Gerald Hawkins Crop Circle Theorums

[Mat]

Hello Forum!

I would very much like to have Gerald Hawkin's Crop Circle Theorums debunked/settled as they are the most astounding mystery that I am aware of.

Has anyone else heard of them and if so what do you think?


Thanks

Mat

 

[Mick West]

Here's a link explaining the "theorems"

http://cropcircleconnector.com/ilyes/ilyes16.html

Looks like nonsense to me. A circle inscribed in a shape in a circle is not a theorem, it's just a very simple geometric design, and easy to construct with rope (knotted at a few intervals) and two guys. The ratios arise naturally, and there's no measurement needed. There have been vastly more complex designs done since then.

 

[Mat]

Looks like nonsense to me. A circle inscribed in a shape in a circle is not a theorem, it's just a very simple geometric design, and easy to construct with rope (knotted at a few intervals) and two guys. The ratios arise naturally, and there's no measurement needed. There have been vastly more complex designs done since then.

OK. So let's see what the reality might be rather than the "Looks"

I think you have underestimated the significance of the mystery here, though I am very happy to be shown wrong!

Can we agree on these facts?


1: 2,300 years ago Euclid came up with many geometrical Theorums.
2: Twenty years ago Professor Gerald Hawkins discovered five unknown New Euclidian Theorems. (NETs).

There is no question that these NETs are new information to the world, the modern world, at least.
Caveat: One of the five NETs has something that seems similar in a theorm posited by I think Weil(?) but the five NETs are new to the world since the 1990s. There is no question that these NETs belong in Euclid's work. Hawkins is able to say where: "They should be in Book 13, after proposition 12".

Are you happy to agree on the above posited facts?



If you will grant me these I think it becomes clearer that the medium of the message, ie, Crop Cirlces, is actually less signifigant than the message.
The mystery would be significant if Hawkin's received the diagrams from an anonymous postcard.

Looking forwards to your thoughts

Mat

 

[Mick West]

No. They are not new theorems. They are just simple geometric relationships. If you take a circle, and inscribe a regular polygon inside of it, and then inscribe a circle inside of that, all you have done is precisely that. It's not a theorem and more than a square inclined and inscribed inside another square is Pythagoras' theorem, or a circle bisected by a line represents pi.

The people who made the circles just started with simple shapes because they looked nice and were easy to do.

 

[Mick West]

Therems II, III an IV are simply special cases of a trivial property of inscribed (diameter r) and circumscribed (diameter R) circles around a regular N-gon, with n = 3, 4, and 6.

r is proportional to cotangent(180/n) R is proportional to cosecant(180/n), the ratio of the area of the circles is proportional to the ratio between the squares of these proportions. For example, with n=6 (a hexagon), it's:

((cosecant (180/6 degrees))^2)/((cotangent(180/6 degrees))^2) = 4/3

This is a high-school geometry exercise. It's not a new euclidean theorem.

 

[Mat]

No. They are not new theorems. They are just simple geometric relationships. If you take a circle, and inscribe a regular polygon inside of it, and then inscribe a circle inside of that, all you have done is precisely that. It's not a theorem and more than a square inclined and inscribed inside another square is Pythagoras' theorem, or a circle bisected by a line represents pi.

Hi

Can you tell me if what you are saying above is a known fact or a speculated opinion?

My Opinion on your statement above:

• I do not see how you can hold this position.
  • Given that a considerable number of mathmaticians were unable in one year to deduce the fifth theorem from the first four.
  • This was a challenge made my Professor Hawkins in.
    • I think either Science News or Mathematics Teacher.
  • You seem to be suggesting that this is Trivial when it is clearly not.
    • My Opinion: If you wish to debunk/refute this claim you simply need to show me other cases where there where what I am considering probabilistic exotic statements of fact.
      • Let us be clear here that what is missing from Euclid, these New Euclidean Theroem's are special only in their rareness or commoinality.
        • If this is the first time this has happened, if these are the first NETs in 2,300 years then I think these NETs we have in the circles (but could have as grafitti) are astounding irrespective of how they got there. If you do not see this then I think you need to really look into why that is, relevant to the obvious and undisputed points that Hawkin's makes.
        • This point could be done in a few mentions of other NETs being found over the millenia.
  • In terms of how special they are as crop circles, I would say they are by far the most important and unexplained, including the Grassdorf Plates which, even as a hoax are an absolutly mind-boggeling crop-circle story, to my mind at least.
• Much like hawkins I have little idea about who made the Circles and I am not prepared to overly speculate.
  • I do think that if you were an unknown intelligence trying to contact the people of earth in a way that could not be debunked then encoding a "lost" suit of theorems from the very core of mathematics would be a good way to play ball, as it were.

