Dichotomy

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dichotomy summary

Dichotomy summary

The normal world, according to the way we learn maths at school, exists on a continuum of infinitesimally small to infinitely large. There are instantaneous and eternal concepts, but one doesn't hear much of them: this is because time inheres in the structure of what we all perceive to be the world of objects moving through time as a one-way arrow into the future.

In the Fourier domain that axis is radically shifted away from space to become one of time, where infinintesimally small times become the ultimate of fine-grainedness in perceiving the frequency spectrum, and the infinitely long period being the biggest window, or, if you like, the standing wave of the universe (see Hubble constant).

Given that the Heisenberg Uncertainty Principle, or at the very least observer error, seems to defy all attempts to inquire into the nature of things, it is instructive to know where our observations must be incomplete. In the Fourier domain, our information is going to be limited both by an inability to fine-grain the frequencies enough (can't plot infinitesimals) or to observe over a sufficiently long wavelength (window problems). In the space-time world, these roughly equate to the problems respectively of quantum mechanics and cosmology.

Dealing with the continuous Fourier transform, though alien at first, does not pose any special problems that are not normally present. Continuous observation in spacetime requires a perception that does not reduce the content into the spacetime artifacts of things in time - continuous observation of the Fourier domain requires that we do not grade frequencies.

While it is customary to note that the trick of continuous perception is easier said than done, in either case it is not so - it absolutely cannot be said (by its very nature) and hence can only be done! It is instructive to note that each visual perception actually constitutes a reverse Fourier transform,

For a person, however we look at the world, we are constrained perception-wise by the sum total of knowable local information. The beauty is, of course, that our species has developed the capacity to generate information by description (incomplete though that must necessarily be) as well as the ability to theorise about such things.

The two domains in binary

A binary word is a sequence of binary digits of a given length. A binary space is the set of all distinct binary words of the same length. Each of these binary words represent a unique location which exhibits a state. The state is a binary word of the same size as the address. So for example if we have a 10 bit binary space, then we effectively have a set of 1024 locations each containing a 10 bit value. Since the addresses and values are the same size, values can represent addresses.

In such a space, we already have the two domains at play with addresses (unchangeable) each paired with a value (changeable). But the two domains come into play far more strongly when we introduce the concept of sequence into the structure. By introducing a successor operation, every location is given two unique neighbours. This is the basis of interpreting the binary word as a unique number and the whole set of possible words of that length as a dimension (actually a loop because the successor of 111 is 000).

This idea of dimension is derived from interpreting a binary word as a sequence and reading it from one end to the other to yields a specific location. But here's where the two domains come in to play, because the successor process on which this interpretation is built is actually geometric in nature, and as such comes which a geometric compliment. The complimentary interpretation is multiplexing and is obtained by reading the address sequence the other way (reverse addressing, see FFT for more detail about this reversal).

Dimension and multiplexing are isomorphic with the two domains because forward addressing yields a linear "location" space, and reverse yields the energy distribution, or "executional focus" over the entire space. Or to say it in a more dichotomous way, forward addressing divides the space into a sequence of locations and reverse addressing divides the focus (one point-of-view) up into many focus's (many parallel points of view). Forward is division of object or state, and reverse division of subject.

Heaven & Earth

Lets have a closer look at the e^ix formular to see how it splits into by odd and even into the two domains. First here's its taylor-series expansion (I've separated the i and x powers out for clarity)...

[math] e^{ix} = \sum_{n = 0}^{\infty} {ix^n \over n!} = {i^0 \cdot x^0 \over 0!} + {i^1 \cdot x^1 \over 1!} + {i^2 \cdot x^2 \over 2!} + {i^3 \cdot x^3 \over 3!} + {i^4 \cdot x^4 \over 4!} + \cdots [/math]

Now because i² is -1, successive powers of i lead to the terms alternating between real and imaginary as follows...

[math] = {1 \cdot x^0 \over 0!} + {i \cdot x^1 \over 1!} + {-1 \cdot x^2 \over 2!} + {-i \cdot x^3 \over 3!} + {1 \cdot x^4 \over 4!} + {i \cdot x^5 \over 5!} + {-1 \cdot x^6 \over 6!} + \cdots [/math]

As you can see, all the terms have become a sequence of alternating positive and negative i or 1, so this means we can separate all the terms containing i and factorise them out,

[math] = \left( {x^0 \over 0!} - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} \cdots \right) + i \left( {x^1 \over 1!} - {x^3 \over 3!} + {x^5 \over 5!} - {x^7 \over 7!} \cdots \right) [/math]

And so you can see that all the even terms have grouped together to make the real component, and all the odds make the imaginary, so where x is angular, e^ix yields the rectangular components, hence:

[math] \!\, e^{ix} = \cos x + i \sin x [/math]

or more generally, when e is raised to the power of any arbitrary complex number,

[math] \!\, e^{a + bi} = e^a (\cos b + i \sin b) [/math]