Generating a mathematical sequence

Number Sequence

We want to define a number sequence, an ordered set.

In the CA chapter we said that we need 4 things:

  • A number system to describe values and coordinates
  • A structure, defining the predecessors of each point
  • A formula that calculates the value at each point depending on the values of its predecessors
  • An initial condition

We have defined a multidimensional number system in the mathematics chapter.

So what structure do we use?

We just reuse the exact structure of the hierarchical number system we defined, so we have the same concept of predecessors.

Dimension Hierarchy

In order to get all the possible dimensions we will hierarchically nest them by using a generating function W. Each order Wn references the previous order Wn-1 recursively for n >1. This creates a unique hierarchy of dimensions. In order to differentiate the un-nested from the nested dimensions we use the term pure order for the former.

In this table dim refers to the dimensions generated at pure order n, while new-dim is a list of the dimensions that have not been made at previous orders.

order n dimensions new dimensions min number of dim
1 1 1 1
2 1 1 1
3 1,j,j2 1,j,j2 3
4 1,i 1,i 6
5 1,h,h2,h3,h4 1,h,h2,h3,h4 30
6 1,j,j2 1 30
7 1,g,g2,g3,g4,g5,g6 1,g,g2,g3,g4,g5,g6 210
8 1,f,i,fi 1,f 420
9 1,e,e2,j,je,je2, j2,j2e,j2e2 1,e,e2 1260
10 1,h,h2,h3,h4 1 1260
n A(n)=|rn|k B(n) M(n)

(Where g=r7, f=r8, e=r9.)

The minimum number of dimensions M(n) that such a structure is expressible in is shown in column min-nr. It is called the minimum number of dimensions because you can also express the same structure in more dimensions, as a multiple of the minimum. To do that would embed the structure symmetrically into a higher dimensionality.

By symmetrical we mean that we want to keep all the dimensions of the destination structure interchangeable. As an example consider a one dimensional structure in the “1” direction. How can we embed this into a 2-dimensional structure in the “1” and “i” directions, without making the “1” dimension somehow special and no longer interchangeable with the “i” dimension? The answer is to embed it diagonally along the “1+i” direction. This way the permutability of the dimensions is maintained.

Simple function

As a practice run we start with the simple function W that generates an interesting hierarchy of dimensions.

One very simple function to define the point by its predecessor is to simply copy the same value.

W(p)=W(p-1)

We would like to parametrise this to any axis, so we introduce a superscripted parameter.

W^{u}(p)=W(p-u)

For example we could set u to be 1,i,j2 or any other multidimensional axis.

We start with this as our first order function:

W_{1}^{u}(p)=W^{u}(p)

For higher orders we simply add up the values of all the immediate predecessors.

W_{n}^{u}(p)=\sum\limits_{k\,in\,B(n)} {W_{n-1}^{ku} }

Definition: k in B(n) iterates through all the new dimensions.

Let’s go through the first few orders.

In orders 1 and 2 every point other than the initial condition has one predecessor.

W_{1}^{u}(p)=W_{1} (p-u)

and

W_{2}^{u}(p)=W_{1}^{u}(p)

W3 is 3-dimensional. The coordinates of p and the value W3(p,1) are of the form a1+ja2+j2a3. Each point p has 3 predecessors. To be precise, the points that have one or two coordinates that are zero have fewer predecessors, but we are mainly concerned with points that are in the middle of the structure, so we will ignore the corner cases as to not make the equations too unwieldy.

W_{3}^{u}(p)=W_{2}^{u}(p)+W_{2}^{uj}(p)+W_{2}^{uj^{2}}(p)

This expands to:

W_{3}^{1}(p)=W(p-1)+W(p-j)+W(p-j^{2})

At order 4 a point has two immediate predecessors.

W_{4}^{u}(p)=W_{3}^{u}(p)+W_{3}^{ui}(p)

Order 4 on its own contains complex numbers, but since W4 references W3 the values are 6-dimensional. Coordinates and values are of the form a1+ja2+j2a3+ia4+ija5+ij2a6.

You can look at a 6-dimensional number as two sets of 3-dimensional numbers like depicted in this tree diagram:

tree6d

Expanding the formula for W4 gives:

\begin{array}{l} W_{4}^{1}(p)=W(p-1)+W(p-j)+W(p-j^{2}) \\  +W(p-i)+W(p-ij)+W(p-ij^{2}) \\  \end{array}

This means that each point W4(p) is defined by six predecessors one generation previously. These six immediate predecessors should look familiar. They are the neighbours we ran into in here.

Order 5 is simply this:

W_{5}^{u}(p)=W_{4}^{u}(p)+W_{4}^{uh}(p)+W_{4}^{uh^{2}}(p)+W_{4}  ^{uh^{3}}(p)+W_{4}^{uh^{4}}(p)

Where h = r5.

We get 5 groups of 6-dimensional structures and we add them. A graphical representation of the hierarchy would look like this:

tree30d

The formula for W5 expanded looks like this:

\begin{array}{l} W_{5}^{1}(p)= \\  W(p-1)+W(p-j)+W(p-j^{2})+W(p-i)+W(p-ij)+W(p-ij^{2}) \\  +W(p-h)+W(p-jh)+W(p-j^{2}h)+W(p-ih)+W(p-ijh)+W(p-ij^{2}h) \\  +W(p-h^{2})+W(p-jh^{2})+W(p-j^{2}h^{2})+W(p-ih^{2})+W(p-ijh^{2})+W(p-ij^{2}h^{2}) \\  +W(p-h^{3})+W(p-jh^{3})+W(p-j^{2}h^{3})+W(p-ih^{3})+W(p-ijh^{3})+W(p-ij^{2}h^{3}) \\  +W(p-h^{4})+W(p-jh^{4})+W(p-j^{2}h^{4})+W(p-ih^{4})+W(p-ijh^{4})+W(p-ij^{2}h^{4}) \\  \end{array}

Each point p has 5 sets of the 6 neighbours it had at order 4. The values at the points as well as the coordinates are 30 dimensional.

At order 6 we don’t get any new dimensions, because all the dimensions created by r6 already exist in W5, so W6 is still 30-dimensional.

W_{6}^{1}(p)=W_{5}^{1}(p)

W7 is 210-dimensional. In this diagram each of the 7 big branches contains the 30-dimensional tree from above.

tree210d

W8 is 420-dimensional. The formula for V8 is then:

W_{8}^{1}(p)=W_{7}^{1}(p)+W_{7}^{\mbox{r8}}(p)

The total nested dimensions required for Wn is given by Cn.

So far so good, but all W does is add up vectors, losing the nice hierarchy of the dimensions; every dimension is interchangeable with any other. We need complexity gradients.

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