Silly φ and π crackpottery

Over time, I’ve come to really, really hate the number φ.

φ is the so-called golden ratio. It’s the number that is a solution for the equation (a+b)/a = (a/b). The reason that that’s interesting at all is because it’s got an interesting property when you draw it out: if you take a rectangle where the ratio of the length of the sides is 1:φ, then if you remove the largest possible square from it, you’ll get another rectangle whose sides have the ratio φ:1. If you take the largest square from that, you’ll get a rectangle whose sides have the ratio 1:φ. And so on.

The numeric value of it is (1+sqrt(5))/2, or about 1.618033988749895.

The problem with φ is that people are convinced that it’s some kind of incredibly profound thing, and find it all over the place. The problem is, virtually all of the places where people claim to find it are total rubbish. A number that’s just a tiny bit more that 1 1/2 is really easy to find if you go looking for it, and people go looking for it all over the place.

People claim it’s in all sorts of artwork. You can certainly find a ton of things in paintings whose size ratio is about 1 1/2, and people find it and insist that it was deliberately done to make it φ. People find it in musical scales, the diatonic and pentatonic scales, and the indian scales.

People claim it comes up all over the place in nature: in beehives, ant colonies, flowers, tree sizes, tree-limb positions, size of herds of animals, litters of young, body shapes, face shapes.

People claim it’s key to architecture.

And yet… it seems like if you actually take any of those and actually start to look at it in detail? The φ isn’t there. It’s just a number that’s kinda-sorta in the 1 1/2 range.

One example of that: there’s a common claim that human faces have proportions based on &phi. You can see a bunch of that nonsense here. The thing is, the “evidence” for the claim consists of rectangles drawn around photographs of faces – and if you look closely at those rectangles, what you find is that the placement of the corners isn’t consistent. When you define, say, “the distance between the eyes”, you can measure that as distances between inner-edges, or between pupils, or between outer edges. Most of these claims use outer edges. But where’s the outer edge of an eye? It’s not actually a well-defined point. You can pick a couple of different places in a photo as “the” edge. They’re all close together, so there’s not a huge amount of variation. But if you can fudge the width a little bit, and you can fudge other facial measurements just a little bit, you’ve got enough variation that if you’re looking for two measurements with a ratio close to φ, you’ll always find one.

Most of the φ nonsense is ultimately aesthetic: people claiming that the golden ratio has a fundamental beauty to it. They claim that facial features match it because it’s intrinsically beautiful, and so people whose faces have φ ratios are more beautiful, and that that led to sexual-selection which caused our faces to embody the ratio. I think that’s bunk, but it’s hard to make a mathematical argument against aesthetics.

But then, you get the real crackpots. There are people who think φ has amazing scientific properties. In the words of the crank I’m writing about today, understanding φ (and the “correct” value of π derived from it) will lead humanity to “enter into a veritable Space Age”.

I’m talking about a guy who calls himself “Jain 108”. I’m not quite sure what to call him. Mr. Jain? Mr. 108? Dr 108? Most of the time on his website, he just refers to himself as “Jain” (or sometimes “Jain of Oz”) so I’ll go with “Jain”).

Jain believes that φ is the key to mathematics, science, art, and human enlightenment. He’s a bit hard to pin down, because most of his website is an advertisement for his books and seminars: if you want to know “the truth”, you’ve got to throw Jain some cash. I’m not willing to give money to crackpots, so I’m stuck with just looking at what he’s willing to share for free. (But I do recommend browsing around his site. It’s an impressive combination of newage scammery, pomposity, and cluelessness.)

What you can read for free is more than enough to conclude that he’s a total idiot.

I’m going to focus my mockery on one page: “Is Pi a Lie?”.

On this page, Jain claims to be able to prove that the well-known value of π (3.14159265….) is wrong. In fact, that value is wrong, and the correct value of π is derived from φ! The correct value of π is $\frac{4}{\sqrt{\phi}}$, or about 3.144605511029693.

