There are a few things to share here, most are results that can be readily confirmed. The exact identify 𝜋=5*arccos*((w(1+sqrt(5))/ln(10)) is one of them. If you know a mathematician, ping them.
w can be understood in the context of a chalk board loadd with equations. In third grade I had a math teacher that wore the same purple dress every day and the back got covered with yellow chalk. But I digress.. If you can understand that w= 24 +10 =34, the magic number. w in so many equations=34 so the equation should read w =7+3+24 =in a radian degree onversion= 34 felons.
You are bringing it and removing it at the same time. I could say e.g. 1+1=2+w-w. Same thing, expect it is removed via multiplication/division rather than additatively. or e.g. 1+1=2+w-ln(10)/4
It's not the way I FEEL about it, it's how it IS. This whole w thing is embarrassing. It undermines the intellectual credentials of us vaccine skeptics. Please, I am a professor with a PhD in a STEM field. You really don't understand what you're talking about.
I think, the only valid reason for introducing w would be if one wishes to build the theory of exponentiation from scratch without using the Napier constant. i.e use decimal logarithms, power 10^x as your fundamental exponential function, and pretend that e doesn't exist or hasn't been discovered yet. Then you will need w to differentiate 10^x and log10 (x). If you let w play such a fundamental role in the build up of the theory, then it is e that appears to play a secondary role and comes across as redundant. However, when w is mixed up with the exp or the ln function in some expression, it defeats the whole point, and in some of these expressions it can be basically cancelled out.
One way to calculate π is to find the perimeter of a polygon with 2^n sides. Maybe not the coolest way, but the easiest to explain and the most intuitive.
I will study this approach of which I am aware, to see if it can be explained or informed specifically in terms of w. I've been studying triangles circumscribed by circles already. There may be a link.
I don't expect w to be relevant in this. The magic trick is that if you have the trig numbers for some angle, then you can get the trig numbers for one half of that angle. So you can start with a square and then go, 8-gon, 16-gon, 32-gon, 64-gon, etc.
Not sure this write-up conveys fully, but since 10 segments result directly from w, I have found that the base of each of 10 isoceles triangles ( i.e., each side of the resulting 10-sided polygon) can be expressed as an exact function of w; that the area of the hemicircle is also determined by w (the arc length is of course a function of theta, determined by w from the 0.62832 radians (also pi/5 radians, neatly); b = 2r sqrt[((ln(10)+w(1+sqrt(5)))/(2ln(10)))]... And of course (in excel format) where A1 = pi/5 radians, the arc length of each hemicircle, determined by w.... =A1 * ACOS((LN(10)/4 * (1 + SQRT(5))) / LN(10))... (remember ln(10)/4 = w) ... The more general problem of polygons acquiring -> inf sides leading to pir^2 is informed here as (Sum of Areas of 10 homomorphic isoceles triangles) + (Sum of Areas of 10 Hemicircles) = pir^2 always, replace 10 with with n, even (I would say trivially) with infinite triangles, because (Sum of Area of Hemicircles) goes to zero. So the interesting case in the limit is not a special case, just an outcome of whatever determined theta, the interior angle, which in the case of w = ln(10)/4, is pi/5 radians. Thanks for the challenge, I have the workup, it's neat how w determines theta, then both arc length and b. It's a lengthy workup for which I have solutions that are like threads... w has an interesting relationship with pi/5.... given that the fifth segment starts from 144 degrees and spans an additional 36 degrees, reaching 180 degrees; This is 5 × 𝜋/5 radians, which simplifies to 𝜋 radians (a half-circle). So we might better appreciate w, and 2w for that matter. There are many "also trues" that do not detract, but rather confirm (because I reached them via w)....
