Ah what a fantastic presentation. It answers a lingering question about the use of spectroscopy and apparent brightness to determine the distance to a star. The key step that I had missed was using the stars with distances measurable by parallax to calibrate the color/apparent brightness so that it could be used for stars that were beyond the distance for parallax. Obvious in hindsight, but wonderfully and clearly explained here.
“Ironically, when Aristarchus proposed the heliocentric model, his contemporaries dismissed it, on the grounds that they did not observe any parallax effects...”
“ so the heliocentric model would have implied that the stars were an absurdly large distance away.”
> Tycho Brahe (1546-1601) proposed an experiment that would determine whether or not the earth goes around the sun. Basically, if the Earth orbits the sun, nearby stars should periodically "move" back and forth in their position with respect to more distant stars every 6 months. If the Earth was stationary (at the center of the Universe, this wouldn't occur.
Yes, but historically, scientists ended up accepting the heliocentric model long before the first stellar parallax was measured.
The heliocentric model's parsimony (explaining many different phenomena, such as retrograde motion of the inner planets, with the fewest assumptions) and various supporting observations (such as Venus' phases, which rule out any strictly geocentric model), combined with a physical theory that grounded the model (Newton's laws) rendered any alternative implausible by the late 1600s at the latest. It would be nearly 200 years before parallax was first detected.
I’m not familiar with the Ancient Greek objections here, but I remember Renaissance scientists being hung up on this same thing.
They had measured the visible diameter of stars and without any parallax, would’ve also meant that the stars were absurdly large in addition to far away.
I don’t think they had a problem with the distance, per se, but the implied size.
(You have to treat stars as points - any perceived diameter is an illusion - but nobody knew that at the time.)
Yeah, using basic geometry to calculate the size of a star from it's distance and the size of the dot it makes in your telescope, doesn't give sensible answers when the size of the dot is due to quantum diffraction (an Airy disc). And since that's named after a guy who lived in the 1800's, you can guess how much the ancients knew about it.
If we had to prove which is right without access to prior wisdom, most of us would fail miserably. Our current understanding of the world can't be built in any single person's lifetime.
But it's easier than ever to do your own research in the modern age. Heliocentrism vs. geocentrism is a well established topic. It doesn't take a genius to reach the right information. You can even conduct your own experiment if you want to.
The only thing stopping people from learning besides poverty is their own free will. I think the reputation is well deserved.
> Try to correct people to say, "What a beautiful earth rotation" instead of "sunrise".
That's just figure of speech. I don't see how it's relevant.
> Doing research means believing
The thing about good science is that it's verifiable, no need to believe. In this case, anyone with a cheap telescope can conduct their own experiment. There's just no excuse for believing in geocentrism in 2023.
As I mentioned, this is a well established scientific fact we're talking about here. People believing in geocentrism failed at multiple levels. For starters, they failed to distinguish trustworthy information from outright misinformation. They also failed to comprehend the details of the arguments being made, or haven't even bothered. On top of that, they also failed at critical thinking too.
That's more than enough reason to draw conclusions about one's ability to learn.
> works written by old dead men who held power
People so powerful that they were persecuted for their work by the church.
There are also countless people, currently alive, of various age and gender, independently verifying scientific facts and writing about it. Nobody gets their facts solely from "old dead men."
I agree with the geocentristic mindset being a product of limited vision.
What is verifiable? A well written paper on some particles being smashed together has to be taken on some level of faith unless I have the means and methods to reproduce the event. I do not have the means so I just believe it.
The thing that stood out to me was the second rung of the ladder, the Moon. Having a large companion satellite is not a prerequisite for life (probably), so how would aliens without one bridge this gap? It seems like it would take a lot more physical theory beyond simple geometry to get there.
If you can find a copy I can recommend The Tragedy of the Moon which covers such questions (from a 1973 perspective), also raising the advantages to land based life moving from the oceans with the assistance (or forcing) of the tides and tidal zones.
As a sub thread bonus I've got a sneaking suspicion Terence Tao read this collection of essays about the same time he met Paul Erdős in Australia in the early 1980s.
> Having a large companion satellite is not a prerequisite for life (probably)
I don't know, there's a sibling comment to this one that brings up some proposed benefits to having a moon.
