Persons who study the finer details of pyramids are sometimes derisively referred to as “pyramidiots” (the term “pyramidologist” has probably attracted an almost equally unflattering reputation for being someone who dabbles in “pseudoscience”, in spite of the fact that although measurement errors can happen easily enough (easily demonstrated by comparing various data sources for the proportions of ancient monuments), interpretive approaches such as mathematics and geometry are very much actual science, and assertions to the contrary must be questionable at best.

I could only wear the name “pyramidiot” proudly knowing what good company I am in for thinking there is something remarkable about the designs and proportions of pyramids and other ancient architecture, a proposition that seems to grip many of us almost intuitively even though we are often on many different wavelengths, with many of us setting out to prove this in our own ways.

I’ve already briefly described some of the vital contributions of Carl Munck and Michael Morton, and pointed out where Sir Flinders Petrie might be considered in his own way something of a kindred spirit and a predecessor, as well as a uniquely valuable source of carefully gathered data from pyramid measurements.

Perhaps it’s a lesser known fact, but the renowned Sir Isaac Newton can be counted among those who approached the pyramids with the expectation of making mathematical findings that would perhaps demonstrate a level of ancient mathematical and geodetic skills that rival or even exceed those of contemporaries.

Since some of this fascinating and thought-provoking background is less commonly included in reference works, I hope in the very near future to update this entry with a proper write-up on this subject once and for all.

I’m going to jump right into it here, since the most important thing I have to share with anyone might be the Remen. For background, which might be starting backwards, let’s look at probably the famous ancient Egyptian unit of measurement, the Royal Cubit, of about 1.72 feet. If you look at books on archaeology or Egyptology, you’ll quite frequently find the interior and exterior measures of Egyptian pyramids given in Royal Cubits in addition to meters or feet, for intents and purposes invariably in Royal Cubits.

There are some problems with this, one of the being that the Royal Cubit is not the only unit of measure the ancient Egyptians used. One of the most renowned and trusted sources of data used in pyramid studies is Sir William Matthew Flinders Petrie, who over the course of his publications reminded us on occasion that there was at least a second unit, which was 1/2 of the diagonal of a square with sides of one Royal Cubit, called the Remen.

Even though he advocated and publicized the idea of the Remen, even Petrie seems to have been curiously timid about applying the Remen to his own archaeological findings, instead expressing the measurements he took in Royal Cubits, which may have contributed to the difficulty he seemed to have in keeping Royal Cubits to a particular length.

Petrie and many others also seemed timid when it came to applying theories about the origin and meaning of the Royal Cubit, at least when it came to giving the ancients credit for precision.

Because of a very limited number of surviving mathematical papyri, the world seems to be well saturated with the idea that the ancient Egyptians knew the meaning of Pi (3.141592654…) no better than to only be able to express it as the simple fraction 22/7 (3.142857143…), and because the supplemental evidence that gives them credit for more may lie in architecture itself, where measurement alone may be unable to sort out the minuscule but tell-tale differences between the two.

Munck may have been the first person to insist that the ancients knew the value of Pi to at least ten decimal places, and Morton the first to give them credit for therefore having the value of the Royal Cubit to at least ten decimal places, since Morton’s Royal Cubit value is directly related to the Pi ratio. Morton used the very same type of equation that Petrie and others were toying with the thought of accepting as the geometric origin of the Royal Cubit, expect that Morton was far more decisive – and generous – about it.

Hence even though the was not the first to consider it, I call Morton’s cubit after him.

Morton’s cubit value can be obtained as .03 x (360 / 2 Pi) = 1.718873385 feet.

The interesting thing to note here is that the Royal Cubit only has this remarkable circular geometry-related value when expressed in modern feet. That’s part of the suspicions about the nature of the Royal Cubit that Petrie and others entertained.