The people who made the circles just started with simple shapes because they looked nice and were easy to do.

My Opinion: This seems demonstrably false to me because you are implying that it was a coincidence that five new NETs were discovered by accident, and that these NETs coincidentally also had the diatonic ratios, unlike any other of Euclid's many theorems. is that really what you are saying?

[Caveat: I am pretty sure but haven't checked recently about this last statement. Question: Are there any other Diatonic Euclidean Geometries?].

I will be honest and frank here, this is my first day and I would like to start of speaking openly and frankly and unemotionally, and in accord with the good guidelines on this site.

I think that what has happened here is that you have made the mistake of arguing without all the information. We do this all the time! Even the most ardent skeptics, I would imagine So I am saying to you that, expectedly, you are ignorant of the key facts of the story and that I am sure you can bring your self up to date with these facts in a matter of minutes in de google I guess I have outlined them prety much above. Most people do not know about this, that is a demonstrable fact

Anyways!

Looking forwards to your well reasoned reply

Mat

 

[Mat]

This is a high-school geometry exercise. It's not a new euclidean theorem.

I can only disagree.

Question: If this was "high-school geometry" then how do you explain how no mathematician got theorem five, in one year?

 

[Mick West]

What is the fifth theorem?

 

[Mat]

It's not a new euclidean theorem.

Question: When hawkin's can describe in which section of Euclid's works are missing from ("They should be in Book 13, after proposition 12") what is he saying, then?

Is he mistaken? It should be easy to see if he is?

Why didnt one of these 250K+ "math-fans" just call him up on it as you have done just now?

Do you think you could have got the fifth theorem from the other four, in 1992?

I look forwards to your considered replies

Mat

 

[Mick West]

So does this fifth theory actually exist? Because I could just say:

V: The ratio of the area of a circumscribed and inscribe circle of a regular N-gon is proportional the ratio of the squares of the cosecant and cosine of half the Nth segment of a circle.

and that would fit.

or actually since cosec(x)/cotan(x) = secant(x) it resolves to the simpler:

V: ratio = (secant(pi/N))^2

or using a more familiar trig function:

V: ratio = 1/sin(pi/N)^2

 

[Mat]

What is the fifth theorem?

I do not know. I do not that it was published in Mathematic's Teacher in the early nineties. You can order it from the publisher. I emailed them years ago about its availability.

But here is the important point:

My Opinion:

it does not matter about the content of the theorem; What matters is the the historic and epistemic nature of the theorms.

That is, their newness, rareness, commonality, difficulty, probability of various items of knowlege. Frankly, it doesnt matter if it is geometry theorems or cupcake recipies so long as we can guarentee some kind tamper proof knowledge filter.

Finding Euclids theoreums can be traced back consitently to euclid's time either in actuallity or causality. Finding Euclid's cupcake recipies would be interesting, but these theorems, that are locked into the our common understandong of how the world is both real and mathematical.

It doesnt matter in this sense if hawkin's theorems trace their lineage by connection or coincidence, what matters is that as symbols, they have a special, mathematically embedded certainty about them.


Cheers

Mat

 

[Mat]

So does this fifth theory actually exist? Because I could just say:

V: The ratio of the area of a circumscribed and inscribe circle of a regular N-gon is proportional the ratio of the squares of the cosecant and cosine of half the Nth segment of a circle.

and that would fit.

Hey Mick, I have stated very clearly in my last post that I don't think the content of theorems are that important as much as their historic/epistemic status and specialness.

Please acknowledge that you understand that I mean by that

 

[Mick West]

They are neither historic nor epistemic. They are trivial geometry exercises.

 

[Mat]

They are neither historic nor epistemic. They are trivial geometry exercises.

Will you please answer at least some of he questions above?

 

[Mick West]

I'm sorry, but I don't follow what the questions actually are. You claim that he came up with five new theorems. I showed that three of them were just a trivial geometric relationship, and the fifth does not seem to exist. So what is the question?

 

[Mat]

I'm sorry, but I don't follow what the questions actually are. You claim that he came up with five new theorems. I showed that three of them were just a trivial geometric relationship, and the fifth does not seem to exist. So what is the question?

Hiya

Question: When hawkin's can describe in which section of Euclid's works are missing from ("They should be in Book 13, after proposition 12") what is he saying, then?

