Of Surreal Geometry 2

Now, even oscillating curves have a rate of change. It may not be constant, but it exists. Take, for example, the harmonic series. We know it diverges. If we speed things up, we can ask how “fast” is a task approaching infinity. How fast does the harmonic series diverge?

The zeroes of the sine function approach infinity at a constant rate equal to a wavelength per unit. Here, we assume as our standard, the rate of change of the natural numbers, which is one per unit.

Definition: A discrete supertask is as sequence of events whose number of elements approach infinity at a rate smaller than that of the natural numbers.

If we make a transformation of the natural numbers such that its velocity equals ½, then we have a supertask, and have compressed the whole natural number line between 1 and ω/2.

Such a transformation can be a projection which will preserve a constant rate of change in the projected space.

You may ask what does it mean to reach or approach infinity. With a normalized arctangent function, we can map the whole (and real) numbers between -1 and 1.

Here, -∞ lies at -1, and ∞ at 1.

In this figure, we approximate the sum of natural numbers from -100 to 100 with the following code in GNU Octave:

function ret = natan(x)

ret = atan(x)*2/pi

endfunction

x1 = y1 = linspace(-natan(100), natan(100), 201)

[xx,yy] = meshgrid (x1,y1)

z1 = (xx +yy)

            mesh (x1,y1,z1)

As you can see, the space of sums of whole numbers under a normalized arctangent transformation looks just like a plane, preserving the rate of change of the sums of whole numbers.

Back to the harmonic series, even though it diverges, the difference between consecutive elements converges as it approaches infinity.

One question that arises is if the exponential curve is self-similar under this projection. If we project the exponential curve between -1 and 1 with the inverse of our transformation, do we get the exponential curve?

Perhaps we should formalize this system next to find out!