This is highly consistent with Munck’s idea that the primary unit of measure used by the ancients was the modern foot, which has somehow resurfaced with the more complete story of its origins having been lost to us, aside from what can be deduced from architecture, geometry, and mathematics.

I have had the honor of proposing that the true value of the Remen is 1.216733603 feet. That’s another remarkable number which is related to geometry far beyond simply being half the diagonal of a square with sides of 1 Royal Cubit in length.

The number isn’t my discovery, 1.216733603 is half of a value used by Munck and others (I’ll skip the sordid history of e’/sqrt 5 here, suffice it that ancients knowing what a decimal place is or discovering long division earlier than history books give them credit for is apparently too “far out” for even some of the “far out” themselves).

After frequently marveling at the success of Morton’s 1.718873385 ft Royal Cubit in the realm of “Munck’s” mathematics, I became more curious about other ancient units of measure and wondered whether they too might be some missing part of the puzzle we were working on.

Reading Algernon Berriman’s book Historical Metrology, which links various units of measure though fractions of the Remen, and details Penrose’s work on ancient Greek temples, the pieces were rapidly falling into place. Using Munck’s style of mathematics, I obtained a value for one form of the ancient Greek foot of 1.013944669 ft, and when I went to convert it into inches out of curiousity,

1.013944669 ft x 12 = 12.16733603 inches

That’s the first time I was looking at that figure close to its proper context.

Again, technically speaking, the Remen is the diagonal to a square with sides of 1 Royal Cubit in length, but a problem is the lack of consensus as to what a Royal Cubit is exactly, Flnders Petrie himself being a notable – and possibly extreme – example of this. Using a common rounded value of 1.72 feet,

1.72 x sqrt 2 = 1.216223664 x 2

It took a long time to accept this, though, because using Morton’s Cubit value of 1.718873385 ft,

1.718873385 ft x sqrt 2 = 1.215427027

Since absolutely accuracy is the usual standard for Munck and his students, this was puzzling, and made it seem like the correct value might be 1.215854024 = 12 / (Pi^2). However, 1.215854024 has failed ever since I started this to establish itself as a legitimate value for the Remen, even after possible additional values have surfaced.

What we learned is that the ideal values for ancient metrological units may not be related to each other geometrically with absolute accuracy. Indeed, Munck’s system of numbers excludes lower square roots of whole numbers (at least in terms of their exact values), like sqrt 2, sqrt 3, sqrt 5, sqrt 6, and etc.

The lowest whole number whose square root is represented in Munck’s system of numbers is 15, found in the mathematics of Stonehenge. Stonehenge was thought by Munck to indicate the applicability of square root 15 to its mathematics to through the unusual display of 15 stones in the trilithon “horseshoe”, and I am in agreement, as may be local legends as well.

It’s curious that debates continue over both the validity of the Remen as metrological unit even after Petrie’s endorsement of it, and over its origins when it’s only too easily explained as simply the diagonal length of a square with sides of 1 Royal Cubit – mystery already solved?

However, since I began pursuing that idea that ancient architecture attempts largely to express numbers important to calendar systems, I finally had to stop brushing something aside that those of us working with these numbers must have thought was sheer coincidence (I thought exactly that for a long time).

Munck once introduced a number, 365.0200808, and wondered if it were the length of the Solar Year year long ago, perhaps prior to some earth-changing event.

What I now know this number as is the ideal length of the calendar year for mathematical purposes, and we have only try to divide the 365 days in a year into a simple number of divisions such as trimesters, or into months of 30 days, and we get

365.0000000 / 30 = 12.16666666

365.0200808 / 30 = 12.16733603

Hence, not long after man has first counted out 365 pebbles or beans or scratches to identify the number of days in a year as 365, he is very plausibly looking at the very same number (give or take decimal placement, which is irrelevant in my work) that the ancient Egyptians so dutifully preserved by keeping it as a metrological unit, even for the inconvenience of using more than one metrological unit at a time.

