douglas Triangle, Khufu and Khafre

The Douglas triangle is a special right triangle using φ and √3. It appears that its proportions and peculiarities impressed the Giza planners so much that they used it, when scaled by 50π, for the base lengths of the Khufu and Khafre pyramids. This paper presents the triangle and discusses some of its interesting features.

The Beautiful System

A one-page summary version of my “The foot, cubit, metre, and φ, π and e” paper, suitable for wall mounting or as a handy reference.

https://doi.org/10.5281/zenodo.6038713

The six pyramids and planets at Giza

In a previous paper, Zep Tepi Mathematics 101, and the ones it references, I showed how Giza had six main pyramids, with a mathematically designed layout, based on a stellar arrangement around 55.5k BCE. One “problem” with that proposal is that the sizes of the pyramids do not match the brightness of the stars. In this paper I show that the pyramid sizes actually correlate to the sizes of the six planets closest to the sun, with the pyramids arranged from west to east in increasing size order. This implies that the builders could accurately measure the diameters of the five planets plus Earth, as well as being aware of the four Galilean moons, and Saturn’s moon, Titan. This implies they had optical instruments like telescopes.

https://doi.org/10.5281/zenodo.6374281

π, φ and e between Khufu and Khafre

Diagrams and calculations showing how the dimensions and relative locations of the Khufu and Khafre pyramids provide reasonable approximations for the π, φ, and e. The pyramid sizes and spacing meets other criteria, so this is either co-incidental, or the designers showing off.

https://doi.org/10.5281/zenodo.6387552

The Original Location of Khufu’s Coffer

A geometric and astronomical determination of the probable original location of the coffer in the Great Pyramid.

https://doi.org/10.5281/zenodo.6549912

The Writing is on the Wall: The King’s Chamber Game

There is an at times heated debate between mainstream archaeologists and historians, and alternative researchers, about the age of the monuments at Giza. Mainstream produces Carbon 14 tests and ancient histories, written long after the fact, as proof. Alternative researchers propose much earlier dates. Neither side has rock-solid evidence. Part of the problem with Giza is the total lack of inscriptions on the walls. This is contrary to typical dynastic burial practices. Luckily, Robert Bauval offers a strong clue as to what we should be looking for. There are no inscriptions in the pyramids at Giza, because writing on the walls would have obscured the actual message, which has literally been hidden in plain sight all along. “He that hath eyes, let him see.” There is no writing <em>on</em> the walls, because the walls themselves are the message. We just need to look at them in the right way, and we will see π, φ, e, √2, √3, the Fibonacci series, and possibly even the speed of light, in metres/second, all spelled out in base 10 digits. This advanced mathematical knowledge proves that the Great Pyramid can not possibly be from the 4th Dynasty.

https://doi.org/10.5281/zenodo.6570342

The King’s Chamber Game Diagrams

Addendum to The Writing is on the Wall: The King’s Chamber Game (https://doi.org/10.5281/zenodo.6570342), with all the diagrams.

https://doi.org/10.5281/zenodo.6590181

Pahl’s Point and the Giza Star Map

An alternative method of aligning Giza with the stars. Pahl’s Point is the midpoint of a line between Khufu and Khafre, dividing it into two pieces each 100π cubits long. Pahl’s Point provides the necessary geometry to solve two of the four problematic stars discussed in 55,500 BCE and the 23 Stars of Giza, and Zep Tepi Mathematics 101. A 1000φ ₢ circle is key to solving the other two. Along with adding the coffer to the plan, we can now get simpler stellar alignments than shown previously.

https://doi.org/10.5281/zenodo.6794296

The King’s Chamber Floor, decoded

The King’s Chamber floor appears to be randomly-sized blocks, perhaps the forerunner for the “crazy paving” décor style. Like everything else about Giza, it is actually mathematically designed, incorporating almost all the favourite irrationals used extensively elsewhere around the site. The floor includes π, φ, φ², e², √2, √3, √5, √7, √π, √φ, √e, as well as c² , g₀ and α. References to the speed of light allow us to potentially also model Einstein’s famous equation, using stone lengths.

