So, upon having examined a 'special case' of 'ultimate symmetry' (Part 2), I now broaden the scope of the investigation. Again, for reasons of conciseness, I am selective, as otherwise a considerable 'book length' treatment would of necessity be required. In this instance, rather than showing all four posibilities (1-,2-,3-,4-colourations) I show only a 3-colouring type. The reason for such an arbitrary choice is that of examining the most potentialy the most complex of the four.
For the study, the method is as before, as with Part 2, one of showing successive stages (in theory Stages 1-4, albeit Stage 4 was curtailed), using the ‘quadrant' method, showing only the permissible colouration examples (map colouring).
As for the remit of the study, I instead begin simply. For this, I begin with a single motif, No.1 from the Set of 144 individual motifs. In contrast to showing only those examples from the same families, I am less restrictive, permitting any example from the Set of 144. Note that this beginning consciously omits others from the 6 different families (examined, but not shown here). This took the form of placing each of the four colours (of heads as in the 4-colouring) at the centre vertex by the final stage. Furthermore, I relax the conditions of Part 2, in that here I permit both 'balanced' and 'unbalanced' examples to occur. As a consequence, as may readily be imagined, this greatly increases the number of blocks at each stage.
However, as I show by abstract analysis at Stage 3, by the final stage, Stage 4, this would have resulted in a Set of 784 4 x 4 blocks, which is simply an overwhelming number, given that most of these would be of trivial colourings. Therefore, upon Stage 4, I curtailed the study, instead showing the first few examples as representative samples.
Stage 1, a Set of 1
Stage 2, a Set of 25 (for reasons of conciseness I show only the first 12 members)
Stage 3, a Set of 308 (for reasons of conciseness I show only the first 12 members)
Stage 4, a Set of 784 (for reasons of conciseness I show only the first 12 members)
Upon examining the Set of 784, it can be seen that in terms of aesthetics these leave a lot to be desired. The overwhelming majority aer of the 'unbalanced type' with only a few balanced. Furthermore, those that are balanced are not necessarily from members of the same family, thereby resulting in inferior examples when in comparison to 'same families'. Therefore, having examined just an initial, single family member, the same process, to be all encompassing in its remit should be undertaken for the 6 or 7 family members. Furthermore, all this only concerns the 3-Colouring - to be thorough, the 1-, 2- and 4-colourings should likewise be examined. However, such a task is somewhat disproportionate as to worth - many trivial examples would occur that are simply not worth the time and trouble in determining.
Now, up to this point all the examples have been consistent in that for the centre vertex all four colours are required. Again, further possibilities occur if this rule is relaxed:
(i) For example what about different combinations at the centre e.g. '2, 1, 1' (e.g. two blue, 1 yellow 1 red) and ; '2, 2' (e.g. two blue two red). Again, this would thus result in further extensive studies. Two respective arbitrary examples are shown below.
(ii) All the above has been previously consistent with the 1, 2, 3, and 4 colours. Why not include examples from different colourations e.g an 2- and 3-colouration? This has not been looked at.
At this point I stop in the analysis. Quite simply, the number of possibilities is becoming overwhelming, way beyond the bounds of practicality. To determine the matter at the current rate of progress would perhaps take many years, of which 'just' for a colouration problem is an abuse of time. What is obviously needed is a different approach. Possibly, rather than using colour per se, one could instead use numbers. However, although I have indeed attempted this, this appears to cause problems per se, as envisaged by the dual findings as regards distinctness of the 1- and 2-colourings (Part 2). Certainly, this method is obviously less intuitive than with colour itself.
So some unanswered questions are:
1 Exactly how many 'all encompassing' colourations are there?
2 What is the best process to determine this matter. Almost certainly, the only practical solution would be to use a computer. However, I am lacking in this aspect, and have absolutely no idea how one would approach this.
3 Extending the problem - the same principle of multiple colourations can also be applied to other tessellations, for example based upon equilateral triangles. What is needed is a 'theory of everything' that can be applied to this and other such tessellations.
Bridges 2008 Art Exhibit Colouration - An Introduction > Bridges 2008 Art Exhibit Colouration 2 - The 4 x 4 Colourations >