There are Egyptian pyramids whose passages may measure x number of Cubits wide and y number of Remens high, or z number of Remens long. That’s rather like a modern architect designing a building whose hallways are measured in feet for their height and meters for their width. Even we with pocket calculators and vast mathematical and architectural skill would likely consider this an rather inconvenient challenge. What could possibly motivate anyone, particularly ancient people, to actually rise to such a challenge?

The best I have ever been able to imagine for why we would find such a thing in ancient pyramids, is because those who designed them prided themselves on preserving through their diverse units of measurement at least two numbers that may be literally almost as old as time itself, numbers that may have appeared shortly after the passage of time began to be marked and quantified.

Something else I like to point out, is that in spite of the number of authors who like to tell us that the planetary cycles express the “Golden Ratio” Phi (often cited as if the result of intelligent design of the solar system), they in fact do not. Using more accurate values from Wikipedia, the ratio between the Venus Orbital Period and the Earth Year (365.24219 days / 224.701 days) is

365.24219 days / 224.701 days = 1.6254586761, rather than Phi ( 1.618033988)

For the “canonical” simplified calendar values, 365 and 225,

365 / 225 = 1.6222222222 – not Phi either, but compare this to the value 1.622311470 that Munck introduced, wondering if could have been “an ancient form of Phi”. 1.622311470 is an amazing number and one THE most useful numbers for analyzing unfamiliar numbers with through multiplication and division.

1.622311470 x 1.5 = 1.216733603 x 2, the diagonal of a square, in modern feet, with sides of 1 Royal Cubit in length.

My opinion has to be the Royal Cubit is descended from the Remen, rather than things being the other way around.

I have a lot more to say about the Remen and the Royal Cubit in general.

In the meantime, even if I am not incomplete agreement, here is one of the most admirable things I have ever seen anyone write about the Remen

http://grahamhancock.com/phorum/read.php?1,1211234

You can read more of the things I’ve written about Munck’s work, Morton’s, and mine here:

http://grahamhancock.com/phorum/read.php?1,1198957

Granted I am not easily swayed by the mainstream or by orthodoxy, but comments are welcome nonetheless.

— Luke Piwalker

Be yourself; Everyone else is already taken.— Oscar Wilde.

Hello! I’m Luke Piwalker. “Archaeocryptographers” Carl P. Munck and Michael Lawrence Morton are my “Yoda” and “Obi-Wan” respectively. Being people, we have all had our differences in the ways we work with and interpret ancient numbers and architecture, but what we have most had in common is that we have been seekers of answers to ancient mysteries, particularly the mysteries of ancient mathematics and metrology (the study of ancient units of measure). If you’ll bear with me, that’s neither as dry or boring as it must sound.

All three of us in our own way, like so many others, have held the conviction that ancient peoples possessed unappreciated and demonstrable mastery of mathematics, mathematics that is deeply intertwined with other ancient mysteries. Our group, however, often seems particularly inquisitive. We can for example (and do), ask not only why Stonehenge or the Great Pyramid were constructed, but why they were designed and proportioned the particular way that they were. Some of the possible answers may surprise you – they certainly surprised me.

For the record, I decline the use of the expressions “archeocryptography” or “code” to describe the numbers I work with. Quite the contrary, I think the ancients very plausibly went out their way to make things “readable”, presumably so to anyone who was versed in the units of measurement of their day. I can’t imagine that anything was so hidden as to merit terms that seemingly imply they were concealed as if for some select elite few.

For background, Carl Munck held the view that the geographic coordinates of ancient monuments are encoded in some of the details of their composition to tell us “why they are where they are”. Michael Morton and I worked with this view for a decade or more, and he eventually attempted to plot out a stellar cartography grid to mirror and supplement the geographic earth grid that Munck espoused, called the “Archaeo-Sky Matrix”, the celestial equivalent of Munck’s “Pyramid Matrix”.