It's possible that the foundation of life itself relied on having tidal variation of some kind, since AFAIK the equivalent of early amino acids were formed in tidal pools cut off from the main ocean and cooked under the stress of UV and evaporation.
Obviously not something we'll know for sure until/unless we find other organic life, but it's interesting to think about something like having a large orbital influence being a requirement.
Plug that into the fermi equation with observed planets with a moon >0.2% the size of the parent body and it has a definite damping effect on the probability of observing complex life.
Edit: Also shocked I had to revise my numbers, the moon is only 1.2% of the weight of the earth. I thought it was something more like 10% - so you can probably put "tides too strong to allow life" under the other side of that fermi equation factor as well!
Aliens might live on a planet without a large moon but it’s pretty unlikely for their planet to be the only object in orbit around their star. The easy way to measure the Earth-Sun distance today is to use radar to precisely measure our distance from Venus and triangulate our position from that. Aliens with radar could do the same trick using other objects in their home planetary system.
But even before radar, we had parallax methods for figuring this stuff out. Radar just brings the error down into the low parts-per-billion territory.
The way a few very very smart ancients figured out how things actually are in the solar system with just trivial observations and feeble measurements makes me realize that is exactly what modern astronomers are doing with the universe.
Thousands of years from now someone will do a presentation on how a few of us "ancients" correctly estimated the distance to other solar systems with such poor instruments and how their modern measurements from actually going there show we were only 10% off!
I had always associated him with the earth's measurement and didn't know the others. It was great reading through all the other things this person had done in his time including mapping the known world, and the prime number sieve.
A truly fascinating polymath, it must have been so satisfying to identify previously unsolved problems, and come up with a solution for them that were more or less 'good enough'. I wonder what he would have made of the way we are today, or if he were born in this era, what kinds of problems he'd have identified that needed solving.
This article had one detail that was not included in the video that had always bugged me. Carl says "how could it be .. that at the same instant there was no shadow at Syene and a very substantial shadow at Alexandria". That seemingly requires coordination between 2 people across a vast distance and accurate time measurement. I rationalized that it could be accomplished with people at each location, each with a sundial, making records of shadow length, and later comparing their measurements, but it still seemed like a messy explanation.
The detail of no shadow at the zenith on a specific day solves that problem. That removes the complication of the coordination of 2 observers and the lighting of the well better explains why the phenomenon was noticed to begin with.
The other unresolved problem for me is that it still requires the assumption that light is parallel i.e. the sun is (relatively) incomprehensibly far away, and that was not established fact at the time afaik.
> The other unresolved problem for me is that it still requires the assumption that light is parallel i.e. the sun is (relatively) incomprehensibly far away, and that was not established fact at the time afaik.
That is clever! Thanks for sharing these. It's endlessly fascinating to read about how such accurate conclusions were made via simple observation and deduction.
>I rationalized that it could be accomplished with people at each location, each with a sundial, making records of shadow length, and later comparing their measurements, but it still seemed like a messy explanation.
I'd like to know how ancient people figured out how to make a sundial. They need to point the stick called a gnomon north if they're in the northern hemisphere or south if in the south. Not just pointed an old way it needs to be tilted at the angle according to their latitude it's not just a vertical stick in the ground.
I think that approach might only work if both locations are directly north/south from each other, unless I'm thinking about this wrong. If the locations are east/west relative to each other, the shortest shadow won't occur at exactly the same moment.
edit: I guess that would still be a problem if you're using the zenith to determine the time of measurement. The best map I can find of the 2 locations used is [here](https://mathigon.org/step/circles/eratosthenes) and seems to indicate that Alexandria and Swenet/the other location are relatively north/south to each other.
edit 2: some more thoughts. If 2 points at the same latitude but on the opposite sides of the world were chosen, Eratosthenes method wouldn't have worked. The sun would have the same position in the sky at the zenith, but they'd be separated by thousands of miles, implying a flat world. Whether by design or luck, it seems that Eratosthenes experiment only works if the same longitude is used for each location, and otherwise he would have arrived at a very different answer for the circumference of the earth.
