The Geometry of Euler’s Formula – part 5
by greg ehmka
Euler’s Imaginary Exponential in Multiple Uses
In the previous blogs reference was made to two coordinate systems. Those being ‘3Di’ and ‘4Dii’. There are several more and the eBook is organized around these various categories or groups of functions. The underlying principle is that the number of dimensions is equivalent to the number of variables. These variables can then be in parametric, or “embedded” mode, or they can be in graphed, or “observable” mode.
The following table lists the various categories of functions by the number of dimensions. The number of dimensions/variables includes both the input ones and the output ones.
3Di helixes, polynomial and conic bifurcation
4Di function morphing
4Dii helix morphing
4Dii complex natural logs
4Dii complex imaginary logs
4Dii closed and open surfaces
5Dii circular surfaces
6Dii closed and circular surfaces in motion
With five and six variables, x becomes exclusively an output variable, whereas, in lower dimensions it is usually both an input and an output dimension.
The following example is from section 15.33 of the eBook and is of a ‘Circular Helicoid Standing Wave’ and is in the ‘6Dii’ category. In this example Euler’s formula (the imaginary exponential) is used no less than five times:
– as a surface operator in both terms of the function G,
– as an orbit operator in the function O,
– as a synchronizing rotator in the function R,
– as a spin coefficient on the first term of G,
– and, as a reciprocal spin coefficient on the second term of G.
Depending on the application O and R may have different expressions.
These functions are then combined as follows:
If the two terms of G, with their different spin coefficients, are graphed separately they generate two circular helicoid waves rotating in opposite directions around the vertical axis.
And, if these two waves are summed, as in G above, the standing wave results:
In section 15.0 of the eBook, with the same process, a circular transverse standing wave is generated along with other waves.
Also in section 15.0 of the eBook, the complex exponential is used with closed surfaces on polynomials with complex coefficients to generate circular trajectories along which the closed surface objects can travel. The imaginary exponential is also used as a coefficient which causes bodily rotation. These are then used in both two-dimensional and three-dimensional solar system models
All of this demonstrates wonderfully why the complex exponential has been hailed for centuries for its beauty and inspiration.
In the other blogs of this series Euler’s Formula is demonstrated as follows:
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.