For reasons that will be soon explained, traditional Pi is deficient because historically it has awkwardly used logical straight lines to measure illogical curvature. Thus, by using the highest level of mathematics known as Intuitive Maths, the True Value of Pi must be a bit more than anticipated to compensate for the mysterious “Area Under The Curve”. When this is done, the value, currently known as JainPi, = 3.144… can be derived, by knowing the precise Height of the Cheops Pyramid which is based on the Divine Phi Proportion (1.618…). Instead of setting our diameter at 1 unit or 1 square, something magical happens when we set the diameter at the diagonal length of a Double Square = 2.236… which is the Square Root of 5 (meaning 2.236… x 2.236… = 5). This is the critical part of the formula that derives Phi $\frac{1+\sqrt{5}}{2}$, and was used by ancient vedic seers as their starting point to construct their most important diagram or ‘Yantra’ or power-art called the Sri Yantra. With a Root 5 diameter, the translation of the Phi’s formula into a geometric construct derives the royal Maltese Cross symbol, concluding that Phi is Pi, that Phi generates Pi, and that Pi must be derived with a knowledge of the Harmonics of Phi. When this is understood and utilized, we will collectively enter into a veritable Space Age.

How did we get the wrong value? It’s based on the “fact” that the computation of π is based on the use of “logical” straight lines to measure “illogical” curvurature. (From just that one sentence, we can already conclude that Jain knows nothing about logic, except what he learned from Mr. Spock on Star Trek.) More precisely, according to Jain:

In all due good respects, we must first honour Archimedes of Syracuse 2,225 years ago, who gave the world his system on how to calculate Pi, approximated to 22÷7, by cutting the circle into say 16 slices of a pizza, and measuring the 16 edge lengths of these 16 triangular polygons (fig 3), to get a good estimate for the circumference of a circle. The idea was that if we kept making the slices of pizza smaller and smaller, by subsequently cutting the circle into 32 slices, then 64, then 128 then 256 slices, we would get a better and more accurate representation for the circumference. The Fundamental Flawed Logic or Error with Archimede’s Increasing Polygon Method was that he failed to measure The Area Under The Curve. In fact, he assumed that The Area Under The Curve, just magically disappeared. Even in his time, Archimedes admitted that his value was a mere estimate!

This explanation does a beautiful job of demonstrating how utterly ignorant Jain is of math. Archimedes may have been the first person from the western tradition to have worked out a mechanism to compute a value for π – and his mechanism was a good one. But it’s far from the only one. But let’s ignore that for a moment. Jain’s supposed critique, if true, would mean that modern calculus doesn’t work. The wedge-based computation of π is a forerunner of the common methods of calculus. In reality, when we compute the value of almost any integral using calculus, our methods are based on the concept of drawing rectangles under the curve, and narrowing those rectangles until they’re infinitely small, at which point the “area under the curve” missed by the rectangles becomes zero. If the wedge computation of π is wrong because it misses are under the curve, then so will every computation using integral calculus.

Gosh, think we would have noticed that by now?

Let’s skip past that for a moment, and come back to the many ways that π comes into reality. π is the ratio of the diameter of a circle to its radius. Because circles are such a basic thing, there are many ways of deriving the value of π that come from its fundamental nature. Many of these have no relation to the wedge-method that Jain attributes to Archimedes.

For example, there is Viete’s product:

$\frac{2}{\pi} = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2 + \sqrt{2}}}{2}\right)\left(\frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2}\right)(...)$

Or there’s the Gregory-Leibniz series:

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - ...$

These have no relation to the wedge-method – they’re derived from the fundamental nature of π. And all of them produce the same value – and it’s got no connection at all to φ.

As supportive evidence for the incorrectness of π, Jain gives to apocryphal stories about NASA and the moon landings. First, he claims that the first moon landing was off by 20 kilometers, and that the cause of this was an incorrect value of π: that the value of π used in computing trajectories was off by 0.003:

NASA admitted that when the original Mooncraft landing occurred, the targeted spot was missed by about 20km?
What could have been wrong with the Calculations?
NASA subsequently adjusted their traditional mathematical value for Pi (3.141592…) by increasing it in the 3rd decimal by .003!

Let’s take just a moment, and consider that.

It’s a bit difficult to figure out how to address that, because he’s not mentioning what part of the trajectory was messed up. Was it the earth-to-moon transit of the full apollo system? Or was it the orbit-to-ground flight of the lunar lander? Since he doesn’t bother to tell us, we’ll look at both.

