The Geometry of Euler’s Formula – part 4
by greg ehmka
Euler’s Formula as Points on an ‘Exponential/Natural Log Surface’
In section 12.4 of the eBook we define and explore a ‘rectangular complex region’ (u + iv) with intervals:
which, if graphed in 3Di coordinates, appears as:
In the previous blogs, the input domain of the complex exponential function was what we might call ‘line–angle input’, or line input with an animation variable.
If, rather than line-angle input, we use the rectangular complex region, defined above, as input to the function:
then, using the above intervals, a remarkable thing occurs – an exponential/natural log surface of revolution results!
the exponential/natural log surface appears as:
Both line-angle input and complex regional input produce a four-coordinate system referred to in the eBook as ‘4Dii’, three dimensions of which are spatial, and one of which is the animation variable. Further, two of the dimensions are real and two are imaginary. The variables are related to the geometric axes by:
This tells us that the four variables in:
or, depending upon input:
each have specific geometric interpretations! In section 11.3 of the e-Book, in another version of ‘4Dii’ the same basic idea is applied to generate an imaginary exp/log surface.
In this context Euler’s ‘famous five’ identity:
specifies a point in 4Dii coordinates! Meaning that with:
by slightly rearranging Euler’s identity to:
then it will be in the four coordinate form:
and so the four coordinates become:
and specify a point, (red ball), on the Exp/Ln surface:
In section 11.4 of the e-Book there are examples of additional surfaces that can be generated in this way. The concept is generalized for different bases and for functions in the exponent for the real and imaginary parts. Meaning:
And last, with this notion of a “4Dii” coordinate system there is a choice of which input variable to graph on the x axis. To generate the exponential/natural log surface above we used:
If instead we used:
another remarkable results occurs. The resulting surface is a certain kind of helicoid wave!
If the v interval is increased to:
This amazing result of transitioning between a surface object and an associated wave using the same function is explored with many examples in section 13.0 “Object-Wave Duality Surfaces”.
And more to come!
In blog #1 Euler’s formula is used as a rotation of the exponential graph though 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.