2nd Generation
Upon completion of a finished design, as exemplified on the home page, this can be described as a 'first generation' parquet deformation. As such, this is all well and good and is perfectly acceptable as a finished design in its own right. However, for any one such finished parquet deformation, it is possible to use this as a framework or springboard for a variety of further 'second generation' possibilities (the concept is of my own devising), hence the title.
The broad aim here is to see any trends/tendencies, both favourable and unfavourable (i.e. aesthetic and non-aesthetic), and then apply the favourable instances to other parquet deformations of a broad-like nature. To this end, I use three different parquet deformations as exemplars, with the tiles being asymmetric, and one- and two lines of mirror symmetry. The format adopted is two-fold: first, I show the first-generation parquet deformation, and second, the various 'extra possibilities' (as outlined above) as a second-generation parquet deformation (as such, there can be one or more such second-generation instances). Such a format makes clear the 'before and after' aspect at-a-glance. This involves applying standard mathematical symmetry operations to it, with overlays of translation, reflection and rotation. Typically, with the subsequent (second generation) symmetry operations, the parquet deformation becomes more involved, with a broad doubling of lines. As alluded to in a dedicated mini-essay on the subject, the computer is a great aid here and in such instances is where it really comes into its own, enabling the initial drawing to be used time and again, as against tedious successive drawing by hand which mitigates against experiment. In short, there is after all such a thing as a free lunch! The possibilities are just about endless, but the outcomes are not as aesthetic as another. Judgements must be made in the matter, otherwise the results become trivial. Although each of the examples shown here is practical by hand, much more can be accomplished by computer, in both speed of execution and design.
Although the premise is easily stated, it is not perhaps as straightforward as may otherwise be thought to present in a consistent and ordered manner. First, there are the first-generation parquet deformations. In themselves, these can be described as of symmetrical and asymmetrical tiles. And then for the second generation, there is the matter of choosing a symmetry operation point; different places, e.g. vertices, mid-side, the midpoint of the cell all have implications as to the outcome. And how should the analysis be set out? With so many variations, it is not easy! Therefore, the presentation given is subject to considerable revision. That said, it is 'reasonably' consistent for any one section. A 'perfect' presentation may never happen! The main point here is to outline some possibilities. Ideally, 'all' would be covered, but this reality is so difficult to accomplish that it may never happen! As is my usual policy, by far the best is to put something out (in a reasonable state), rather than wait for some ideal that may, or may not, eventually arrive, perhaps many years down the line.
1. SUBDIVIDING
1.1 Asymmetric Tile
1.2 Symmetrical Tile, One Axis of Reflection
1.3. Symmetrical Tile, Two Axes of Reflection
2. TRANSLATION
2.1 Asymmetric Tile
3. MIRROR OVERLAY - HORIZONTAL AXIS
4. ROTATION OVERLAY - 180
1. SUBDIVIDING
By suitably subdividing the tiles, new parquet deformations can be formed. A variety of lies are used, including diagonals, from left to right and right to left and then both diagonals. A particular favorite is a double basket weave structure. This gives a very pleasing overlay effect at right angles. Other subdivisions are less well defined, selected as according to the parquet deformation. The outcome can depend on the nature of the parquet deformation. For instance, some parquet deformations have asymmetric tiles, of mirror symmetry, or order 4 rotation, all of which influence on the process as to aesthetics. To this end, I show examples of all three instances, with typical subdivisions. As can be seen, only a few can be described as aesthetic.
Fig. 1a. First Generation Parquet Deformation
Fig. 1b. Second Generation Parquet Deformation. One diagonal subdivided from left to right
Fig. 1c. Second Generation Parquet Deformation. One diagonal subdivided right to left
Fig. 1d. Second Generation Parquet Deformation. Two diagonals subdivided, left to right and right to left
Fig. 1e. Second Generation Parquet Deformation. Alternate diagonals subdivided, left to right and right to left
Fig. 1f. Second Generation Parquet Deformation. Double Basketweave
1.2 Symmetrical Tile, One Axis of Reflection
Fig. 2a. First Generation Parquet Deformation
Fig. 2b. Second Generation Parquet Deformation. One diagonal subdivided right to left
Fig. 2c. Second Generation Parquet Deformation. Double Basketweave
1.3. Symmetrical Tile, Two Axes of Reflection
Fig. 3a. First Generation Parquet Deformation
Fig. 3b. Second Generation Parquet Deformation. Basketweave 1
A beautiful design, with the pleasing feature of overlapping blocks of four tiles at right angles, in the same manner as the famous Cairo tiling
Fig. 3c. Second Generation Parquet Deformation. Basketweave 2
Another beautiful design, with the pleasing feature of overlapping blocks of four tiles at right angles, in the same manner as the famous Cairo tiling.
2. TRANSLATION
Fig. 4a. First Generation Parquet Deformation
Fig. 4d. Second Generation Parquet Deformation
Fig. 5a. First Generation Parquet Deformation
Fig. 5b. Second Generation Parquet Deformation
3. MIRROR OVERLAY - HORIZONTAL AXIS
Fig. 6a. First Generation Parquet Deformation
Fig. 6b. Second Generation Parquet Deformation
3. MIRROR OVERLAY - VERTICAL AXIS
Given that this symmetry operation results in the second generation parquet deformation being the same at both ends, this is lacking aesthetically, and is not to be compared to the 'standard model', where the tiles differ at the ends (this negates instances where the same tile arises, but through part of its natural cycle, which is aesthetic)
Fig. 7a. First Generation Parquet Deformation
Fig. 7b. Second Generation Parquet Deformation
4. ROTATION OVERLAY - 180
Fig. 8a. First Generation Parquet Deformation
Fig. 8b. Second Generation Parquet Deformation
Created 3 June 2024. Last Updated 4 June 2024