Reminds me of the Cavendish experiment to measure the Earths mass in 1798 which got to within 1% if the correct value (or the currently accepted value)
The Greeks (or, more precisely, the Eastern Mediterraneans, since a ton of these guys were Phoenician) were jaw-droppingly awesome at this stuff. I wish I could have seen it. It makes you wonder how much knowledge from other places - places with wetter climates, and/or further from the colonial powers of the 19th C - has been lost. One of the contributing reasons heliocentrism disappeared from view was that a large quantity of the Pythagorean and Neo-Pythagorean texts were systematically destroyed. When the texts from Umayyad Spain were translated back in Italy, there were simply very few heliocentric-themed writings among them. It's somewhat remarkable we even got mention of Anaxagoras and Aristarchos.
There's a growing suspicion among some in physics, so far as I can tell from my layman's chair, that many qualities like distance (i.e., the spatial dimensions) could be emergent phenomenon, resulting from a sort of bulk degree of freedom exhibited in macro structures (aka "Space from Hilbert Space"). It's an evocative notion, along with MOND and LQC and suchlike; it sometimes does rather seem like we're looking with the wrong set of eyes, or, rather, assuming things we perhaps shouldn't assume. I wish I had a time machine to, say, 2523, to see how this all resolves. How does it interact with the measurement methods cited in this presentation? Whether the cosmos is far larger than we think, or far smaller, or - perhaps most likely - that the thought of a cosmos being "larger" or "smaller" was an absurd starting point to begin with.
I had a history prof address these feats-of-observation of these primordial scientists: you've got a group of totally idle wealthy, with up to 16 hours of day time to fill, across millions of people and thousands of years. Eratosthenes could've thought about this problem — and nothing else — for a thousand hours in just six months, with plenty of time off to pursue other hobbies. When's the last time you dedicated 1000 hours to a single problem?
Absolutely! That's absolutely a prerequisite for these periods of very sudden advancement. Look at the 19th century! Very nearly the entire world's capacity was pulled into a very small geographic area, which then made the modern world.
It's a bit depressing because of the brutality that's involved, but let's not write off Europe and Greece quite so quickly. There's plenty of powers in world history that established a similar level of "excess capacity" re: labor but didn't engage in anything that could be interpreted as "societal advancement". England, during the rise of its supremacy, decided to end slavery. Yeah yeah yeah I know - there were very good realpolitik reasons for this, and England over the centuries has had some very good PR agents, but in the end, they did it. The Enlightenment was, compared to the theocratic/autocratic fusion of the early modern era, an advancement[1]. I'll die on that hill. It wasn't possible without the blood and labor of countless aboriginal peoples, who we should honor in the same proportion, of course.
[1] There's some disagreement about this in the current era, which I won't express an opinion on. "Palingenesis" is an applicable cautionary term.
I'm glad I'm not the only one where my mind splits from the rest of the group and goes off in tangential directions.
One of the video links here explaining triangulation to calculate distances showed examples with one of the grand canyon measurements. I stopped paying attention, and started wondering what the first person to find the grand canyon thought. "shit, I guess we've got to go around THAT!?!?"
Land along the Nile was heavily surveyed, yearly, to ensure its course changes were recorded in landownership.
> Long distances were measured by professional distance walkers, called bematists, who walked at a very regular pace and counted each step. Shorter distances were measured with lengths of knotted rope by men called harpedonaptai, which means “rope stretchers”
Staking claims, where corner posts were placed at half kilometer intervals, were usually paced on foot, estimating for canyons, stream crossings, and other diversions. Some claims were continuous up to 20km by 20 km.
This was in complete wilderness, in the Yukon, British Columbia and so on. Pre GPS. Not that long ago!
They used topographic maps for reference. It is amazing how accurate the claims were, when transferred to modern mapping systems.
5% error is incredible. I wonder what Eratosthenes thought of his own measurement. Did he believe it was accurate? Or questioned if unknown factors could have thrown off his calculation?
He measured it in "stadia" and there was no consensus about exactly how long a stadion was. The definitions ranged widely, from ~150m to ~210m.
The low error assessment comes from magnanimously (but baselessly) choosing to use the definition of a stadion which assigns Eratosthenes the lowest error.
