So my aim was not to define pi alternatively on the basis of (root) phi for some mythical (numerical) reason; in the contrary. Knowing that the square of pi is round about 9.87, which can be used to represent the 17th Fibonacci nr. , it was just meant to be a way of showing how the two constants are numbertheoretically or even mathematically related despite its sort of randomness. The mystical (Egyptian) 4/rootPhi is simply a consequence of taking the square root (out) of the approximation 15,97/9.87. ≈ 4/3.14. or the other way around.
Dividing a square amount of 4 by pi will result in a diameter of a circle of round about root phi. That’s all.
One could alternatively divide 5² by ‘π’ to get a diameter of ≈ 8

Or roughly ‘decimal’ Phi: 16/1.62 , which of course is the square of 4/√1,62

But the derivation of π 3.1446 as 4b=π is NOT flawless. You make an invalid, unproven assumption with no mathematical validation in your construction that invalidates the results.

And the fact that entire industries rely on the traditional value of Pi to do what they do without catastrophic results should be enough to say to you that maybe, just maybe, there’s an flaw in your construction that you need to see and correct.

The difference between 3.14159… and 3.14460… is huge in this of these applications—managing satellites in orbit, calculating when an eclipse will happen, determining your position with GPS.

As a NASA engineered shared, “If Pi were off we would know it.”

Isn’t it much more incomprehensible yet that thousands of years of mathematical research and countless applications in real world industries would be in error, and your golden ratio based Kepler triangle in a half circle redefines all human research, knowledge and experience on this topic?

I agree, it is indeed more like numerology, My aim was to trace back where that suspicious relationship between pi and phi is based upon. and to point out that it makes ‘a lot more’ sense to use actual pi instead of adjusting it in order fior the product of root phi and the adjusted pi being 4 (a square)
According to many numerologist it can be found in the great pyramid because of the unit base to twice the height ratio being about root Phi. (14/11)
A lot of esoteric wanna be mathematician state it to represent the squaring of a circle with the peremiter of the unit base of the pyramid being 4.
Nonsense of course. Why not adjust phi instead to make it 4 ?
If one wants to relate 2 definite constants, one should use both.
Their product is just not four, but fourish:
15.97 / 9.87 = 4² / π² (approx)
9.87 / 1.44 = π² / 1.2² (approx)

I (also) just find it fun to figure out such obscure connections. I really like this one (square loop) I sort of created (or knocked off) from the (Kepler) root phi-sort of pi association.

(√1,618−1,272)÷2 = 0,0000031446463138
4/0,0000031446463138=
1272003,144646
4/1272003,144646=0,0000031446463138
Etc..
Makes T. Howard’s ‘impossible’ root2 loop look paile in comparison. (Maybe that’s just an exoression that is used solely in my country)

*No one is saying that the polygon IS the circle or that its perimeter IS Pi. Talking about the differences between a circle and a polygon is a meaningless defense.*

You’re just confirming that 3.1415 is not defined as the ratio btw P and D.

*The polygon approach is just one of a NUMBER of MATHEMATICALLY VALID ways to estimate the TRUE value of the circumference of the circle, which when compared to its diameter GIVES us an approximation of Pi.*

As you say, it is an estimate. The amount of methods yielding 3.1415 as a result boil down basically to only one method of adding an ever growing number of ever smaller straight segments of line. The polygon method.

*And you don’t have to go very far in this mathematically valid exercise to see that that ratio is way LESS than 4/√φ.*

As Plato said back then, “the circle we see is not the real circle”.
The derivation of π 3.1446 as 4b=π is flawless and points out to still unknown characteristics of the Circle and of Space.
Why do you try to set aback these possibilities, is incomprehensible to me.

No one is saying that the polygon IS the circle or that its perimeter IS Pi. Talking about the differences between a circle and a polygon is a meaningless defense.

The polygon approach is just one of a NUMBER of MATHEMATICALLY VALID ways to estimate the TRUE value of the circumference of the circle, which when compared to its diameter GIVES us an approximation of Pi.

