What I term as the dilution effect applies to a set of certain circumstances in tessellation, as regards specific tilings. Aside from he task of creating life-like tessellations using underlying arbitrary polygons, such as squares, equilateral triangles, and hexagons, is the possibility of using more specific sets of tiles, that have some ‘special’ mathematical properties, such as the Penrose tilings, Isohedral tilings, Archimedean tilings or the 14 known types of tiling pentagon, to name but few.
Aside from their ‘special’ nature, when considering using these to form life-like tilings a feature what I term as the dilution effect can occur, which somewhat negates their appeal. This occurs when the tile in question is subdivided into two or more regions i.e. motifs. Ideally, one should simply use the one tile in question, in its ‘purest’ state, without having to subdivide it. As an example, the Penrose tiling of thin and thick rhombi, where ideally one would have two motifs of the respective rhombs. Although a subdivided example is still ‘permissible’, this should be looked upon as inherently weaker, in that a single tile is superior, as it more obviously represents the given tile. For an example, see my own Penrose Bird and Fish, of thin and thick rhombi, of one tile, one motif - a superior example. Here the underlying tile is more evident, thus giving more aesthetic appeal.
Some effects of the dilution process can be seen in Marjorie Rice’s life-like pentagon tilings (see The Mathematical Gardner, pages 164-166), where she uses a variety of subdivided motifs for each tile, thereby ‘diluting’ the tiling. For example, Fig. 16A has 2? Motifs, Fig, 16B has 3, Fig.16C has 5? I qualify two of these with question marks, as the motifs are not readily visible. When viewed as a tessellation, the single, underlying pentagon is lost in a sea of motifs. Again, stronger would be one tile, one motif.
Note that this goes against my advice in Essay 15 that two or more motifs are ‘better’ in a generalised sense. However, here this premise does not apply, as the challenge is different, to more clearly view the single tile, without additional, in effect, distracting, extraneous motifs by the necessity of subdivision.
In short, the stronger and more effective test of ability here is the ‘one tile, one motif’ type. Only as a last resort, where one has tried and failed to come up with the desired type should the lesser quality example be used. An analogy can be seen with the two different types of tessellation, of ‘whole motif’ and ‘heads’. You are very much playing in the second division if your work contains more than a few examples of the latter. Do you want to be in the first or second division here? The same principle applies to the dilution process. You know what to do…
Created: 16 April 2010