The Geometry of Euler’s Formula – part 1
by greg ehmka
Happy News Mr. Pierce!
Benjamin Pierce is often quoted, relative to Euler’s formula, as saying:
“Gentlemen, that is surely true, it is absolutely paradoxical;
we cannot understand it, and we don’t know what it means.
But we have proved it, and therefore we know it must be the truth.”
More recently Scott E. Brodie has remarked:
An intuitive understanding of Euler’s formula for the complex exponential,
remains elusive, notwithstanding hundreds of years of contemplation.
Well, I’m pleased to report that the moment for an intuitive understanding of the meaning of Euler’s formula has arrived! The Muses have been quite generous over the past many years bestowing a great number of math innovations all included in my new eBook: A Three Dimensional Coordinate System for Complex Numbers.
In section 2.0 of the eBook a new coordinate system is constructed called, ‘3Di’. This is an acronym for ‘Third Dimension Imaginary’. The intuitive geometrical understanding of Euler’s formula uses this coordinate system. This coordinate system allows for a great many wonderful new understandings and innovations, and, as I’ve commented in a few places, is very likely a good example of what Victor Hugo called, “an idea whose time has come.” The point there is that since the moment has arrived (i.e., “the time has come”), while the Muses have been speaking to me with many gifts, they are likely speaking to others as well.
Everything that will be in this series of blogs is covered in great detail in the eBook, so we can move fairly quickly with the key points.
Beginning with the usual exponential graph in the real plane:
we specify the coordinates x and y as the horizontal and vertical axes respectively. To this we define the third-dimension, depth as z, and make this dimension imaginary, iz.
If we then add an imaginary input to the function, we can write:
While , the graph is just that of the usual two-dimensional exponential function. But if x takes on other values, the exponential graph is seen to rotate! Theta becomes an ‘animation variable’.
Euler’s formula has , and so:
will rotate the point, , through an imaginary circle of radius = 1.
As x takes on other values, -1, 1, 1.5 e.g. , etc., Euler’s formula rotates each of these points through an imaginary circle, the radius of which is
These are “imaginary circles” because, other than the two points, (x, ex) , they do not show up in the two-dimensional real plane, or what we refer to in the eBook as the ‘Front Real Plane’ (FRP). The imaginary circles show up in the side view, or what we refer to in the eBook as the ‘Right Side Imaginary Plane’ (RSIP), which has coordinates: (horizontal, vertical) = (iz, y).
The FRP and RSIP along with the ‘Top Imaginary Plane’ (TIP) are the front, side and top “projection planes” also constructed in section 2.0 and used throughout most of the eBook. They show what any given three-dimensional curve appears like, when projected to each of these planes, with useful and often surprising results.
So, this is the first geometrical understanding of Euler’s formula: that of rotating each point 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.