π does matter when computing the trajectory of the earth-to-moon trip – because it involves the intersection of two approximate circles – the orbit of the earth around the sun, and the orbit of the moon around the earth. (Both of these are approximations, but they’re quite useful ones; the apollo trajectory computations did rely on a value for π.

Let’s look at earth-to-moon. I’m going to oversimplify ridiculously – but I’m just trying to give us a ballpark order-of-magnitude guess as just how much of a difference Mr. Jain’s supposed error would cause. THe distance from the earth to the moon is about 384,000 kilometers. If we assume that π is a linear factor in the computation, then a difference in the value of pi of around 1 part in 1000 would cause a difference in distance computations of around 384 kilometers. Mr. Jain is alleging that the error only caused a difference of 20 kilometers. He’s off by a factor of 15. We can hand-wave this away, and say that the error that caused the lander to land in the “wrong” place wasn’t in the earth-moon trajectory computation – but we’re still talking about the apollo unit being in the wrong place by hundreds of kilometers – and no one noticing.

What if the problem was in the computation of the trajectory the lander took from the capsule to the surface of the moon? The orbit was a nearly circular one at about 110 kilometers above the lunar surface. How much of an error would the alleged π difference cause? About 0.1 kilometer – that is, about 100 meters. Less than what Jain claims by a factor of 200.

The numbers don’t work. These aren’t precise calculations by any stretch, but they’re ballpark. Without Jain providing more information about the alleged error, they’re the best we can do, and they don’t make sense.

Jain claims that in space work, scientists now use an adjusted value of π to cover the error. This piece I can refute by direct knowledge. My father was a physicist who worked on missiles, satellites, and space probes. (He was part of the Galileo team.) They used good old standard 3.14159 π. In fact, he explained how the value of π actually didn’t need to be that precise. In satellite work, you’re stuck with the measurement problems of reality. In even the highest precision satellite work, they didn’t use more that 4 significant digits of precision, because the manufacturing and measurement of components was only precise to that scale. Beyond that, it was always a matter of measure and adjust. Knowing that π was 3.14159265356979323 was irrelevant in practice, because anything beyond “about 3.1416” was smaller that the errors in measurement.

Mr. Jain’s next claim is far worse.

Also, an ex-Engineer from NASA, “Smokey” admitted (via email) that when he was making metal cylinders for this same Mooncraft, finished parts just did not fit perfectly, so an adjusted value for Pi was also implemented. At the time, he thought nothing about it, but after reading an internet article called The True Value of Pi, by Jain 108, he made contact.

This is very, very simple to refute by direct experience. This morning, I got up, shaved with an electric razor (3 metal rotors), made myself iced coffee using a moka pot (three round parts, tight fitted, with circular-spiral threading). After breakfast, I packed my backpack and got in my car to drive to the train. (4 metal cylinders with 4 precisely-fitted pistons in the engine, running on four wheels with metal rims, precisely fitted to circular tires, and brakes clamping on circular disks.) I drove to the train station, and got on an electric train (around 200 electric motors on the full train, with circular turbines, driving circular wheels).

All those circles. According to Jain, every one of those circles isn’t the size we think it is. And yet they all fit together perfectly. According to Jain, every one of those circular parts is larger that we think it should be. To focus on one thing, every car engine’s pistons – every one of the millions of pistons created every year by companies around the world – requires more metal to produce than we’d expect. And somehow, in all that time, no one has ever noticed. Or if they’ve noticed, every single person who ever noticed it has never mentioned it!

It’s ludicrous.

Jain also claims that the value of e is wrong, and comes up with a cranky new formula for computing it. Of course, the problem with e is the same as the problem wiht π: in Jain’s world, it’s really based on φ.

In Jain’s world, everything is based on φ. And there’s a huge, elaborate conspiracy to keep it secret. Any Jain will share the secret with you, showing you how everything you think you know is wrong. You just need to buy his books ($77 for a hard-copy, or$44 for an ebook.) Or you could pay for him to travel to you and give you a seminar. But he doesn’t list a price for that – you need to send him mail to inquire.

34 thoughts on “Silly φ and π crackpottery”

1. There’s a much simpler rebuttal, if you’re worried about missing the “area under the curve” by using the perimeter of an inscribed polygon, as in Archimedes’ method, just use the perimeter of the circumscribed polygon as well. The former will be an underestimate for pi, the latter an overestimate, bracketing pi in between. And they will both converge to the same value.

