The Geometry of Euler’s Formula – part 2
by greg ehmka
Euler’s Formula as an Orbit Operator
As we saw in part 1, Euler’s formula:
will rotate the point, (x, y) = (0, 1), through an imaginary circle of radius = 1.
and will rotate all points (x, ex ):
through imaginary circles if x takes on values other than 0. If we slightly rearrange this function to:
and then specify an ‘orbit operator’ O:
by adding an additional term with a coefficient b in the exponent e.g.:
the orbit through which each point is rotated takes on an amazing array of curves depending on the value of the coefficient.
With b = -2, a trifolium:
With b = -3, a quadrifolium:
With b = -4, -5, -6, a 5-folium, 6-folium, 7-folium etc.:
Amplitude coefficients give further possibilities. For example:
With b = -3 and B = .125, the quadrifolium becomes virtually a square:
In section 3.0 of the eBook dozens of historical curves are generated with new equations, in just this simple way, in both RSIP view and in helix form.
Additional terms may be added to the orbit operator:
With b = -5, c = -7:
With b = -1, c = -7:
Next, the orbit operator can be applied to the whole exponential function:
Each point of the function is rotating through the same orbit.
Next, In section 2.82 of the eBook the Lambert W function is generated in this way as a two-dimensional projection of the three-dimensional exponential graph at a fixed position of an orbit, meaning theta is a constant. So, the Lambert W curve can be generated by:
and then by arranging the axes so that the two-dimensional projection shows up as desired.
And last, the orbit operator can be applied to any function. With the “square” orbit operator above, and the half-circle function also shaped into a “square”, i.e.:
will outline a cube:
In section 12.0 of the eBook, by altering the input domain to a complex region, many closed surfaces are generated including this cube and a sphere surface function that allows for a one-to-one inverse function.
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.