Question: Why didnt one of these 250K+ "math-fans" just call him up on it as you have done just now?

New Question: Are you saying Hawkins is a Liar or mistaken?

Cheers

 

[Mick West]

Question: When hawkin's can describe in which section of Euclid's works are missing from ("They should be in Book 13, after proposition 12") what is he saying, then?

He's saying that if Euclid though they were interesting enough to put in the Elements, then that is where he would have put them, as 13.12 is

If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle.

Which is a similar type of relationship.

Question: Why didnt one of these 250K+ "math-fans" just call him up on it as you have done just now?

I'm sure they did. He's a little vague on what actually transpired. He claims:

ML: You found Euclidean theorems demonstrated in these other patterns?

GH: These are plane geometry, Euclidean theorems, but they are not in Euclid's 13 books. Everybody agrees that they are, by definition, theorems. But there's a big debate now between people who say that Euclid missed them, and those that say he didn't care about them — in other words, that the theorems are not important. I believe that Euclid missed them, the reason being that I can show you a point in his long treatise where they should be. They should be in Book 13, after proposition 12. There he had a very complicated theorem. These would just naturally follow. Another reason why he missed them was that we are pretty sure that he didn't know the full set of perfect diatonic ratios in 300 BC.

ML: These are theorems based on Euclid's work, but ones that Euclid did not write down himself. But they are widely accepted as fulfilling his theorems on geometry?

GH: Only widely accepted after I published them. They were unknown.

ML: Based on your analysis of these crop circles, you discovered the theorems yourself?

GH: Yes. A theorem, if you look it up in the dictionary, is a fact that can be proved. The trouble is, first of all, seeing the fact, and then being able to prove it. But there's no way out once you've done that. The intellectual profile of the hoaxer has moved up one notch. It has the capability of creating theorems not in the books of Euclid.

It does seem that senior high school students can prove these theorems, but the question is, could they have conceived of them to put them in a wheat field? In this regard, we've got a very touchy situation in that there is a general theorem from which all of the others can be derived. I stumbled upon it by luck and accident and colleagues advised me to not publish it. None of the readers of Science News [which published an article on this subject] could conceive of that theorem. In a way, it does indicate the difficulty of conceiving these theorems. They may be easy to prove when you're told them, but difficult to conceive.

New Question: Are you saying Hawkins is a Liar or mistaken?

No, I'm saying he seems to be ascribing meaning where there is none. Cherry picking ratios that appear on the diatonic scale : 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 - ratios which are bound to appear in many geometric patterns.

 

[Mick West]

I would say though that he's mistaken when he says "They were unknown." his middle theorems are so simple, and such common geometry questions nowadays, that it's ridiculous to say they are "unknown". See, for example, all these Yahoo Answers on the topic:

https://www.google.com/search?q=rat...p.r_qf.&fp=4376dff589751a20&biw=1043&bih=1019

It seems like it's a standard high school geometry question.

 

[Mat]

No, I'm saying he seems to be ascribing meaning where there is none. Cherry picking ratios that appear on the diatonic scale : 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 - ratios which are bound to appear in many geometric patterns.

I dont understand? Are you saying that you can find other examples comparable with his? That seems to be what you are saying? Can you show me one other example of this? if you cannot you simply are not entitled to make that claim as if that closes the case.

Where is your evidence, Mick?



Cheers

Mat

 

[Mick West]

I dont understand? Are you saying that you can find other examples comparable with his? That seems to be what you are saying? Can you show me one other example of this? if you cannot you simply are not entitled to make that claim as if that closes the case.

Where is your evidence, Mick?



Cheers

Mat

Well, you could find all the same types of things (and many more complex geometries) in arabic and Islamic architecture.

http://catnaps.org/islamic/geometry2.html

 

[Mick West]

And I'm a little unsure why you are banging on this, when two years ago you said:

http://www.skepticforum.com/viewtopic.php?p=211274#p211274

Thank you for all your help. It has taken a while (as skeptics we should never be sure of anything until we have reason to be sure) and there are still specks of doubt in my mind about the the "discoveries" Hawkin's made and about how these were perceived by those involved.

I now think simply that the Hoaxers were smart, inventive but not the originators of "missing" theorems. I am pretty sure the "features" are not accidental. I think that there remains some mystery about the significance of the diatonic ratios and hawkin's calculation of their inclusion being an incredibly unlikely accident. There also remains some mystery, to me at least, about the "Hawkin's Challenge" and the significance or otherwise of that and its failure to be answered, if it existed at all.

