**Under Construction**

## Part 1

### Calculating of Possibilities for 1- 4 Colours

Below I show the possibilities for all colourings of 1 to 4 colours, both individually and in combination. Note that I do not explain how this was determined. An explanation would, in this instance, be somewhat lengthy and convoluted, and for such a 'simple' piece of mathematical colouring is judged unecessary. The phrase 'a picture is worth a thousand words' comes to mind. Without doubt, the mathematics here is correct.

As such, I restrict the investigation up to and including four colours, this reflecting the number of distinct regions of a bird and fish (the two most frequently occurring motifs) outlined above. Although five of more colours can be included, by 'varying' the regions, such as beak, head, body, wings and tail, such numbers greatly increase the amount of work involved at the tessellation stage (Part 2). Indeed, the task by hand would become somewhat lengthy and tedious, in effect an abuse of time, with any potential gain offset by the relative triviality of such examples which would run into the *thousands *of diagrams*.* Indeed, as discussed elsewhere (Essay 8, Colouration and Contrasts of Motifs), for reasons of discernment of motif, as a rule it is best to limit the number of colours for any one motif to at most four. Therefore by so restricting the analysis to four colours is not in any way a shortfall in a practical sense.

An advantage of the colourations shown is that these are transposible can thus be applied to *any *tessellation based upon the same symmetry arrangement as regards the line arrangement.

The colourations are shown in three distinct ways, with successive, finer distinctions:

1. The Set of 256. An 'all encompassing' set of colours, of all combinations, from a pool of four colours

2. 1-, 2-, 3- and 4-Colourings derived from the Set of 256. These are simply arranged as according to the number of colours for each motif

3. 'Same family' colourations, derived from 2. above. For each of the 1-, 2-, 3- and 4-colourings, examples can be seen where the colouration is of an like type as regards ordering, specifically of the head region, which acts as a base colour.

## The Colourations

### 1. The Set of 256

The Set of 256 is based upon showing all possibilities, i.e. permutations, arising from four colours, from which other, smaller sets can be derived. All permutation, involving one to four colours are shown. The 256 diagrams are shown as four successive blocks of 64 (i.e. sub sets of 64, 128, 192, 256).

### 2. The 1-, 2-, 3-, 4-Colourings Derived from the Set of 256

As the original set is somewhat unwieldy to work from, I now differentiate into sets of colourings as according to the number of colourings, i.e. of Sets of 1, 2, 3 and 4 colours. Note that the small numbers directly below the motif are the source number, derived from the Set of 256. These are thus renumbered as according to their new ordering.

### 1-Colouring, a Set of 4

### 2-Colouring, a Set of 84

### 3-Colouring, a Set of 144

(note that this is shown, of necessity in two parts, rather than the ideal one 4 x 36 strip. As such, (b) attaches to the end of (a)

(a)

(b)

### 4-Colouring, a Set of 24

### 3. 'Same Family' Colourations

Rearranging the above sets as according to like intrinsic colouring

### 1-Colouring, a Set of 4 (no change in circumstance)

### 2-Colouring, a Set of 84 (A-G)

### 3-Colouring, a Set of 144 (A-E)

### 4-Colouring, a Set of 24 (no change in circumstance)

For the 4 x 4 colourations of Part 2, specifically, No.1 is used as the core. Each of the different head colours is then included once.