The Geometry of Euler’s Formula – part 6
by greg ehmka
Euler’s Formula Upgraded for any Base
In the eBook excerpt below we are applying a relatively simple intuitive observation to upgrade Euler’s wonderful formula for any base, without formal proof. It is easy enough to show that the upgraded formula is true using a calculations and a simple proof follows.
More specifically, for any base B:
with a and b real numbers, except both not equal to zero. Then, Euler’s formula:
for any base B becomes:
Additionally, by a similar intuitive observation, we derive a relationship between base and wavelength for helixes and spirals that includes negative, imaginary and complex wavelengths. Definitions and examples of these are in the eBook following this excerpt.
The base-wavelength relationship is:
which in the form:
is seen to be identical to wave number:
And then, combining these two new relationships, we can conclude that Euler’s Famous identity itself:
is valid for any base and its associated wavelength. I.e.:
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6.3 The Helix Base and Wavelength
The Euler Helix given by:
can be applied in a more general way to helixes of any base B:
In a helix, altering the base changes the wavelength. But, most often, the e-base is retained, and wavelength and frequency characteristics are changed by adding a coefficient to the exponent. I.e.:
This is, of course, fine for many applications. But, keeping the concept of different bases allows for an easier understanding of the greatly expanded range of helix and spiral functions that come with negative, imaginary and complex bases, as well as negative, imaginary and complex wavelengths – for all of which, geometric interpretations will follow. Rather than only frequency changes to the e-base helix, the relationship between base and wavelength will serve to easier analyze what’s going on with all of the new and different possibilities.
To derive this broader view of the base/wavelength relationship, we go back to Euler’s identity:
and, take natural logs without simplifying:
Then we can observe that, in this particular case, e is the base and is the wavelength divided by 2. I.e.:
So, a more general statement of Euler’s identity can be written:
On the left side:
Simplifying and rearranging, we have the direct base-wavelength relationship:
The Base–Wavelength Relationship
which effectively expands the definition of wavelength to include negative, imaginary and complex bases in addition to real ones.
Note that in the form:
this is the same as the wave number equation(*)
And so, in this context, wave number is, in fact, equal to the natural log of the base!
And, when needed, we can write this in exponential form:
6.31 The New Famous Five Identities
Noting in Euler’s identity:
as we did above, that in this particular case, e is the base , and is the wavelength divided by 2. I.e.:
And so, as we concluded above, a more general statement of Euler’s identity, can be written:
If, this time, we convert this back to exponential form:
then, we have stated Euler’s identity for any base and its associated wavelength – this tells us that Euler’s Famous Five equation is true for all bases! This means that when any helix or spiral, base B has its wavelength in the exponent, the result is always equal to 1; and, if the wavelength over 2 is in the exponent, the result is always equal to -1.
Additionally, there is an identity for:
The New “Famous Five’”Identities:
This will be seen to be true for all base-wavelength pairs including real, imaginary and complex.
For example, the i-base:
And, a random complex base :
6.32 Euler’s Famous Formula Upgraded
combined with the laws of exponents and de Moivere’s theorem(*):
allows for some interesting insights. First, if n = 1 we have, of course, the ‘Euler Helix’ that we have referenced several times.
then, in the FRP:
we have a helix with both wavelength and frequency equal to 1. This will be seen to be true any time a base and wavelength pair are used for a helix, meaning:
will give a helix with wavelength and frequency = 1. So, for example, an arbitrary helix, referencing the above complex base:
will also give a helix with wavelength and frequency equal to 1.
Now, the fun part: when we derived the initial base-wavelength relationship, we observed a deeper meaning in We then made use of this fact. So, in the equation above which used the laws of exponents and de Moivere’s theorem:
When n = 1, it is also true that n = 1 = ln e. We then have:
And then, just as with the base-wavelength relationship, we can speculate that this is true for any base. If so, we have:
The Euler formula upgraded for any base B
Or, if the helix is regarded as a wave, since the wave number
And then, in wavelength form:
So, except for:
‘Euler’s Formula Upgraded’ makes his formula valid for all bases – positive, negative, real, imaginary or complex.
A relatively simple proof for:
begins by substituting:
on the right side giving:
and then converting the left side to e base by solving for y in:
Euler’s formula itself which, of course, is already proved.
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This is the last blog of this series. The others are available at the links below
In blog #1 Euler’s formula is used as a rotation of the exponential graph through an imaginary circle.
In blog #2 Euler’s formula is used as an ‘orbit operator’.
In #3 as a helix with both fixed three-dimensional geometry, and in four dimensions with an animation variable.
In #4 we show that Euler’s identity actually maps a point on an ‘exponential/natural log surface’!
In #5 a ‘circular helicoid standing wave’ is shown in which Euler’s imaginary exponential is used no less than five times in the equation: as a surface operator, an orbit operator, a synchronizing rotator, a spin coefficient, and as a reciprocal spin coefficient!
In #6 we present the equation that updates Euler’s formula for all bases: positive, negative, real, imaginary and complex.