You’re getting closer to understanding, but there’s still a misunderstanding of both my model and the flaw in your own proof.

Archimedes’ goal was to approximate Pi by using polygons with more and more sides, gradually approaching the circumference of the circle from both inside and outside. This method established upper and lower bounds for Pi and has been proven accurate, supporting the value of approximately 3.14159… for nearly 2,500 years. Since then, every mathematical method, including calculus, infinite series, Monte Carlo simulations, Fourier series, fractal geometry, and continued fractions, has confirmed this value.

My model offers yet another new approach made possible by modern graphing calculators and computing power. Unlike Archimedes, who approximated the circle from within and outside, my model deliberately moves away from the circle’s circumference. The purpose is to show that π=4/√φ fails repeatedly because it overestimates the area — calculating an area larger than what is already more than sufficient to completely cover the circle.

For instance, in my model, when the radius is increased to 8,000, the circle is covered with an extra four or five unit squares beyond its circumference. Despite this, the area calculated by π=4/φ is still greater than the area that fully covers the circle. You can see this in action at this Desmos Model.

The flaw in your approach lies in assuming that 4b=π, without mathematical validation and proof that side b of your golden ratio-based Kepler triangle is actually equal to one-quarter of the circle’s circumference. In a Kepler triangle with a hypotenuse of 1 unit, side b has a length of 1/√φ. By assuming 4b=π you’re essentially claiming that 4×1/√φ=π, or 4/√φ=π, without any mathematical proof to support this assumption.

This simple step of substitution doesn’t serve as a proof. For your claim to be valid, you must mathematically demonstrate that 4b=π, rather than just making the assumption based on your inferences from the diagram you’ve created. Drawing an illustration and stating that 4b=π without any mathematical validation cannot overturn 2,500 years of solid mathematical reasoning and evidence.

Moreover, you’d also need to identify fallacies in every method that consistently confirms the traditional value of Pi. These methods all converge on the same value of π=3.14159…, further undermining the notion that π=4/√φ is correct.

Now I get it. But this is nothing new.
2000 ys ago Archimedes set up the lower and the upper limit for the π value and yes, 3.1446 falls beyond the upper limit. But even though, 4b=π proves it to be right.

The model makes the results perfectly clear.

The quarter circle can be COMPLETELY covered by 3,143,587 unit squares.

The TRUE area of the quarter CIRCLE must be LESS than the area that MORE than completely covers it.

The area of 3,144,605 based on π=4/√φ fails to meet that test, but the area of 3,141,592 based on π=3.141592… passes the test.

There is no other way to interpret this result.

Please watch the video at https://youtu.be/9Z9yfLMvAJQ to see this clearly illustrated.

*The model does not say that the correct value of Pi is 3.143587. It says that the correct value of Pi must be less than that.*

The model doesn’t say anything, it is the way you interpret it.
If the circle is already covered by 3’143’587 unit squares we can consider it the closest π value and less the 3.1415, which falls short of it.
And to that it shows that 3.1435 is closer to 3.1446 than to 3.1415, as a matter of fact.
I can’t see why we should reject it just because it is bigger. The value is closer.

See my responses below. I’ve not overlooked anything. You just don’t understand the model yet. Please watch the YouTube video again.

You misunderstand the model. The model doesn’t attempt to estimate the true value of Pi. It determines an area that exceeds the area of Pi, which allows us to see that the greater area value defined by π=4/√φ CANNOT be the value of Pi.

No, you’ve misunderstood the model. It’s not a matter of getting closer to one value of Pi than another. It’s a question of how many unit squares completely cover the circle. If the circle is already completely covered by 3,143,587 unit squares, the area based on π=4/√φ cannot be the area of the circle. The model does not say that the correct value of Pi is 3.143587. It says that the correct value of Pi must be less than that.

*That was a lengthy explanation, but this is as simple as draw a circle, color in the squares that completely cover it, count the squares and compare the count to the area measure for each value of Pi to see which one can be true. Only ONE can, and it’s 3.14159…*

Well, this is still a guess. Just because 3.1415 is less than the „standard Circle“ you did draw doesn’t mean it is the real value. Seen this way every value below the standard circle could be considered the true value.

*Note that this is the SAME value for Pi that EVERY mathematical method has derived for almost 3,000 years,….*

Well, the methods used were not that different btw them. Basically adding the value bit by bit in endless steps. It yields always almost the same result, which is not necessarily π but the result of the method.

*…..and now we’ve confirmed it with a high precision digital representation of a simple physical experiment providing empirical evidence.*

That is interesting.
Could you give some links.