PHEW IS OFFICIALLY VERIFIED

Clue: drag up the curve as well as the corner base line as far as 0.29289 up to obtain an area.

From 0 to sin45 you will get 0.7071068 × 1 unit of area

plus

From sin45 to sin90 you will get 0.29289 × 0.29289 = 0.08578 unit of area.

Total = 0.79289 unit of area

Area (2D) is the Helper for length (1D) calculation.

Translate it into 1D, the result is 0.79289 unit of length.

To explain more simple:
Draw every point of curved line from 0 to 0.7071068 (i.e. 0 to sin45) as far as 1 up. The area beyond x=1 gets moved to the base line under the curve of 45° of arc. You got 0.7071068

The points of base line from sin45 to sin90 also got drawn as far as 1. BUT it cannot equals 0.29289 × 1. It is only as high as 0.29289. Why? Because the cursor of 45° vertical move is limited only as far as 0.29289.
So... The gain equals base line times height = 0.29289 × 0.29289.
You got 0.08578

Total = 0.79289 unit of area.

Translate it into 1D. And you got 0.79289 unit of length.

Again, don't forget to celebrate Phew Day at March, 17 next year. 👌

The formula: arc length = sin45 + (1-cos45)² = 0.79289

From where?

Clue: drag up the curve as well as the corner base line as far as 0.29289 up to obtain an area.

From 0 to sin45 you will get 0.7071068 × 1 unit of area

plus

From sin45 to sin90 you will get 0.29289 × 0.29289 = 0.08578 unit of area.

Total = 0.79289 unit of area

Area (2D) is the Helper for length (1D) calculation.

Translate it into 1D, the result is 0.79289 unit of length.

Or to summarise:
Throw in a bunch of random numbers with no justification at all to get an entirely meaningless number out.

Pythagoras

Gives you the hypotenuse for a right angle triangle. Not an arc.
Just how are you applying this here?

Quote from: Danang on April 04, 2025, 06:51:30 PM

Clue: drag up the curve as well as the corner base line as far as 0.29289 up to obtain an area.

From 0 to sin45 you will get 0.7071068 × 1 unit of area

plus

From sin45 to sin90 you will get 0.29289 × 0.29289 = 0.08578 unit of area.

Total = 0.79289 unit of area

Area (2D) is the Helper for length (1D) calculation.

Translate it into 1D, the result is 0.79289 unit of length.

To explain more simple:
Draw every point of curved line from 0 to 0.7071068 (i.e. 0 to sin45) as far as 1 up. The area beyond x=1 gets moved to the base line under the curve of 45° of arc. You got 0.7071068

The points of base line from sin45 to sin90 also got drawn as far as 1. BUT it cannot equals 0.29289 × 1. It is only as high as 0.29289. Why? Because the cursor of 45° vertical move is limited only as far as 0.29289.
So... The gain equals base line times height = 0.29289 × 0.29289.
You got 0.08578

Total = 0.79289 unit of area.

Translate it into 1D. And you got 0.79289 unit of length.

Again, more nonsense.
Try a diagram with the explanation, clearly explaining just what you are doing.

Calculating the length of ⅛ of a circle (in order to calculate the total length of the circle).
To prove phew = 3.17157, and cancel phi = 3.14159.


We try to calculate the magnitude of length (1D) with the HELP of the magnitude of area (2D).

For the foundation, let’s take an illustration of a 2×4 plane (length unit × length unit).
If the area is calculated, it will be 2×4 = 8 (area unit)

The lengths of 2 & 4 are valid because of their perfect shape.

The details:
Each point on the line along 2 (length units) if drawn upwards, ALL of them will reach a length of 4 perfectly, so that 2 meets the requirements to be multiplied by 4.

And 4 is also VALID as a full 4, because at all points along the line 4 (length units) a horizontal line can be drawn along 2 (length units).




Likewise, how to calculate the length of the curve along 0 to sin45.




First Stage:
on the curve along 0 to sin45, which is 0.7071068.
At all points on the curve, a vertical line is drawn along 1 (unit of length).

Then there will be an excess in “area 1” outside the 1×1 plane.

then area 1 is MOVED to the empty space under the curve, which is “area 2”.

Then the total area (first stage) obtained is 0.7071068 (unit of area).

Second Stage:

From the tangent point of view, to reach 45°, line X must reach a full length of 1 (unit of length).

Then the remaining line along 0.29289 (unit of length) will be calculated upwards, which is also 0.29289

Why 0.29289 upwards?

Because the cursor along 45° goes up to the position 0.29289 AKA “1-cos45”.

Because the shape is square (equilateral), then the multiplication is valid to be carried out, which is 0.29289 × 0.29289 = 0.08578.

The total area obtained is 0.7071068 + 0.08578 = 0.7928932 (area units).

This area justifies that the length & width have met the requirements to be multiplied.

And the magnitude of the area is translated into the dimension of length, which is 0.79289 (length units).

Total length = 0.79289 × 360/45 = 6.34314 (length units).

And circumference/diameter AKA Phew = 6.34314/2 = 3.17157.

Likewise, how to calculate the length of the curve along 0 to sin45.

And your first problem here is you have gone from a straight line to a curve, and from a rectangle to a non-rectangle.

e.g. you can take your rectangle, and skew it and the area remains constant while the length of the lines do not.
You can do this at different rates along the rectangle to get a curve, and the area still remains constant.
So you likely wont get anywhere useful with this.