In answer to the question why did Hawkin's miss the things he missed, I have been thinking about this lots lately and the best answer I can come up with is that he probably wouldn't have missed them today, in an internet post Google etc. I am not that happy with the answer but it seems the most reasonable.

So for me the key mystery, the one that might have lead to evidence for exotic intelligences being involved has now been extinguished... oh how this mundane world we live in rolls ever on....

namaste,

Mat

So how exactly is this "the most astounding mystery that I am aware of"?

 

[Spongebob]

And I'm a little unsure why you are banging on this, when two years ago you said:

http://www.skepticforum.com/viewtopic.php?p=211274#p211274

Thank you for all your help. It has taken a while (as skeptics we should never be sure of anything until we have reason to be sure) and there are still specks of doubt in my mind about the the "discoveries" Hawkin's made and about how these were perceived by those involved.

I now think simply that the Hoaxers were smart, inventive but not the originators of "missing" theorems. I am pretty sure the "features" are not accidental. I think that there remains some mystery about the significance of the diatonic ratios and hawkin's calculation of their inclusion being an incredibly unlikely accident. There also remains some mystery, to me at least, about the "Hawkin's Challenge" and the significance or otherwise of that and its failure to be answered, if it existed at all.

In answer to the question why did Hawkin's miss the things he missed, I have been thinking about this lots lately and the best answer I can come up with is that he probably wouldn't have missed them today, in an internet post Google etc. I am not that happy with the answer but it seems the most reasonable.

So for me the key mystery, the one that might have lead to evidence for exotic intelligences being involved has now been extinguished... oh how this mundane world we live in rolls ever on....

namaste,

Mat

So how exactly is this "the most astounding mystery that I am aware of"?

Amazing the trails people leave - talk about coming back to "bite yer bum"

 

[Mat]

Amazing the trails people leave - talk about coming back to "bite yer bum"

No really, not at all. I have since realised that what is important is not the content of the theorems but their status as items of knowledge. To be frank, I got bamboozled by the maths in that thread and (I think graciously enough) acknowledged so.

Now I see that, as said, it doesn't matter, fundamentally, if its new euclidean theorems or new euclidean cupcake recopies.

For the record:

I still agree with:


I am pretty sure the "features" are not accidental.

I think that there remains some mystery about the significance of the diatonic ratios and hawkin's calculation of their inclusion being an incredibly unlikely accident.
There also remains some mystery, to me at least, about the "Hawkin's Challenge" and the significance or otherwise of that and its failure to be answered, if it existed at all.

I hope that this forum would allow a fresh discussion of these issues to those parties who would be interedted in doing so.

Let us try to debunk there, rather than deny or derail, as per the last forum, which i reckon was a few years ago now.

Sound interesting?

mat

 

[Mat]

And I'm a little unsure why you are banging on this, when two years ago you said:

http://www.skepticforum.com/viewtopic.php?p=211274#p211274

So help me through my renewed doubts, rather than just poopooing the doubts existance

So how exactly is this "the most astounding mystery that I am aware of"?

Because one can ask: Where did this encoded information come from?

And there is no answer that you or anyone can give.

The best you have done so far attack the status of the encoded information without any comparative evidence that suggests its insignificance.

 

[Mick West]

Encoded information that a school child could figure out?

Diagrams of a really really simple form, of a circle inside and outside a regular polygon?

You are asking me where the encoded information comes from, when that information is the stuff of high school geometry lessons?

I'm afraid I'm going to have to assume at this point that you are trolling. Unless you can demonstrate some statistical significance in the amount of diatonic ratios in the crop circles, then it's just cherry picking.

 

[Mat]

Encoded information that a school child could figure out?

But that is just bad arguing. You have stipulated this condition but offer no proof of it and expect that to close the argument?

If you are not interested in this because it is insignificant to you, sure, that's fine . I get that. But then it seems against your own principles the way you poopoo its discussion.

In my opinion:You speak with a certainty that you have not justified to have. And you certainly have shown no such evidence about its specialness or mundanity.

I am saying it is special you are saying it is mundane. Kindly show me why you are right and I am wrong

Diagrams of a really really simple form, of a circle inside and outside a regular polygon?

Problem: You are misrepresented me after repeated informings. for the third time: My claim is that what is special about the theorems is their epistemic/historic status, not their mathematical structure and principles. Please can you not misrepresent me about this again and focus on showing why you think it is mundane not speical. For example, anything comparable.

