# Dichotomy

## Origins

We see these two fundamental aspects in every system having names like, *potential/actual*, *field/point*, *analogue/digital* or *class/instance*. Many philosophers and traditions have realised that this pair are common to all systems and experiences and have built their views of their ontologies (description of the universe) on them - the Taoist tradition reflects the pair and their relation to each other in its logo, and names them *yang* and *yin*.

But how can these concepts help us in defining a functional ontology on which our code is based? A good portion of this answer is found in ancient Taoist texts that offer us surprisingly specific insight into the functioning of these two aspects.

## 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 represents 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 yield a specific location. But here's where the two domains come into play, because the *successor* process on which this interpretation is built is actually geometric in nature, and as such, comes with a geometric complement. The complementary 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) into many foci (many parallel points of view). Forward is division of object or state, and reverse, division of subject.

The following diagram shows how the single binary digit representing "1" (the unified undivided Whole) can be split in such a way as to create an infinitely-divisible continuum both spatially and temporally.

The tree can divide into any number of levels; here three levels are shown, with a single node dividing down one level further as an example. The meaning of the numbers in the diagram are as follows:

- The numbers down the left hand side show what proportion of the whole each item in that layer represents.
- The numbers below the nodes show the numeric value of the node by summing each rightward branch starting from root
- The numbers inside the node represent the order in which the nodes are visited in multiplexed order starting at zero

Starting with 1 at the top level and halving the value down each successive layer allows any level to divide to any depth without repeating values from higher levels and disrupting the continuum.

## Dichotomy and Duality

The Foundation Ontology (and the older nodal model) are founded on what could be called a *meta-dichotomy*, or a dichotomy of dichotomies which we'll call *dichotomy* and *duality*. Both are recursive and so in a sense give rise to a discrete division process implying two kinds of *continuous dimension*. The *dichotomy* dimension is the kind which have opposites at each end like *black* and *white* or *negative* and *positive*, and they concern sequence, order and time. This dimension is thought of as "horizontal" because both ends are equally powerful and part of the same domain.

The *duality* kind of dimension is about *containment* like *foreground and background* and it forms the "vertical" dimension of hierarchy. It's been called *duality* because its the kind of dichotomy that gives rise to the wave-particle duality where one is the environment and the other an object within it. Some more abstract ones are *subject/object*, *potential/actual* and *field/point*.

## Heaven & Earth

Let's have a closer look at the *e ^{ix}* formula to see how it splits 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 - see also the local e article)...

[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 *i ^{2}* 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]

## See also

- FFT
*- the Fast Fourier Transform* - Binary traversal
- e