Then the remaining line along 0.29289 (unit of length) will be calculated upwards, which is also 0.29289

You are now starting to describe lines you have not explicitly labelled.

So why don't I complete the diagram for you?


A1=A2 by the argument above. But this isn't really needed at all.
A2+A3+A4+A5=1*(1/sqrt(2)) =~ 0.7071068

Then A6+A7 = (1-1/sqrt(2))^2 = ~0.2928932^2 =~ 0.0857864.

Then we get A2+A3+A4+A5+A6+A7 = 0.7928932.
This doesn't correspond to any length.
And importantly, this doesn't have any curve so there is basically no connection to the curve or arc length at all.
This is simply a rectangle and a square joined together.

So this does not justify any length.

So how do you get from this to the length of the curve?
Justify the connection.

C/D = 3.17157 is already validated by directorate general of copyright.

Phew is not blurry stuff anymore. It's been verified by experts. 👌

Are there any extra digits in Phew? (or formula to recreate it?)

Quote from: Danang on August 29, 2024, 02:40:36 AM

C/D = 3.17157 is already validated by directorate general of copyright.

Phew is not blurry stuff anymore. It's been verified by experts.

Are there any extra digits in Phew? (or formula to recreate it?)

Phew has so many digits as well, here is an example: 3.17157287525

Formula has to do with ratio, arc : radius
Many ways can be applied, here's some:
r=1
angle = 45°

ratio = sin45 + (1-cos45)² = 0.79289

or

ratio = (sin45)² + (1-cos45) = 0.79289


Phew = 0.79289 x 360/45 : 2 = 3.17157

Many ways can be applied, here's some:

Yet you can't provide any with a justification.
For example, what you provide now has nothing at all to do with the arc length.

You are taking a right angle triangle with the other angles being 45 degrees, and the hypotenuse being 1.
You are then taking the side length, i.e. 1/sqrt(2) = sqrt(2)/2, and 1 minus that, i.e. 1-sqrt(2)/2=(2-sqrt(2))/2; and then combining them.
(1/sqrt(2))^2 = 1/2
((2-sqrt(2))/2)^2 = (4 - 4*sqrt(2)+2)/4 = (3-2*sqrt(2))/2

So your first line of BS becomes:
sqrt(2)/2 + (3-2*sqrt(2))/2 = (3-sqrt(2))/2

And your second line of BS becomes
1/2 + (2-sqrt(2))/2 = (3-sqrt(2))/2

So all you are showing is (3-sqrt(2))/2 = 1.5-1/sqrt(2)

But again, this has absolutely no connection to an arc length.
It is entirely useless.

Like I said, justify the connection. Until you do, you just have garbage.

Jack, everybody is impressed by Phew, except you. Why is it? ~

But it's good if Jack keeps on critisizing Phew. Critique is important. If not him, who else? ~

Jack, everybody is impressed by Phew, except you. Why is it? ~

Really?
I'm yet to see anyone who is impressed by it.

They might be impressed by your persistence in continually spouting crap without any sense of reason; but I'm not impressed by that because any idiot can do it.

If you want people to be impressed, try justifying your BS.

e.g. pi can be justified by lookiing at the difference between the chords and the lines created from the tangent.
e.g. starting from a circle with radius 1 we can draw a square that the circle just fits inside (with perimeter 4*2=8), and one that just fits inside the circle (with perimeter 4*sqrt(2)).
The one that fits inside has a perimeter which is clearly shorter than the perimeter of the circle, as it is going in a straight line between the points.
And the perimeter of the circle is clearly shorter than the perimeter of the outer square, as the circle is cutting the corner.
This allows us to place an upper and lower limit on pi, noting the perimeter of the circle is 2*pi*r = 2*pi in this case.
i.e. this allows us to know that 2*sqrt(2) <= pi <= 4.

We can refine this down by using shapes with more and more sides, to get closer and closer to the circle.
And from the first digit, all digits that agree up until the first point of disagreement must be correct.

Now you could try doing it your way and just go straight to the sine and tangent functions, but they use pi, so you would be using pi to calculate pi.
To do it properly, you need to use the half angle formula, or some other method to cut the angle into smaller pieces.

e.g, for the above, the angle we were using internally was 45 degrees.
And the lower bound was 4*sin(45 deg), and the upper bound was 4*tan(45 deg)

And we start noting that sin(45 degrees) = 1/sqrt(2) = sqrt(2)/2 = cos(45 degrees).
and tan(45 degrees) = 1
Then using the half angle formulae:
sin(theta/2) = sqrt((1-cos(theta))/2)
cos(theta/2) = sqrt((1+cos(theta))/2)
than(theta) = sin(theta)/cos(theta)

And by applying that 4 times you end up with an angle of 2.8125 degrees, with a lower bound of 3.1403... and an upper bound of 3.1441....
And this lets us know that pi starts with 3.14

Quote from: Danang on May 25, 2025, 02:37:56 AM

Jack, everybody is impressed by Phew, except you. Why is it? ~

Really?
I'm yet to see anyone who is impressed by it.

The only 'impressed' feeling I get is how 'impressive' it is for a fictional number debate to be so endless.
The only way to make it (barely) usable is to create a square Phew * Phew and call it a magic square.
Then the square root of the magic square will be Phew. (Doesn't give it any other purpose besides content creation)

But I can do the same thing with Pi (approx 3.14159), e (approx 2.718) and any other non-negative number (negative numbers squared will be positive, but the square root will appear positive in a calculator) 

Thank you guys... You're such secret admirers. What a wonderful you are