12 comments

[–] Womantree1 2 points

u/frontdirect7269 nice work bro

[–] [deleted] 1 points

[deleted]

[–] emperorbma 2 points

π(area) = 3.1750342893406196275691162777; Or more simply:

π(area) = 4/(ln(ϕ)*ϕ2)

π(circumference) = 4ln(ϕ)ϕ (3.1144...)

How did you derive the new numbers?

The classical definition of π is as a convergence of the ratios of the perimeters of circumscribed polygons to the circumference. Since measuring a circle itself is trickier, Archimedes also approximated using circumscribed and inscribed polygons about the circle to the same effect. Values such as these invariably converge to 3.1415926535898...

How does this new value correspond to geometry?

Furthermore, later mathematicians such as Leibniz found associations to many other principles that are mathematically critical as well.

It can be summed as a definite series:
∫ [From 0 to 1] 2/2+x^2 dx = π/4

Or as an infinite series:
∫ [k from 0 to infinity] (-1)^k / 2k+1 = π/4

Naturally, these summations also imply the inverse tangent function from trigonometry:
tan^-1 (1) = π/4

And as a relation of the exponential series to the root of -1 which has indispensable geometric consequences in radial coordinates:
e^iπ = -1
or ln(-1) = iπ

With π/2 being directly correspondent to the value used to represent orthogonality as a 90 degree rotation, and π being the value used to represent negative values as a 180 degree rotation.

This relationship is functionally critical in defining orthogonal relationships through the root of -1 and the periodicity of trigonometric functions. Functions in form re^Θi are polar and generate the components of x and y coordinates as cosine (real) and sine (imaginary) parts: cos(Θ)+i*sin(Θ). This integrates the fields of trigonometry, coordinate plots and circle mechanics as well as providing an intuition about what the "root" of a number actually is geometrically: specifically that it's a subdivision of the value of the angles around the origin point.

Are you suggesting that these newer values are an additional parameter for the original base function which isn't yet included in the structure of the derivation?

Edit: tagging u/frontdirect7269

[–] FrontDirect7269 2 points

(when I type φ I mean the Golden Ratio, for whatever reason the alt code seems to represent different)

I could walk you down the path I took (maybe eventually I will), but the most important part is understanding the spiral and how we place pi into polar coordinates. When you realize that pi is in places it need not be you can realize that it self-normalizes the math in a way that makes the error difficult to detect.

The concept in my proof above is fairly simple, perhaps I did not explain it very well.

When you look at the Golden Spiral in polar form, r(t) = φ^(t) every value for r(t) is φr(t-1), The obvious corollary is that on a polar plot (0:3) each crossing of the axis has an incident angle that is a right angle to the neighboring ones...it has to be to come full circle and maintain the ratio of growth so that every spiral is similar.

It is also important to understand how the spiral going from 1:φ^(-1):φ^(-2):φ^(-3) diverges from the circle in a similar fashion to the one that goes from 1:φ:φ^(2):φ^(3). The area under the smaller spiral is 1/φ the area of the circle it diverges from, and the area of the larger spiral is φ times the area of the circle it diverges from.

This allows us to not only calculate for the area of the spirals, but also the area of the circle that 2 complimentary spirals represent.

All this without once invoking pi. That is the simplest derivation. The math is to take the derivative or integrate the equation for the spiral.

The Derivative of r(t), r'(t) = ln(φ) * φ^(t)

The Integral of r(t) is 1/ln(φ)*φ^(t)

which leads to the nth derivative that I listed above due to the property of evaluating this exponent:

nth derivative = [ln(φ)^(n)]*φ^(t);

If you are calculating perimeter, slope, area, volume, etc...you can hopefully see how the value of pi must change depending on the order of calculation you are using. This is the important part, pi is not needed using the math above...it is just something you can derive if you feel the need to define pi.

These Golden Identities exist for every derivative and integral, it is what makes the math powerful. Calculating a complex system of particle trajectories, for instance, becomes quite simple as long as you have a good value for φ...then you just have to set limits of granularity (i.e. how many derivatives deep you want to go).

Some background.

I gave myself a challenge of trying to calculate pi without using trig (since pi is inherently built into it). This was in part prompted by the fact that my calculated angle of departure of the Golden Spiral at the cardinal points wasn't 90 degrees from the neighboring cardinal crossing(89.66...). The original method of exhaustion by Archimedes gives the same result everyone is used to.

My next attempt was to calculate using the geometric definition of a square root:

https://janmr.com/media/root1.svg

I reduced this into a generalized cord (2*sqrt(x)) that I swept (integrated) along the diameter of the circle, this too resolved into the accepted value for pi. If it interests you, I can provide that proof...it doesn't bridge the inconsistency, but it does point to the problem. Some of our assumed relationships between straight line geometry and curved geometry are flawed.

I had originally thought that perhaps the infinity that Archimedes was approximating was flawed by virtue of either a sharing of points or some other abstract mixup that only manifested itself when dealing with different sets of infinities. A good example is sweeping out pie slices to determine area, but the inner most point must be shared...when there are infinite slices, you can introduce error.