According to this nifty little calculator I found, already at 128 sides, the circumscribed polygon has a smaller perimeter than what this guy claims.

1. Interesting….Archimedes used 64 sides, both inner and outer, to obtain 64.sin(Pi/64) < 223/71 < Pi < 64.tan(Pi/64) < 22/7. …A good approximation is 3 + (16/113) = 3.14159292… as Pi = 3.14159265… Of course Jain would say your calculator's calculations are based on "the lie".

2. dtm0 says:

The page http://www.jainmathemagics.com/category/13/ and the lack of any evidence of an Australian DGR with the word “Mathemagics” makes me wonder what the penalties are in Australia for soliciting donations, claiming them to be tax-exempt, when in fact they aren’t. Were he in the US, the IRS doesn’t mess around with false claims of 501(c)(3) status; I don’t know how strict the Australians are on that.

I also don’t know if Australian tax law requires the same sort of registration and extensive paperwork for something to be able to receive tax-exempt contributions as the US does, so my inability to find any registration might not mean anything.

(There is an ABN assigned to the business name “Jain 108 Mathemagics”, but that isn’t a DGR so far as I can tell, it’s an ABN for an individual)

1. decourse says:

I don’t see a DGR either. Someone may wish to bring this to the attention of the ATO.

2. I\m Canadian but i get the impression from recent news items that Australia is more stringent on tax-exempt permits and more vigorous in enforcing compliance than America.

3. An excerpt from the interview in the video on the main page of his site:

“I was one of the victims of the state where I got high grades in mathematics but I literally was forced to study higher-level mathematics.
A bit like a robot, I had to pass calculus and all this difficult algebra but I honestly really didn’t understand what I was learning.
So I entered a quest, my personal quest was to understand the truth of mathematics.”

He then goes on to explain how his heritage and experience in brick-laying gave him a good intuition for areas and volumes. Huh.

4. JoeBussen says:

His real name seems to be given in his “about” section. He “recently” visited the US (in 2005), and opposes No Child Left Behind, because of a related Bill that requires that any child scoring below 50% be given Prozac!

5. bryanmcc says:

Not a mathematician, but doesn’t the fact that pi is transcendental, while phi is merely irrational, pretty much throw all of his (already garbage) theory out the window? Isn’t it impossible to construct transcendentals from a less-than-infinite series of algebraic numbers?

1. markcc says:

Yes. But that’s less fun to explain.

2. That’s kind of begging the question, though.

Let’s say there’s some ratio R of a circle’s circumference to its diameter (we know this is π, but go with me here). We know that some constant R exists because both are linear dimensions and all circles are similar; Euclid covers as much.

This guy claims that R is a particular number X, but you complain that his X is algebraic while π is transcendental, so they can’t be the same. He responds that of COURSE X isn’t transcendental; that’s the whole point of his system. Not only is X simpler than π, it’s even ruler-and-compass constructible; he can square the circle!

Just saying “X is algebraic while π is transcendental” doesn’t really get there. What you need is a proof that whatever R is, it must be transcendental, and that’s a lot harder than, say, circumscribing an N-gon for sufficiently high N and showing that he misses the upper bound.

6. Phil Koop says:

I blame “Donald in Mathemagic Land” in both cases.

7. Sean Holman says:

There are actually subtle issues with the Archimedes calculation of pi, but they are related to (potentially) miscalculating the circumference rather than missing area. See this:

Of course Archimedes got the correct value for pi, but it just shows there is a bit more to it than one might think at first.

8. David Starner says:

@Sean Holman That drove me nuts in high school, and I never did resolve it. My problem I think was clearer, since there was no pi involved; take the diagonal of a unit square, and approximate it by vertical and horizontal segments. You can get arbitrarily close, but the vertical and horizontal segments obviously add up to 2, whereas the diagonal is sqrt(2). I guess I learned something today.

1. Sean Holman says:

@David Starner You might be interested in this paper by Mandelbrot:

http://faculty.washington.edu/joelzy/howLongIsTheCoastOfBritain.pdf

He asks the seemingly simple, but actually fairly deep question:

How long is the coast of Britain?