>An empirical determination of the length of the stadion was made by Lev Vasilevich Firsov, who compared 81 distances given by Eratosthenes and Strabo with the straight-line distances measured by modern methods, and averaged the results. He obtained a result of about 157.7 metres
Sounds like there are other lengths given by Eratosthenes which could be used to tell what he meant by one stadion.
How did Eratosthenese synchronize the measurements? That seems like the biggest challenge: sync'ing two clocks and transporting one to a place 500 miles away without losing time? 2000 years ago?
The podcast Song of Urania does a super deep dive into the history of astronomy, including this specific measurement. One of the insights the podcast brings to light is that there were many such estimates; relative size of the Earth, Moon and Sun - distance to the Moon, etc. By reviewing the distribution of these estimates, and the methods applied, the host of the podcast suggest that Eratosthenes was lucky to get quite so close.
A woman from my small town was one of the folks who originally explored the Cepheid variables which would end up as sort of a cosmic yardstick. I saw a couple plays about this a few years ago. https://freedomsway.org/story/henrietta-swan-leavitt/
Somewhat surprised the answer isn’t based on the angle to the North Star - that’s my preferred approach and it seems less explainable by a flat earth model under assumption the stars are further than the sun
Unless the sun was VERY close, the actual distance doesn't matter. What matters is that the sun (or a star, or whatever you use as a reference) is so far away that there is no practical difference between "far away" and "even further".
For practical purposes, we treat the sun as being an infinite distance away, and the error caused by it being ONLY 93,000,000 miles away is much smaller than Eratosthenes could measure :)
It's similar to drawing perspectives, you draw perspective lines to join at the distant horizon. You don't care how far to the distance horizon.
(Those ancient Greeks had amazing understanding of geometry. Much better than most of us.)
Good answer but weirdly judgy last sentence. More like, a handful of ancient Greeks had a better understanding of geometry than most of us, just like a handful of us has a better understanding of geometry than most of us. And of course, a handful of us has a better understanding of geometry than most ancient Greeks.
Apparently the ancient Greeks didn't have much, if anything, in the way of symbolic algebra or calculus - what we mostly think of as "maths".
Instead their normal tool for mathematical logic was geometry. An educated Greek would have used geometry much like we use algebra.
For most of us, given a non-maths problem (eg physics, chemistry, almost anything in real life...) the first thing we do is express it as an algebra problem, and solve that. The ancient Greeks used geometry in a similar way.
This estimation method would only work under the assumption that earth is a spherical. Now that we do know that it's a sphere. But how could you figure that out back in 2000 years ago?
That’s the entire thing! He read an account of a city so many miles away from where he was that on a specific day of the year had no shadows at noon. On that day in his city, shadows still were cast. If there was no discrepancies in the shadows that would make sense for a flat earth. But the discrepancy made him determine it had to be sphere and then he went out and calculated the circumference.
Huh? If I put a lightbulb above something, the place it’s above won’t have shadows, but there will be shadows at places that aren’t directly underneath.
Only places near the equator will be parallel with the sun. Anywhere else, because the curve of the earth, there will always be an angle, and thus even at high noon on the summer solstice, there are still shadows in Alexandria. So I don’t understand your analogy. It doesn’t apply because the earth isn’t flat.
Whoops, I was thinking "solids that were highly significant to Plato" (for which spheres definitely qualify) but of course, as you point out, they're not actually Platonic solids. Still Plato relate-o though:
> Plato's Timaeus proposed that the body of the cosmos was made in the most perfect and uniform shape, that of a sphere containing the fixed stars.
(Wikipedia, referencing F. M. Cornford, Plato's Cosmology: The Timaeus of Plato)
Well, the time is obviously noon by definition(*), and the date would have been well known as it's entirely predictable. Syene is about a degree or so north of the Tropic of Cancer, so essentially the only date the sun is in the zenith is the summer solstice, and that’s the day they should measure shadow length in Alexandria. Had it been farther to the south, there would have been two such days, still well known and understood by the people of the time, and had been for millennia, and it wouldn’t matter which of the days you’d pick.
(*) This was 2000 years before time zones became a thing and local solar times were disengaged from the wallclock time. Not that ancient Greeks had wallclocks.