And you don’t have to go very far in this mathematically valid exercise to see that that ratio is way LESS than 4/√φ.

If you doubt the validity of these approaches please download my Excel model from https://www.goldennumber.net/pi-314159-vs-pi-31446-circumference-solution/. It has a simpler yet estimate of the value of Pi.

Experiment with the input value in cell C5. You only need 500 rows to show that the value of Pi is closer to 3.14159… than 3.14460…

Pump it up to 20,000 rows and truth becomes clearer yet, with a variance from Pi of only 0.00000010394327.

All by using ONLY the formula of a circle and Pythagorean theorem.

Tell me if you can find anything that is incorrect in its assumptions, methods or conclusions.

I didn’t say 112+ is a flaw., it is just the ø of that polygon.

🔹 π is not derived from patterns, polygons, or physical constructions.

Well. In fact it is.

🔹 It is defined mathematically as the ratio of a circle’s circumference to its diameter.

It is not exactly like that. When the approximation is made by polygons you can’t define it that way.

To that, 3.1415 is a number that can’t ”rotate”, like a perfect circle. Even by trillions of sides it is still a polygon.

Thanks for the follow-up, and I appreciate the clarification.

You’re absolutely right to say that a 22-sided polygon with 16mm sides doesn’t yield a perfect diameter of 112mm. That’s exactly the point—approximating a circle using a polygon, even a regular one, can never perfectly replicate the geometry of a true circle.

In fact, this small discrepancy is exactly what led mathematicians like Archimedes to the discovery of π: by comparing the perimeters of inscribed and circumscribed polygons, he could see that the value of π lies somewhere in between—and that it converges toward a unique, constant value.

The difference you’re noticing—that the diameter is slightly more than 112mm—is a feature, not a flaw. It’s the natural consequence of trying to wrap straight-line segments (a polygon) around a curve (a circle). That tiny gap between polygon and circle doesn’t disprove π—it confirms it.

And that’s the bottom line:
🔹 π is not derived from patterns, polygons, or physical constructions.
🔹 It is defined mathematically as the ratio of a circle’s circumference to its diameter.
🔹 It holds up under trillions of digits of calculation, spaceflight, GPS, quantum physics, and calculus.

So while it’s fascinating to explore geometric approximations and relationships with φ or Fibonacci numbers, none of them change the foundational reality that π ≈ 3.1415926535…. It’s not a matter of belief or perspective—it’s a constant baked into the very structure of mathematics and the universe.

Sorry.That comment I refer to, is wrong. But it also has nothing to do with the fact that it is a polygon.
The real problem is that there is no 22-sided polygon with a perimeter of 352 and a major diameter of 112.. The diameter is slightly larger than 112..
I didn’t pay enough attention.

Thanks for sharing this. I appreciate the curiosity behind finding connections between π, φ, and Fibonacci numbers—it’s a fascinating area. But there are a few key issues with the logic here.

When you write expressions like:

(φ×π)²≈15.97≈17th Fibonaccinumber

You’re essentially multiplying π² (≈ 9.8696) by powers of φ (like φ ≈ 1.618…), then noting the result is close to a Fibonacci number. That may seem “neat,” but it’s numerology, not mathematics. If you try hard enough, you can make almost any constant squared or multiplied by irrational values look like it’s near a Fibonacci number.

Here’s the key: π is not defined by proximity to patterns. It’s defined as the ratio of a circle’s circumference to its diameter—a value that emerges naturally from the very geometry of space. Any connections that rely on manipulating π and φ to land near Fibonacci numbers don’t prove anything—they just fit patterns to predetermined outcomes.

If we reverse-engineer formulas using π, then say “only π fits them well,” that’s circular reasoning, not discovery. The same logic would “prove” dozens of other spurious mathematical constants.

So while the numerical coincidences you’re pointing out are interesting curiosities, they don’t disprove or validate π. The true test of π lies in its consistency across trillions of digits, its use in orbital mechanics, engineering, and physics, and its rigorous derivation from calculus and geometry.