About ten or twelve years ago, I came to the harsh realization that none of us working with these global grid were adequately qualified to work with the maps and cartography we relied on, and had likely placed too much faith in the skill of modern cartography to afford us the clarity and quality of mapping and geographic data we relied on. I decided to take a break from the whole thing to return later when I could make a fresh start at it, but many things happened along the way and it’s only the last 3 years that I’ve been able to return to the subject with more of a clean slate.

I have from the beginning based on much experience working with Munck’s style of mathematics, staunchly supported the idea that he has the mathematics and some of most important parts of the ancient metrology right. However, my disillusionment with Munck’s ideas about mapping and global coordinate systems left a burning question in its wake – if the proportions and design of ancient monuments aren’t talking about “why they are where they are” after all, what are they talking about through the numbers built into them?

Three years I was inspired to go back to working with “Munck’s” system of math by a discussion on the Great Pyramid, which brought back many memories about the amount of effort I spent working with associated numbers. My renewed searches for reliable architectural data on ancient Egyptian architecture serendipitously found a wealth of data on ancient Mayan architecture and I began to explore it, hoping that it held the missing key to understanding what ancient architects were trying to express through proportion, ratio, measurement and other areas in which we had already noted consistent patterns.

Even though many years ago, I made a few “discoveries” (I question the idea that I am “discovering” anything, as opposed to re-discovering things) in the data from Mayan temple pyramids at Tikal gathered by Teobert Maler, when I returned to the data to look deeper for patterns and clues, I was perplexed. It wasn’t until the question of what ancient Mayan mathematics might be expressing was reduced to the question of what was important to the ancient Maya that was numerical in nature, that any apparent progress was made.

The answer to that pivotal question is of course calendars. The ancient Maya and Aztecs, to name only two of many peoples with similar inclinations, are duly renowned for their preoccupation with calendars, their skill at keeping and correcting calendars, and for creating remarkable calendar-related artifacts such as the Aztec Sun Stone (frequently seen mislabeled as “The Mayan Calendar”) and the related Tizoc Stone.

To try to keep things short for now, my view the last three years is that Munck was right about the importance of geography and geodesy to ancient architects and about the highly integrated system of numbers that they used; Morton was right that celestial objects command a great deal of importance to ancient architects, albeit that the planets, their cycles, and the inter-relatedness of these cycles, may take precedence over any mathematical attributes of stars.

If we combine the best ideas from both researchers, we may have enough of a foundation to begin to paint a more complete picture for the first time.

Since the logic of what I do with numbers can be elusive to others even with voluminous notes attached, perhaps particularly those who have been tempted incessantly with simple, prefabricated and often rigidly orthodox answers in spite of their admirable open-mindedness, I’ve decided to start this blog to share my ideas and discoveries, which I hope will continue to come along on a regular basis.

Often enough, we have been like exiles, fleeing one Internet forum for another and having to start fresh. Relatively little of Munck’s printed material may be available now, most of what Morton and I posted to the Internet is now vanished. That must be somewhat ironic in its way – we study monuments that have retained their messages written on the “parchment” of stone for thousands of years, yet most of our published work about numbers, their meanings, and their relationships has already vanished into the mist of time in barely a decade, even while many of us must harbor hope that the Internet will serve so that people’s efforts and the histories of those efforts do not so easily disappear.

Since I’m making a new start here, there will be a lot of catching up to do, but I will try my best to share anything and everything that I think people might want to know in order to appreciate what I’m talking about and to be able to experiment with trying out a similar perspective to my own, because it’s been an immensely, if not immeasurably, rewarding perspective.

My search for the meaning of ancient numbers may have taken me back as far as to the very roots of mathematics, to what may literally be the very first numbers known to man as we first began to try to measure the cycles of the heavens and the seasons of their earth, cycles that may be the roots of countless customs and traditions. My search may have led me to units of measure of space and of time that have been handed down to us with astonishing dedication as both heritage and inheritance.

–Luke Piwalker