You don't need to communicate in real time. You just need to agree on which date to do the measurement and that is determined by reference to the stars. Priests had been maintaining the calendar already for quite some time. Then you just wait for the sun to reach the zenith, noon. Now you can measure the angle of the sun from the vertical then it's relatively simple trigonometry to calculate the circumference.
The key is that this phenomenon was well known and predictable. First get the date right, then wait until local noon for the sun to be at its maximum altitude. The Egyptians had long had excellent calendars, so they could use past records to predict the date.
Note that they don't actually need to be on the same longitude. It just makes measuring the north/south distance between them a lot easier.
The angle between the sun and the zenith at local solar noon will be the same everywhere at a given latitude so that part doesn't care if the cities are on the same longitude..
You do need to know the north/south distance between the latitudes of the two cities, and picking two cities on the same longitude makes that easier to measure: just go straight from one to the other and note how far you traveled.
If the longitude was substantially different you'd have to use spherical trigonometry to figure out the north/south distance from the distance and bearing of the straight route between the cities, and for that you need to know the size of the sphere you are on.
Instead you'd have to do something like travel north from the southern city until you are at the same latitude as the other city (probably determined by observing the altitude of the North star), note how far you've traveled, then travel east or west along a line of constant latitude to try to reach the other city. If you miss the other city because you didn't get the latitude quite right, you'd have to move north or south, updating your north/south distance estimate, and try again, repeating until you actually hit the other city.
I don't know whether this is true, but I think I've heard it said in connection with this story.
Egypt had a solar based-calendar, so (to a decent approximation) on every named date the sun would be in the same place in the sky.
So all that would be needed was for it be known that the sun shined straight down the well on (for instance) the 14th day of the 2nd Month of Growth (I had to look up the Egyptian calendar to get that date!), and Eratosthenes just needed to measure on that date.
The trickiest part was to know the effective distance component in the North/South direction. Luckily the Nile is roughly oriented that way anyway so the distance between the two cities is roughly the same as the North/South component
The communication requirements are minimal. The only real requirement is that we agree on which day to perform the measurement.
You and I could sit at Christmas dinner, and conspire to measure the shadow of a 1metre pole at solar noon on Easter Sunday, at our respective locations.
We could then compare those measurements when we meet next Christmas.
I always struggled (and still struggle) with math.
A couple of years ago, randomly browsing YouTube, I came across this home made video asking how they figured out the distance to the moon before modern technology. The host starts out small scale showing he can calculate the distance to things in his back yard using trigonometry and then scales it up to the moon.
My mind was blown, because no one ever told me that. It was simple, anyone could understand it. When I was in school, all I was told was to memorize abstract formulae like calculating the length of sides of triangles based on angles and known length of one side. It was never contextualized to any actual, let alone interesting or fascinating, applications.
I think you've highlighted well an outcome of the modern drive to "learn to pass exams" in many places. The original intention to "learn" has been lost or corrupted over time in a multi-century gradual example of Goodhart's Law in action.
I had a similar experience earlier in my education. "Learning" sine and cosine was nothing compared to understanding it well enough to use in a 2d game. I went from struggling with standard algebra classes to getting 5s on AP Calculus BC and Physics.
You may be interested in an Astrophysics course. I took one in college taught by an Astronomy professor and was surprised most of the content was focused on what we could determine about stars, galaxies and such based on what we could measure from them. In retrospect that seems obvious but I guess I had assumed the content would be like really heavy theories of stellar formation or gravity or something.
In my course we basically progressed from the traditional OG methods of measurement to increasingly sophisticated methods. It’s amazing how much you can learn about stars and galaxies just from combining models of black body radiation, spectral lines, and red shift with the wavelengths of the light they emit.
Yeah, a lot of schools do a poor job of tying math to practical applications. And no those absurd word problems in elementary algebra are not what I mean. Calculus, for example, makes way more sense if it's presented alongside physics.
When I was in school everyone hated word problems, but to me, they were the most clear examples of the answer to "when will I ever use this". Sure, maybe you don't care that a train leaving New York traveling at 55mph while a car leaving Philly traveling at 35mph did any thing, but they were definitely real world examples.
