Introduction

I have learn that Euler discovered a Complex Spiral that is similar to my version of the zylinder from the web page by Charles Douglas Wehner at:

 

http://wehner.org/euler/index.htm

 

Euler did not however see that constructing an entire complex cylinder (zylinder) was important as he would not have considered complex space as a valid entity. Euler did not use powers of i spread into three dimensions instead of two either but he used powers of x. While this makes his spiral different from my zylinder by using both manifold we can determine the paths of new waveforms along the spiral and project the zylinder into real space. Euler used power series whereas I just used the Argand Diagram.

Deriving the Exponential Function by Power Series

Charles Douglas Wehner revisits the McLaurin-Taylor equation, and distributes it into four boxes. This is his FOUR-BOX ALGORITHM:

 

 

 

Where:

Approximating these values we get:

Box1+Box2+Box3+Box4=e^X
Box1-Box2+Box3-Box4=e^-X
Box1+Box3=Cosh(X)
Box1-Box3=Cos(X)
Box2+Box4=Sinh(X)
Box2-Box4=Sin(X)
Box1-Box3+i(Box2-Box4)=e^iX

So approximating we get:

x^1 + x^2  + x^3 + x^4 = e^X

x^0 + x^4 – (x^1 + x^5) + (x^2 x^6) – (x^3 + x^7)  = e^-X

x^0 + x^2  + x^4 + x^6 – (x^1 + x^3 + x^5 + x^7)  = e^-X

Box1+Box3= (x^0 + x^4) + (x^2 x^6) = Cosh(X)
Box1-Box3 = (x^0 + x^4) - (x^2 x^6) = Cos(X)
Box2+Box4 = (x^1 + x^5) + (x^3 + x^7) = Sinh(X)
Box2-Box4 = (x^1 + x^5) - (x^3 + x^7) = Sin(X)
Box1-Box3+i(Box2-Box4) = (x^1 + x^5) + (x^3 + x^7) + i[(x^1 + x^5) - (x^3 + x^7)] = e^iX

 

So we have:

 

e^-X = Cosh(X) - Sinh(X)

 

and

 

e^-iX = Cos(X) + i Sin(X)

 

as before.

These discoveries were made by McLaurin and Taylor, with the final one by Euler (pronounced “Oiler”).

The Euler Spiral for e^ix is shown in the diagram below. I would interpret the red circle as lying in the complex YZ plane and Argand Diagram. Since the complex YZ plane has logarithmic distances I interpret e^ix has representing a radius of ix. This choice of axes is a little confusing and in this case the Y axis has real distances with the Z axis having complex values and shown with an arrow marked i.

The shadow cast by the spiral in the XY plane is a cosine and this means the spiral must be continuous.

The diagram is marked “WEHNER” because there does not appear to be any prior art in explaining the Euler equation as the “EULER SPIRAL”.  

see diargams below

The shadow cast by the spiral in the XZ plane is a sine since it is offset from the cosine by 90 degrees.

Wehner nearly treats e^ix as a radius by saying that the spiral’s magnitude depends on the scaling factor e^A.

Approximating these values we get:

Box1+Box2+Box3+Box4=e^X
Box1-Box2+Box3-Box4=e^-X
Box1+Box3=Cosh(X)
Box1-Box3=Cos(X)
Box2+Box4=Sinh(X)
Box2-Box4=Sin(X)
Box1-Box3+i(Box2-Box4)=e^iX

 

So in the next diagram, the spiral is wrapped around an EXPONENTIAL HORN. To avoid any confusion with perspective, the horn is exponentially DECLINING (A is negative):

see diagrams below

This is similar to the geometry of the ztar (y=1/x) and does have a connection within my description of complex space.

The value for e^x is a polynomial of the form:

e^x=Ax⁰+Bx¹+Cx²+Dx³+Ex⁴+Fx⁵+Gx⁶+......

where A, B, C, D, E, F, G and so on are the reciprocals of the FACTORIALS

0!, 1!, 2!, 3!, 4!, 5!, 6!, and so on

 

The GAMMA FUNCTION

Euler set out to create a CONTINUOUS form of the factorials, which he called the GAMMA FUNCTION.

Reconciling Euler’s Spiral to the Zylinder

There is not an obvious way of relating Euler’s Spiral to the zylinder or even extending it’s usefulness unless the distances upon the complex plane are treated as logarithmic. See the diagram “Comparison of the Euler Spiral” below.

 

see diagrams below

 

I admit that I have had to stretch the description of the zylinder to match it with Euler’s Spiral but it holds well.

Wave forms in Euler’s Spiral.

The diagram “Components of e^x” shows the shadows cast by Euler’s Spiral. These  do not match the power series terms though:

 

Box2-Box4 = (x^1 + x^5) - (x^3 + x^7) = Sin(X)

 

see diagrams below

But in the diagram:

 

x^1 +x^3 + x^5 + x^7+ … = Sin(X)

and

 

(x^0 + x^4) - (x^2 x^6) = Cos(X)

Not as in the diagram:

 

x^0 + x^2 + x^4 +x^6 +… = Cos(X) which actually equals Cosh(X)

 

Perhaps the explanation for this is that the waves need rotating by 180 degrees so that they are multiplied by a factor of i^2 = -1. This suggests that the cosh x and sinh x values lie upon the Euler Spiral (x + p). This is really an arbitrary matter of the definitions for the shadows.

 

see diagrams below

 

The diagram “Components of e^-x” shows the cosine wave acting in an opposing direction as before. Perhaps a better wave for cos x here is to use

cos (x + p) and so the direction would not need reversing. In practical terms we may be able to use both wave solutions.

see diagrams below

The diagrams “Sine Wave Components”  and “Cosine Wave Components” are drawn according to the power series definitions. They should combine into a single wave that spirals inside the Euler Spiral itself.

This suggests using more than one Euler Spiral and if this is done a zylinder starts to emerge.

 see diagrams below

Conclusion

The way Wehner has drawn the Euler Spiral means that one plane along its main axis is real while the other is negative. So in this case we have one real wave and one complex wave.

The end of the zylinder lies in the complex plane and on the Argand diagram so that plane is complex. Since we are generating complex numbers along the main axis all three planes are complex.

This does, however, raise the possibility of having a zylinder on the complex side of the complex plane and the Euler Spiral on the side of the complex plane in real space.

Wehner did not state his reasons for creating his four boxes which may be solely intuitive.