The only problem is that if you replace all ln(10) with 4w, then all the w cancel out. The only way you can get w to stick around and not go anywhere is by rewriting all a^b in terms of exp and log10, which is kind of an arbitrary choice, since epistemologically, you can't get to a definition of log10 without defining ln first, so if once you got ln, you need a good story for why you prefered to skip it and move on to log10. I had this polygon idea in high school, and did it with π/(2^n) angles, and when I showed it to my math teacher he liked it enough to write a nice recommendation letter that got me from Greece to US. But, yes, you don't need to start with π/2. You can use π/5 or other angles and just half them repeatedly, and each one will give a different sequence, all converging to circle's perimeter. I think it all started with some homework assignment to find the perimeter of some regular polygon inscribed in a circle. Then I realized I could indefinitely double the number of sides, and that was the "oh fuck!" moment.
What an interesting story! Yes, if the game is to keep w around, you are right, I think. I will think about that thought. But now we have the link, however emphemeral...
In other words, for some bizarre reason, you chose to take a simple algebraic number, (1+sqrt(5))/4, and to multiply and divide it by ln(10) to create something mysterious. Why didn't you use ln(k), where k is any other positive number (besides 1)? You could have defined w = ln(k)/4, and then told people to consider w(1+sqrt(5))/ln(k), and the numbers would have worked out just the same.
The bottom line is that cos(pi/5) = (1+sqrt(5))/4. No natural logarithms of any other numbers required if you want the answer to be simplified.
What was the year of American independence? Most people say 1776. I take it you would say it's actually w*1776*(4/ln(10)), right?
Logic? Reason? Science? Do you care about squandering your scientific reputation on this?
PS Relax, this is where you show off YOUR skills at reason and logic and the decorum required for rational discourse. Or at least being good-natured. I have other ridiculous things to share embedded in my articles about w about w, they are even more funny than this one.
Go ahead. I dare you to ping me.
Lol
w can be understood in the context of a chalk board loadd with equations. In third grade I had a math teacher that wore the same purple dress every day and the back got covered with yellow chalk. But I digress.. If you can understand that w= 24 +10 =34, the magic number. w in so many equations=34 so the equation should read w =7+3+24 =in a radian degree onversion= 34 felons.
W.T.F. ?
Exactly. WTF?
This is cute, but it is the inverse cosine function that does all the heavy lifting in bringing out π.
Hard to get there without w. That's like saying 2 is the reason why 2 + 3 = 5!
I could just do something like π=3Arccos(1/2). Or π=4Arctan(1). So no need to bring in w.
This is (in the first case) because pi/3 and is 1/2... But no one has proposed to bring w into these specific functions. Maybe I am missing something?
You are bringing it and removing it at the same time. I could say e.g. 1+1=2+w-w. Same thing, expect it is removed via multiplication/division rather than additatively. or e.g. 1+1=2+w-ln(10)/4
Perhaps, but let's see if that is a result than a construction.
w is absolutely unnecessary.
If that's the way you feel about it, then OK, sure
It's not the way I FEEL about it, it's how it IS. This whole w thing is embarrassing. It undermines the intellectual credentials of us vaccine skeptics. Please, I am a professor with a PhD in a STEM field. You really don't understand what you're talking about.
I think, the only valid reason for introducing w would be if one wishes to build the theory of exponentiation from scratch without using the Napier constant. i.e use decimal logarithms, power 10^x as your fundamental exponential function, and pretend that e doesn't exist or hasn't been discovered yet. Then you will need w to differentiate 10^x and log10 (x). If you let w play such a fundamental role in the build up of the theory, then it is e that appears to play a secondary role and comes across as redundant. However, when w is mixed up with the exp or the ln function in some expression, it defeats the whole point, and in some of these expressions it can be basically cancelled out.
1 + 1 = 2 + w - w, for this special case:
w = ln(10)/4 ????????????????????????????????????????????????
One way to calculate π is to find the perimeter of a polygon with 2^n sides. Maybe not the coolest way, but the easiest to explain and the most intuitive.
I will study this approach of which I am aware, to see if it can be explained or informed specifically in terms of w. I've been studying triangles circumscribed by circles already. There may be a link.
I don't expect w to be relevant in this. The magic trick is that if you have the trig numbers for some angle, then you can get the trig numbers for one half of that angle. So you can start with a square and then go, 8-gon, 16-gon, 32-gon, 64-gon, etc.