I had a physics teacher that had a unique way of providing examples that always revolved around a little monkey that he liked to draw on the overhead. The monkey was usually on/in/near a tree, and we had to use those dreaded equations to figure out whatever was being asked. As dorky as it was, it definitely helped illustrate in way it sounds you never got. I always enjoyed his class, and he is definitely one of the three teachers I had that was on a different level from the rest. Each of those three teachers set me on a path of where I am now that none of the others did.
I had a calculus professor that asked a physics problem on a quiz, and all the students that understood physics got the problem "wrong". It was something dumb about pulling a wheeled suitcase up an incline at a constant speed, and wanted to know the total torque on the wheels...
This is an unfortunately very uniform problem with "school", I'd say (in the US system / nomenclature) from about post-elementary up to "undergrad". Too much of the junior high school and high school classes end up as "piles of facts". The vast majority of attempts to improve education* never deal with this underlying issue, and, thus tend to just make the problem even worse. (Most likely, for various reasons including 'difficulty', attempts are made to avoid this issue.)
You can impose any standards you want - if all you're training on and testing for is ~regurgitation of facts, that's what's going to be optimized for - all of the forces at play will push even the best of teachers (those who might try to provide something other than the driest most immediate-term "goal-oriented" course / experience) towards this terrible (minimum) "standard".
At this point, I highly doubt this will ever be fixed - and certainly, can't see that happening in my lifetime. In the past few decades, hostility, and outright MARKETING of hostility, towards education has increased dramatically. Education is perennially underfunded and massively inequitable from locality to locality (and at even "finer grained" levels). Most of the fights around education these days are so far removed from questions of SERVICE and ARE WE DOING THE BEST WE CAN FOR FUTURE GENERATIONS? that there's just no way to imagine any serious or appropriate attempt can be made to address the real inadequacies of our recent & current system.
It's truly a shame. We'd all be far better off if there was more investment in, respect for, etc. education, teachers, STUDENTS (our kids), etc. Partly, this is a generational issue that even gets at the voting power of generations ... It is possible that Gen X, in part by being a smaller "generation" and in part because of their own experiences of being comparatively ignored and pushed to the side by the priorities of other generations [particularly, the older generations] across their own "lifecycles", will actually help swing things back a little towards student-oriented service (so-to-speak). But, I'm not 'optimistic' either regarding intention or, even more so, actual action.
It's a kind of tragedy, blasting people IN THEIR FORMATIVE YEARS with piles of facts in such a way as to kill off INTEREST and the possibility of real UNDERSTANDING, guaranteeing we end up with a far less informed, engaged, and healthy COMMUNITY and PEOPLE than we might otherwise have.
* Most, seemingly, quite ill-advised, unfortunately. Ill-advised based on research and the experiences of people who have spent years studying (sometimes, even, with a methodical empiricism!) "pedagogy" and "child psychology", and those who have worked on and refined models that tend to have real advantages (e.g., "Montessori" comes to mind - the data is mixed but generally supportive of the benefits of this comparatively grounded in science method, see, e.g., https://www.nature.com/articles/s41539-017-0012-7). Part of this is because of various "stakeholders" engaging in the usual tug-of-war BS where only lip-service is paid to the actual target population this essential SERVICE is supposed to be CENTERED ON...
It doesn't imply it, no. There is an absolute true "first human measurement of Earth's size" and it is not fundamentally unlearnable at this point in history. It's just very difficult to prove and we are far from the due diligence necessary for such an extraordinary statement. It should be properly qualified until we piece together a pretty complete picture of lost civilizations.
No matter what we do know, we can never prove there wasn't someone else who figured it out even earlier but never wrote it down, or they did but it and all references to it were lost.
Most "firsts" carry an implicit asterisk that isn't worth mentioning. The first person to run a 4 minute mile did so in 1954. With the asterisk that somebody else might have already done that millennia ago but didn't have a stopwatch. And we'll never know.
Logically, sure. Practically, we've collectively decided to skip words that are redundant in context.
Kind of the same way you don't usually include the country code of your phone number when someone asks for your number (unless you're in an international context).
In all areas of life, we give technically incomplete information, but reasonably expect people to know what we mean. Because we're humans, not computers. And we do it all day long, every day.
