Crate trinoise

Trinoise

A mathematical noise pattern of 3 values based on Number Theory and Set Theory

Properties

• Assigns a value to every natural number
• Fixed interpretation based on aligned positions to identity map
• Value counts the number of successors with decreasing value
• Never repeats the same number twice for bases greater than 2
• Repeats noise pattern after N^N for base N

The value counts the number of successors with decrementing value. One can use it to skip successors and project into 3 values:

0. 0
1. base - 2
2. base - 1

This is done using the tri function.

The frequencies of 0, 1 and 2 is predictable, e.g. [470, 470, 155] for base 5.

The frequency of 0 and 1 are equal for bases greater than 2 (conjecture).

The frequency of 0 or 1 divided by frequency of 2 converges rapidly to base - 2 when base goes to infinity (conjecture).

Inspiration

The powerset operator is important in Set Theory. To generate the powerset of a finite set, one can use a binary encoding, where each bit represents a membership of an object.

One problem with the powerset operator, is that it does not provide information about isomorphisms of sets to themselves. This extra information is desirable when studying equivalences. Therefore, one would like a more "powerful" way of generating sets.

There is a different way of generating sets that respects isomorphisms:

• Starting with an identity map [0, 1, 2, ..., n-1]
• Modify one position at a time, e.g. [0, 1, 2] => [0, 0, 2]
• The generated discrete combinatorial space forms a groupoid
• Redundant members are removed through post-filtering to form subsets

For example, [0, 0, 0] becomes a set which contains only {0}.

Isomorphisms are also generated, e.g. [2, 0, 1].

This means that the same method construct both subsets and isomorphisms. The combination of subsets and isomorphisms is interesting to study for Sized Type Theory, a type theory where functions can be applied to equivalences. It is believed that an equivalence can ensure the existence of a partial normal path, hence not require the function to be an isomorphism.

For example, [0, 0, 1] is mapped differently than [1, 0, 0], but both has the same set {0, 1}. When a function maps to a smaller set, it can not be an isomorphism, but in Sized Type Theory one can use f(a ~= b) == (f(a) ~= f(b)), so this is still meaningful as existence of some normal path f[g_i->n] where g_i(a) == g_i(b) and g_n(f(a)) == g_n(f(b)). Normal paths are commutative squares of functions. In this case, the square commutes by definition.

One benefit of this groupoid structure, is that it represents all possible transformations of sets closed under the category of functions. It is much easier to study this structure than reasoning about families of functions, because in families of functions the sets are repeated many times.

It turns out that the reachability tree with identity map as root, assigns a node depth equal to n minus aligned positions with identity map. When ordering the reachability tree, the nodes form smaller neighborhoods with same node depth, which size is always 1, n - 1 or n (conjecture).

This is because when counting upwards, the following is true:

• For every n cycle, there is at least one disruption
• Any disruption can either collapse 2 positions, swap 1 vs 1, or collapse 1
• Swapping 1 vs 1 never happens twice during n cycle

The order of the identity map is chosen to preserve this property. If one uses 210 instead of 012 in base 3, this property is destroyed.

The nodes in the groupoid can naturally be encoded with numbers in base n. In base n, the signature of ordered neighborhoods with same node depth is encodable in base n by subtracting 1, counting successors. Therefore, 0, base - 2 or base - 1 are the only values.

Functions

aligned	Counts the number of aligned positions to identity map.
next	Returns the number of successors that share number of aligned positions.
signature	Calculates signature of successors with shared aligned positions.
tri	Maps 0 => 0, base - 2 => 1, base - 1 => 2.