As it turns out the length of coastlines isn’t a particularly well defined concept:

The point is that the length of the coastline depends on what scale you use to measure. If you measure at the scale of km, the answer will be smaller than if you measure at the scale of metres or cm because at smaller scales you will capture many more small variations. This is because coastlines are “fractal like”. Mandelbrot’s point is that to measure such curves it is really more appropriate to introduce the concept of “fractional dimension” since these curves are in some sense more than 1 dimensional.

It is the same phenomenon that is going on with this “proof” that pi = 4 and your example with the triangle. The curve you form as the limit using horizontal and vertical line segments is infinitely wiggly just like a coastline (i.e. it is a fractal). (One can argue whether coastlines are really infinitely wiggly when you get to the scale of atoms say, but that’s missing the mathematical point.)

9. “It’s the number that is a solution for the equation (a+b)/a = (a/b).”

This sentence doesn’t make sense. Your equation has two variables, so a single number can’t be “a solution.” I assume one of those is a constant, perhaps 1?

1. markcc says:

I should have said “It’s the unique ratio that’s a solution for…’ There’s only one value where that’s true.

1. Sean Holman says:

Only one positive value 🙂

10. jc says:

I guess the theory is that a portrait which is more than phi in its height:width ratio would look “too tall,” while one less than phi would look “too wide.” In other words, why paintings are generally neither squares nor long bars, but a certain shape.

1. David Starner says:

That’s the theory, but what are the ratios of portraits in practice? To one digit, The Mona Lisa is 1.5, The Girl with a Pearl Earring is 1.2, Van Gogh’s Starry Night was 1.3. The Persistence of Memory is 1.4. Leonardo da Vinci’s Last Supper is 1.9. In fact, not a single painting I looked at had a ratio of phi. I finally found Bottecelli’s Birth of Venus, which does have such a ratio, but it seems to be a rarity.

Movies are neither squares or (with rare exceptions, long bars), but they range from 4:3 (1.33), 16:9 (1.78), 1.85:1, all the way to 2.39:1. 1.66:1 is the closest standard movie ratios come to phi, and that doesn’t seem particularly common or close.

Paper sizes are 8.5 x 11 (1.29), An (n=4 for normal paper) (sqrt(2) ~ 1.414), or legal at 8.5 x 14 (1.65). Legal is sort of phi ratio, and yet no one seems to be demanding to switch to legal sized paper.

11. KnBa says:

This is a minor issue, but the statement that “In reality, when we compute the value of almost any integral using calculus, our methods are based on the concept of drawing rectangles under the curve, and narrowing those rectangles until they’re infinitely small, at which point the “area under the curve” missed by the rectangles becomes zero.” is dubious. As just one obvious example, it’s not what gets used in polar, cylindrical, or spherical coordinate systems.

Since what is at issue is the value of pi in a 2-dimensional flat coordinate system, your example seems a bit pedantic.

12. What is the mean interest in hiding the true value of Pi?

There are BIG interests that oppose to a corrected value of Pi.

1. Khazar academic mafia does not want the freedom for humanity.
2. A corrected Pi collides with the interests of the oil industry and the Khazarian control of petrodollar.

When PI be corrected humanity will discover an endless source of energy that will liberate humanity from slavery and the dependence to other energy sources.

The number Pi have be maliciously obfuscated to prevent the close of a cycle to some energetic resonance phenomenon which will take humanity to an ENERGETIC GOLDEN AGE and the conquest of deep space.

TRUE VALUE OF PI:

Pi = 3.1446055110296931442782343433718357180924882313508929506596078804047281904892436548476515566340325422595160489765784452235018414818847721014580011238453531659969963123944614330895602447224013851373131501976513250168886718624703787313359434961827623424884419929696155384972370055738355223468907453641698014204369640943817463269453772663395414398903709747924249157889297802333906441767084172268827515380592173997026423023851194242244081992685573437499657987944611238911016107551387207358281657572181883283516336139159023992353694690024845170044516992781985453761660350519720800718970644071409668757828437246633219026822340025407725353821526637922670369853908547616452436921953232107331044735525949802311653660216067204763773809792592558234876801085351187469338952701406443781568048374310664077223404139952343917185562861066240175976669357645765480751311418697916950736513185281927426366978973484884146736468201663051035828968367940082442276210780785802770252790792921943126282608098219773061432750203769…

REMEMBER THIS PURSUED NUMBER ABOVE.