You are asking me where the encoded information comes from, when that information is the stuff of high school geometry lessons?

My Opinion: This is demonstrably false and you are completely wrong about this. Soz!

I'm afraid I'm going to have to assume at this point that you are trolling. Unless you can demonstrate some statistical significance in the amount of diatonic ratios in the crop circles, then it's just cherry picking.

I have been polite, honest in my opinions and not at all confrontational, obnoxious, etc. Kindly show me where this is not the case; where I have been trolling?

 

[Mick West]

I am saying it is special you are saying it is mundane. Kindly show me why you are right and I am wrong

Three of the theorems are just plugging in a number to "ratio = (secant(pi/N))^2", hence it is mundane.

 

[Mat]

Three of the theorems are just plugging in a number to "ratio = (secant(pi/N))^2", hence it is mundane.

If this is true then why did Hawkin's not know how mundane it was. Or the editors of these magazines? Or his academic colleagues?

How come you know, and they don't?

 

[Mick West]

If this is true then why did Hawkin's not know how mundane it was. Or the editors of these magazines? Or his academic colleagues?

How come you know, and they don't?

They probably did.

Look, why don't you find the fifth theorem, and then we can go from there? If it does not exist then maybe the whole thing was a joke? Or he was mistaken?

Let me ask you a direct question. Do you think theorems 2-3 are significant in any way? If so, why?

 

[Mat]

If this is true then why did Hawkin's not know how mundane it was. Or the editors of these magazines? Or his academic colleagues?

They probably did.

Why do you say probably here? You have no evidence for that at all.

You say this in yoiur article:

Don't be an expert

It does not matter what your credentials are. Don't assume that makes you right. Being an expert gives you easier access to information, and it provides mental tools to help process that information. But you can't just then say "because I say so". You still have to explain, to demonstrate, and to provide the means of verifying your point of view.

And yet it seems to me this is exactly what you have done in this thread?

Look, why don't you find the fifth theorem, and then we can go from there? If it does not exist then maybe the whole thing was a joke? Or he was mistaken?

Let me ask you a direct question. Do you think theorems 2-3 are significant in any way? If so, why?

As I have said, more than once above, I think it is the existence of the theorems that is they essential significance here.

But we cannot even agree on this:

Fact: Professor Hawkins claims they are Euclidian.
Fact: You claim they are not.

I am mystified why you claim something I have not heard before, even in the heat of the forum thread you posted.

So unless you can agree they are Euclidian or show me why they are not Euclidan - and ergo that Hawkins and others were critically and embarresingly mistaken - we are at an impas, alas.

Anyways, laters

Mat

 

[Mick West]

"no euclidean"? Clearly they are very very euclidean, so I'm not sure what you mean.

I said "Three of the theorems are just plugging in a number to "ratio = (secant(pi/N))^2", hence it is mundane." so why don't you address that.

 

[Mat]

Hi Mick

Clearly they are very very euclidean, so I'm not sure what you mean.

I mean, do you think that the theorems could have been in Euclid's body of work even though they are absent from that body of work?

I said "Three of the theorems are just plugging in a number to "ratio = (secant(pi/N))^2", hence it is mundane." so why don't you address that.

Repeat: I do not think the content of the information is as significant as the existence and, moreover, as admitted, I am no mathematician.

Please, if we are going to do this, which I would like, can we keep focused on trying to establish some agreed facts before we get into opinions This seems the reasonable way to do it.

Mat

 

[Mick West]

I mean, do you think that the theorems could have been in Euclid's body of work even though they are absent from that body of work?

What they are are interesting relationships in geometry, but not, I think incredibly useful. Look at where they supposedly fit, after

If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle.

So let's say Euclid also had after that, theorem II, reworded as he likes it.

If an equilateral triangle is inscribed in a circle, and a second circle is inscribed in the equilateral triangle, then the square of the radius of the outer circle is four times the radius of the outer circle.

Is this something that he should have had? I don't think so, as it's not usefully leading anywhere. I'm not sure exactly what criteria he had for including things, but he's not going to add every single trivial relationship.

I'm quite sure that if you asked Euclid what the ratio was, then he could have worked it out in a jiffy. It's a simple and mundane relationship. Sure it's also a nice round relationship, but is that really the criteria? There's lots of stuff there on inscribed and circumscribed circles, more fundamental and useful.

 

[Mat]

Mick,

I just don't have the maths knowledge to get past your belittling of the significance and you don't seem to be able to get out of the content of the theorems rather than their existence.