Cool. So we will learn something, either way.
Not sure this write-up conveys fully, but since 10 segments result directly from w, I have found that the base of each of 10 isoceles triangles ( i.e., each side of the resulting 10-sided polygon) can be expressed as an exact function of w; that the area of the hemicircle is also determined by w (the arc length is of course a function of theta, determined by w from the 0.62832 radians (also pi/5 radians, neatly); b = 2r sqrt[((ln(10)+w(1+sqrt(5)))/(2ln(10)))]... And of course (in excel format) where A1 = pi/5 radians, the arc length of each hemicircle, determined by w.... =A1 * ACOS((LN(10)/4 * (1 + SQRT(5))) / LN(10))... (remember ln(10)/4 = w) ... The more general problem of polygons acquiring -> inf sides leading to pir^2 is informed here as (Sum of Areas of 10 homomorphic isoceles triangles) + (Sum of Areas of 10 Hemicircles) = pir^2 always, replace 10 with with n, even (I would say trivially) with infinite triangles, because (Sum of Area of Hemicircles) goes to zero. So the interesting case in the limit is not a special case, just an outcome of whatever determined theta, the interior angle, which in the case of w = ln(10)/4, is pi/5 radians. Thanks for the challenge, I have the workup, it's neat how w determines theta, then both arc length and b. It's a lengthy workup for which I have solutions that are like threads... w has an interesting relationship with pi/5.... given that the fifth segment starts from 144 degrees and spans an additional 36 degrees, reaching 180 degrees; This is 5 × 𝜋/5 radians, which simplifies to 𝜋 radians (a half-circle). So we might better appreciate w, and 2w for that matter. There are many "also trues" that do not detract, but rather confirm (because I reached them via w)....
The only problem is that if you replace all ln(10) with 4w, then all the w cancel out. The only way you can get w to stick around and not go anywhere is by rewriting all a^b in terms of exp and log10, which is kind of an arbitrary choice, since epistemologically, you can't get to a definition of log10 without defining ln first, so if once you got ln, you need a good story for why you prefered to skip it and move on to log10. I had this polygon idea in high school, and did it with π/(2^n) angles, and when I showed it to my math teacher he liked it enough to write a nice recommendation letter that got me from Greece to US. But, yes, you don't need to start with π/2. You can use π/5 or other angles and just half them repeatedly, and each one will give a different sequence, all converging to circle's perimeter. I think it all started with some homework assignment to find the perimeter of some regular polygon inscribed in a circle. Then I realized I could indefinitely double the number of sides, and that was the "oh fuck!" moment.
What an interesting story! Yes, if the game is to keep w around, you are right, I think. I will think about that thought. But now we have the link, however emphemeral...
The Ln(10) cancels out in this formula for pi, so w isn't adding anything
Ln/4 does though.
No.
Is this a sick joke?
w(1+sqrt(5))/ln(10) = (ln(10)/4)(1+sqrt(5))/ln(10) = (1+sqrt(5))/4
In other words, for some bizarre reason, you chose to take a simple algebraic number, (1+sqrt(5))/4, and to multiply and divide it by ln(10) to create something mysterious. Why didn't you use ln(k), where k is any other positive number (besides 1)? You could have defined w = ln(k)/4, and then told people to consider w(1+sqrt(5))/ln(k), and the numbers would have worked out just the same.
The bottom line is that cos(pi/5) = (1+sqrt(5))/4. No natural logarithms of any other numbers required if you want the answer to be simplified.
What was the year of American independence? Most people say 1776. I take it you would say it's actually w*1776*(4/ln(10)), right?
Logic? Reason? Science? Do you care about squandering your scientific reputation on this?
You are a riot. "Sick" joke? No. A joke? Yes. Congrats.
PS Relax, this is where you show off YOUR skills at reason and logic and the decorum required for rational discourse. Or at least being good-natured. I have other ridiculous things to share embedded in my articles about w about w, they are even more funny than this one.