If you study the context and development of Greek culture, astronomy, and mathematics, a good case can be made that if it wasn't Eratosthenes, it was another Greek astronomer.
Do you have any sources which discuss this? From what I can tell the Babylonians were really good at mapping the night sky and tracking the positions of the planets and when they reached various waypoints. I hadn't heard of them trying to estimate the relative sizes of the various celestial objects.
I would claim that Ugg’s effort counts as a lower bound on the size of the Earth, and is therefore a legitimate constraint on true measurement. This bound might even be sufficient for some purposes.
It also seems like ancient mariners should have been able to use the visible arc of the horizon to get a rough guess, long before Eratosthenes.
All we are really arguing about is, how good are the error bars?
Technically we don't know how it was first measured and probably never will. The earliest evidence of the Earth's measurements is the Pyramids and good luck finding out how those were built.
I just found a great recap on this topic that also fits nicely with the points your link brings up, shedding a better light on the number 43,200 in other cultures (the scaling factor between the pyramid and the earth dimensions for those who TL;DR).
It also addresses the fact the speed of light in m/s is encoded three times in the pyramid, twice in the dimensions, in cubits and meters, and another times via the GPS coordinates (yes 3 fucking times !). But how would they know about the duration of a second ? I don't know ! What I know is that 43,200 * 2 = 84,600, the number of seconds in a day.
The thing goes deeper with this man and his investigations of the first edition of Shakespeare's Sonnets.
I'm not saying this with sarcasm. Maybe you could take over my attempts at coming up with a custom set of equations for a pyramid, or any other platonic shape, using symbolic regression / genetic algorithms.
"The pyramids give us the dimensions of our planet on a scale defined by the planet itself." - Hancock
It's a bit as if we had never seen a poem and stumble upon some text that harbors rhymes. You'd argue there's no intrinsic link between, say, pickle and tickle, you'd dig in each word etymology to show they wouldn't rhyme in their past forms. And beyond that you would deny the fact words rhymes because they are not in succession, and when I'd point out it's because the rhymes are crossed, you'd laugh it off.
It's not about convincing you (of what ? of some secret intent ? I'm not even sure this is the case). It's about how convincing it is. For that you need to model the cognitive process that interprets these coincidences.
Algorithmic Simplicity and Relevance - Jean-Louis Dessalles
4.1 First-order Relevance
Relevance cannot be equated with failure to anticipate [16]: white noise is ‘boring’, although it impossible to predict and is thus always ‘surprising’, even for an optimal learner. Our definition of unexpectedness, given by (1), correctly declares white noise uninteresting, as its value s at a given time is hard to describe but also equally hard to generate (since a white noise amounts to a uniform lottery), and therefore U(s) = 0.
Following definition (1), some situations can be ‘more than expected’. For instance, if s is about the death last week of a 40-year old woman who lived in a far place hardly known to the observer, then U(s) is likely to be negative, as the minimal description of the woman will exceed in length the minimal parameter settings that the world requires to generate her death. If death is compared with a uniform lottery, then Cw(s) is the number of bits required to ‘choose’ the week of her death: Cw(s) log2(52×40) = 11 bits. If we must discriminate the woman among all currently living humans, we need C(s) = log2(7×109 ) = 33 bits, and U(s) = 11 – 33 = –22 is negative. Relevant situations are unexpected situations.
s is relevant if U(s) = Cw(s) – C(s) > 0 (2)
Relevant situations are thus simpler to describe than to generate. In our previous example, this would happen if the dying woman lives in the vicinity, or is an acquaintance, or is a celebrity. Relevance is detected either because the world generates a situation that turns out to be simple for the observer, or because the situation that is observed was thought by the observer to be ‘impossible’ (i.e. hard to generate).
In other contexts, some authors have noticed the relation between interestingness and unexpectedness [9, 16], or suggested that the originality of an idea could be measured by the complexity of its description using previous knowledge ([10], p. 545). All these definitions compare the complexity of the actual situation s to some reference, which represents the observer’s expectations. For instance, the notion of randomness deficiency ([8], ch. 4 p. 280) compares actual situation to the output of a uniform lottery. The present proposal differs by making the notion of expectation (here: generation) explicit, and by contrasting its complexity Cw(s) with description complexity C(s).