1. markcc says:

In recent years, every mathematical crackpot seems to make random claims like yours, about how their better math will solve all of the worlds problems and give us the key to free energy.

So please, explain how, if the value of π is off by less than 1%, correcting that will somehow produce free energy.

Further, explain to me how, exactly, every company in the world that produces round products hasn’t noticed that they’re using *more* material than their calculations say they should?

Answer those two, and maybe, just maybe, I’ll spend a couple more minutes looking at your rubbish.

Until then, stop wasting my time.

2. unknown says:

you are absolutely right on this , understanding of the circle would lead humanity to understanding of the universe … you are right that , this brainwash is work of the cabal … i just add and vatican to … but this ratio is wrong …. circle has a finite circumference , that’s why it is wrong … this guy who wrote this article is missing one thing to .. our decimal math has nothing to do with nature and geometry .. and for example he has not knowledge that the most important formulas like C=2pi*r and A=pi*r^ have not proofs in geometry , he is just another brainwashed zombie and doesn’t understand that calculus is based on Pi , not Pi on calculus …. our Pi is wrong,no doubt about that ..but Jain Pi is wrong to … i recommend to watch this ..

13. Why does a helpless book (the book of jain pi) makes to tremble your archaic mathematics foundation in such a way!?

Even you wrote rivers of paragraph trying to circularize the book of jain-pi.

That means that this book is true.

1. David Starner says:

Let’s see: if no one said anything, you could say they were afraid to even respond. If everyone dismissed it tersely, you could say they didn’t bother to understand it. When someone wrote 2500 words on it, including quotes, you call that “rivers of paragraph” and say “that means this book is true”. No matter what, you can disregard what anyone actually said and claim the way it was said proves your case is true.

Since this is true if no matter what your claim was, and thus could be used to prove anything, it’s obviously an incorrect argument.

14. Pjs says:

“π is the ratio of the diameter of a circle to its radius.”

Might want to fix that typo, unless pi is now 2.

15. I can see fear and rage on words against Jain… why ? there is no need… you may not agree… but in a friendly way… that anger just demonstrates fear, or even envy. I really appreciate Jain’s works. And finally.. if he’s wrong…that’s it… he is wrong.. what’s the big deal ? but if he is right …. then I can bet this rage will multiply itself… for some period of time… then it will disappear… ignorance and rage have natural and implicit limits.

1. markcc says:

Why, when I make fun of some idiot, do people insist on seeing it as fear or anger?

Jain is full of shit, and I explained why. In great detail.

When you claim something idiotic like “π” isn’t 3.14159…, you’re making a specific, testable claim. That test fails miserably in numerous different ways.

It’s a failure mathematically, because π isn’t just some arbitrary number: it’s a foundational value in mathematics, whose value can be computed analytically.

It’s also a failure in a very practical, physical sense. We manufacture round physical objects every day. If the value of π that we use were off by even a miniscule degree, that would have a very clear and measurable effect on the cost of manufacturing everything from cars to soda cans. And yet, no one has ever observed that “error”.

But yeah, don’t worry about that. Attack the tone of my writing, because you can’t actually criticize the content of my arguments.

16. allen roberts says:

i understand the point of your paper. jain is wrong. but there is a reason to change the value of pi when creating flight trajectories near objects with strong gravitational fields. space itself is compressed by gravity, and over such long distances, that could effect pi, space, distance in calculations. i would argue, though, that until we had earth to moon laser measurements, our miscalculations were very reasonable.

17. The best method for proving that Golden Pi = 3.144605511029 is the correct value of Pi is to apply the Pythagorean theorem to all the edges of a Kepler right triangle after accepting that the second longest edge length of a Kepler right triangle as the diameter of a circle while the shortest edge length of the Kepler right triangle is equal to 1 quarter of the circle’s circumference that has a diameter equal to the second longest edge length of the Kepler right triangle. Also the hypotenuse of the Kepler right triangle divided by the shortest edge length of the Kepler right triangle produces the Golden ratio of Cosine (36) multiplied by 2 = 1.6180339887… : https://www.youtube.com/watch?v=nvja8rGCbzY&t=81s : https://en.wikipedia.org/wiki/Pythagorean_theorem : https://www.goldennumber.net/triangles/ : https://www.facebook.com/TheRealNumberPi/