https://doi.org/10.5281/zenodo.6866852

G1, c and ₢

There is a claim that the difference in metres between the circumcircle around, and the incircle within the great pyramid, closely mirrors the speed of light, when scaled by powers of ten. The accuracy of this claim depends on the value of the cubit in metres, assuming that Khufu’s<br> intended base size was 440 cubits square. This paper examines various popular values of the cubit, against the speed of light claim.

https://doi.org/10.5281/zenodo.6900644

Observations on the King’s Chamber Dimensions

Observations on the dimensions of the Great Pyramid’s King’s Chamber, as they relate to √5, the golden ratio φ, and π.

https://doi.org/10.5281/zenodo.6944186

A Response to Notes Métrologiques: Le Plancher de la Chambre du Roi à la Grande Pyramide by L. Castaño Sánchez

Luis Castaño Sánchez published a response to my paper, The King’s Chamber Floor, decoded. This is my response to that, pointing out where I disagree with his objections, as well as critiquing his replacement Vitruvian Man model. In return, I offer some support for the mathematical cubit.

https://doi.org/10.5281/zenodo.6954734

The Douglas Triangle, Prime Roots, and e

The Douglas Triangle is a special right triangle with sides 1 ∶ φ2 ∶ φ √3, where φ is the Golden Ratio. Joining two of these on the hypotenuse produces the Douglas Rectangle, which has interesting properties. One such property is a simple yet accurate approximation for e, which differs from e by 0.000000979… . Rewriting the formula gives an equivalent using the first three primes.

https://doi.org/10.5281/zenodo.7396042

Rewriting Euler’s Beautiful Equation with the Golden and Plastic Ratios

Some variations on a theme by Euler, in case it is of some use to someone. Alternate versions of The Beautiful Equation, using the Golden Ratio φ and the analogous Plastic Ratio ρ.

https://doi.org/10.5281/zenodo.7477743

The King’s Chamber, the Sphere, and the Cube

The volume of the King’s Chamber in the Great Pyramid is the sum of a sphere and a cube, both based on the golden ratio φ.

https://zenodo.org/record/8056716

# Thoth’s Constant

Just to formally introduce a ratio used previously, and give it a symbol.

I need it for upcoming documents, which links the great pyramid to the speed of light more accurately than any other extant method, and for deciphering the King’s Chamber.

https://doi.org/10.5281/zenodo.8105721

Another reference to the speed of light by the Great Pyramid

There are several known connections between the Great Pyramid and the speed of light.

This paper shows a new method, utilising the three most famous irrational numbers, π, φ, and e, which are all embedded in the Great Pyramid’s dimensions, as well as elsewhere at Giza.

This neatly expresses an excellent approximation of c using Khufu’s base and π, φ, and e.

https://doi.org/10.5281/zenodo.8184640

A preliminary analysis of the unexplored Great Chamber in the Great Pyramid

In 2022, Filippo Biondi and Corrado Malanga published Synthetic Aperture Radar Doppler Tomography Reveals Details of Undiscovered High-Resolution Internal Structure of the Great Pyramid of Giza, which revealed the network of passages and rooms yet to be explored in the Great Pyramid. One particular room is very large, close to the north face, and may be what the less-precise Scan Pyramids project detected as a large void.

Even though we do not yet have access to this room, the published dimensions are enough to “reverse-engineer” the likely internal dimensions. As with everything about Giza and the Great Pyramid, the room appears to be a mathematical tour de force like the King’s Chamber, including the usual π, φ2, and e. These numbers, all over Giza, continue to challenge the historical time-line.

https://doi.org/10.5281/zenodo.8247385

√10π on Khufu’s King’s Chamber Walls

In The King’s Chamber Game, I showed how π, φ 2 , e, and others are encoded on the walls of the King’s Chamber in the Great Pyramid. The depth and brilliance in the design goes even deeper, as the square root of 10π is also separately encoded, by reading each row as a unit fraction.

https://doi.org/10.5281/zenodo.8249701