Penrose Tilings
The title refers to tiles as devised by the British physicist Roger Penrose, who has a keen interest in recreational mathematics, and as may be imagined, are somewhat more involved than with more 'ordinary' examples. More specifically, these tessellate in an non-periodic manner (hence their uniqueness), of which an excellent introduction, at a broadly accessable level is to be found in an Scientific American article of January 1977 'Extraordinary nonperiodic tiling that enriches the theory of tiles'.
Examples of 'Penrose tilings' include:
Kites and Darts
This particular set of tiles consists of two tiles of the above shapes that can be arranged in numerous distinct ways, forming specific patterns of a non-periodic nature which have their own titles, with arbitrary examples such as Cartwheels, Infinite Sun and Infinite Star. These first came to the attention of recreational mathematicians in the above article, of which they are liberally discussed and illustrated.
'Loaded Wheelbarrow' Tile
This was so named by Matin Gardner (on acount of its shape bearing a resemblance, albeit somewhat rudimentary) in an Scientific American article
of August 1975 'More About Tiling the Plane: The Possibilities of Polyominoes, Polyiamonds and Polyhexes' in which a single tile is shown (this being set as a puzzle, with the answer and the tessellation shown in the next months article).
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a |
b |
The digrams above show the 'loaded wheelbarrow' tessellation (a), alongside the underlying rhombuses (b).
'Thick and Thin' Rhombs
These consist of two rhombs which tile in a non-periodic manner, also discussed in the 1977 article (page 120).
The examples below show my efforts at forming life-like creatures with these. As such, these are generally of a inherently lower quality than with my normal standard, this being due to the tiles unique 'arrangement', in which a single line is repeated in a manner that is not conducive to the addition of life-like forms. This being so, their 'limitations' in this matter thus have to be accepted, albeit this should not be seen as an excuse for inferior examples.
Of interest here is the design process, which is as simple (though effective) as it could possibly be. Each of the kites and darts has been replaced by a arc which rotates 180° about the mid-point of the tiles. From the resulting process, it is then simply a matter of utilising ones imagination for suitable (bird) motifs, as shown. Such a process as above can be applied to more orthodox tesselations, as it is not a 'special property' of these non-periodic examples. This will be discussed in a more generalised sense in the next update.
'Loaded Wheelbarrow'
This is based directly upon the 'loaded wheelbarrow', with the bird motifs of my own devising. Such an example is typical of the slightly lower standard as discussed above, as the bird motif is somewhat contrived, albeit remaining of a sufficient quality as to be worthy of inclusion.
The following three tessellations are essentially minor variations concerning the interior detail of the bird motif, in which different views are shown (emphasised in colour), with 'belly' for No.2, 'back' for No.3, and belly and back in combination for No.4.
No.2
'Belly' view of bird.
'Back' view of bird.
'Belly' and 'back' views in combination.
Single Penrose Rhombs
This is based upon the same underlying grid as of the 'loaded wheelbarrow' (above).
Due to the necessity of subdividing the rhombs, this thus effectively doubles the number of motifs, of which as discussed in Essay 1, generally (but not always results) in a diminuation in inherent quality. However, here although the motifs are not of the highest standard are certainly of an acceptable degree, and are not noticably lacking.