Professor Hawkin's thought they were signifigant you repeatedly assure me they are not and this means we are at an impass.

Do you see my point? if my little brother had found these theorems with his geometry set and you said "They are by no means remearkable etc etc" then of course I would go with that view. But that is now the case here. For nearly 20 years Professor Hawkins maintained that this was: " totally absorbing. It's not a joke. It's not a laugh. It's not something that can be just brushed aside." (From the interview you pasted above)

Why should I belive you and not him?

To end, he says an apt thing:

"It's a difficult topic because it tends to raise a knee-jerk solution in people's minds. Then they are stuck. Their minds are closed. One can't do much about it. But if they can keep an open mind, I think they'll find they've got a very interesting phenomenon."

 

[Mick West]

Don't believe me. Go ask some math professors. Report back.

 

[Mat]

Don't believe me. Go ask some math professors. Report back.

I am sure If I reported back in years to come, "Mick, look I have spoken with 50 Professors of mathematics and they concurred with Professor Hawkin's claims," you would still not be persuaded to proceed on with the debate about the possible meanings of the information. Maybe you would even call me a troll again

Laters

Mat

 

[Mick West]

I am sure If I reported back in years to come, "Mick, look I have spoken with 50 Professors of mathematics and they concurred with Professor Hawkin's claims," you would still not be persuaded to proceed on with the debate about the possible meanings of the information. Maybe you would even call me a troll again

Why don't you speak with two, get quotes from them, and report back tomorrow?

I mean really, if it's all so amazing, why doesn't anyone with an understanding of mathematics find it so?

 

[solrey]

I'm a bit rusty because it's been a while but I was in advanced placement math and got a B in calculus in high school. I instantly recognized Hawkin's so-called "theorems" as trivial geometry. Here's a clear and concise explanation I found:

http://www.skepticforum.com/viewtopic.php?f=7&t=13626

First you must ignore all his misdirection about circles or their areas - his "theorems" are stolen from ancient theorems about regular polygons (equilateral triangles, squares and hexagons).

That's the first thing I recognized. It's all about the polygons which define the radii of the inscribed and circumscribed circles which in turn defines the area of the circles as r is the only variable in the formula; area = pi times r squared.

"Theorem" #1
Hawkins' first theorem was suggested by a triplet of crop circles discovered on June 4, 1988, at Cheesefoot Head. Hawkins noticed that he could draw three straight lines, or tangents, that each touched all three circles. By drawing in the equilateral triangle formed by the circles' centers and adding a large circle centered on this triangle, he found and proved Theorem I: The ratio of the diameter of the triangle's circumscribed circle to the diameter of the circles at each corner is 4:3.
Forget the circles and think about what this really is: the incenter to vertex distance of an equilateral triangle and the same distance less the incenter to side distance of an equilateral triangle inscribed in the first equilateral triangle. There's the 4:3 ratio. The circles are misdirection.

"Theorem" #2
The areas of an equilateral triangle's circumscribed and inscribed circles have a 4:1 ratio.
That's all it really says, and it's ancient (at least since the Pythagoreans) knowledge stemming from the fact that the distance from an equilateral triangle's incenter to one of its vertices is twice the distance from the incenter to one of its sides. These distances are also the radii of the triangle's circumscribed and inscribed circles, so the radii have a 2:1 ratio and the circles' areas will obviously have a 4:1 ratio.

"Theorem" #3
The areas of a square's circumscribed and inscribed circles have a 2:1 ratio
Also ancient knowledge. A square's diagonal and side have a sqrt(2):1 ratio as do the circumscribed and inscribed circles' radii, so their areas have a 2:1 ratio.

"Theorem" #4
The areas of a regular hexagon's circumscribed and inscribed circles have a 4:3 ratio.
More ancient knowledge. The maximal and minimal diameters (which are the diameters of the circumscribed and inscribed circles) of a regular hexagon have a 2:sqrt(3) ratio, so the circles' areas have a 4:3 ratio.

I don't know whether Hawkins was a nutter or purposefully perpetuating a hoax but those geometric ratios are trivial and well known.

 

[Mat]

I don't know whether Hawkins was a nutter or purposefully perpetuating a hoax but those geometric ratios are trivial and well known.

Clearly not to the readers of those magazines, the math-fans.

Why did the people in this video not call him out, do you think?

​

And Hi!

 

[Mick West]

Clearly not to the readers of those magazines, the math-fans.

Why did the people in this video not call him out, do you think?

He's been called out now, so what's the difference?