It is why I thought that by sweeping (integrating) this cord and doing it using only the relationship of known triangle geometry (A^(2) + B^(2) = C^(2)) and the square root definition in the figure would eliminate any potential strangeness in the math... it only pointed to the fact that our definition of how to translate from straight lines to curves is flawed.

The next logical step is to eliminate not only pi from the calculation of area, but also straight line geometrical definitions. It is at first daunting to try and think in curves when we are programmed to break everything down into lines and triangles. The math holds though, you can take the basic equation for the Golden Spiral and integrate it to find area, or use the derivative to go after slope. This is where the inconsistency resolves, pi changes depending on what you are measuring since our classical definition of pi (Circumference/diameter, Area/pi^2, etc.) is just an approximation.

The difference is small when looking at things like circumference to area (~0.8%), but as the magnitude and order change it becomes a problem. This, I believe to be driving some of the problems in science and engineering.

I'll stop there, but let me know if the concept is still a bit unclear. I have been looking at this for a while and things that seem obvious to me may require further explanation.

[–] emperorbma 2 points

It looks like it wants to lowercase Phi. Probably need to add a shift to the code somehow.

So it appears what you're working from isn't trigonometric (linear) or classic circular geometry but something that is more approximating spiral geometry and incorporating the additional dimension in. Perhaps the flaw in traditional mathematics here is to assume the pi derivation is the only derivation of the concept. In reality, pi is as simply a candidate from a series of numbers that all describe a similar phenomenon with spiral behaviors.

It is probably a necessary evolution in mathematics since we're using a work-around of double-trig to get the results for this in quantum mechanics. The case of pi is a special case where you are spiraling around the same point rather than moving forward with a velocity as most analogical systems do such as planets.

[–] FrontDirect7269 2 points

This error is quite magnified when looking at small or large concepts (i.e. "quantum" or "cosmological"). It also blows up when dealing with multi-layered issues (i.e. position, velocity, acceleration, jerk... basic Newtonian analysis becomes slightly different).

It is the beauty of the Golden Ratio. If you are able to segment a thing up accordingly, all the identities apply. It also means that even though you are working in a number system that does not go to zero, you know everything from zero to where you are as long as the ratio holds. There is also a pseudo symmetry in that zero is as far away from "one" as "infinity" is, helpful when normalizing to complex problems.

[–] emperorbma 2 points

Hmm. It’s probably why derivatives “lose” a term. And integration “gains” a constant value. The highest value in derivatives go to nullity. Values never quite go away. It’s just entered a scope where the value is in a stasis point. Ultimate we’re playing with a series of quantum values that never quite reach absolute zero but simulate a zero for specific cases. There is some kind of mapping between the derivative roots here that math has yet to fully map.

It’s probably related at some level to how multiple values can be entangled in a system and sent as a composite state.

[–] FrontDirect7269 2 points

It is all pretty nicely handled. The identity of importance here is

ϕ^x = Σϕ^(x-n)|n=2:∞

Take area under the curve for example, if you segment it up from -∞:0, 0:1, 1:2. The area from -∞:0 is equal to the area of the segment 1:2. These various identities that show up in the Golden Ratio are the special sauce that allow for easier computations. Every identity holds true regardless the order (i.e. segmented lines, segmented areas, volumes, etc.), it makes this higher and lower order analysis easier.

[–] FrontDirect7269 2 points

I should also state that our "accepted" definition of pi may exist at a certain order of the nth derivative equation that I have derived above.

I have just not bothered to reconcile it yet since I have a more robust definition for describing curved geometries that does not need pi.

I have written a bunch of code to simulate all of this, perhaps I will go on an expedition to determine where "classical" pi resides. I have a feeling it is in there somewhere...

Again, the main point is that for this flavor of math you do not need pi...at all.

[–] FrontDirect7269 2 points

3rd reply, sorry but I suppose that I should point out that the -1^(th) order and +1^(th) order were used to calculate my listed values.

It leads to reason that the 0^(th) order contains a value for pi that falls between the 2. The direct average is 3.144 which is very close. This is likely where classical pi resides.

[–] FrontDirect7269 2 points

The copy/paste lost some of the exponential formatting, so the equations may be confusing, just look at the original here for clarity:

https://www.reddit.com/r/reptiliandude/comments/ztkvgz/explains_a_lot/

Most everything is exponential, so the identities above:

ϕ0 + ϕ1 = ϕ2

ϕ2 = ϕ + 1

1 = ϕ - 1/ϕ;

ϕx = ϕx-1 + ϕx-2

ϕx = Σϕx-n|n=2:∞

should be:

ϕ^0 + ϕ^1 = ϕ^2

ϕ^2 = ϕ + 1

1 = ϕ - 1/ϕ;

ϕ^x = ϕ^(x-1) + ϕ^(x-2)

ϕ^x = Σϕ^(x-n)|n=2:∞

Area_of_unit_circle = 4/[ln(ϕ)*ϕ^(2)];

Nth Derivative = nth derivative = [ln(ϕ)^(n)]*ϕ^(t)

[–] Seeker_1337 1 points

Cool