## Lattice Labyrinth Tessellations for Apple’s Steve Jobs – a cryptic iPhone cover design?

So far as Apple’s logo is concerned, it can be any colour so long as it’s black, to paraphrase Henry Ford. One colour makes it difficult to shade the tiles of a tessellation, so I’ve had to borrow some other colours approximated from their earlier rainbow version. As you well know, anything goes well with black! Steve Jobs was born on February 24, 1955 and died on October 5, 2011. Gosh, it’s already year 4 PJ (post Jobs). So, below is a union of Trefoil Lattice Labyrinth (2,24) and Trefoil Lattice Labyrinth (10,5). I had some trouble finding both, essentially because in each case the separation parameters (a,b) have a common factor. The overall shape might fit onto your iPhone cover.

These are tessellations based on birthdate number pairs. I’ve used the word “cryptic” in that few would guess or think to observe that the pattern will repeat if you count a triangle-sides across and b up at an angle of 60°, an observation not made easier by the omission, for effect, of the triangle boundaries.

It is sometimes possible to construct a Lattice Labyrinth such that the supertiles are comprised of a specified number of tiles (or sometimes a small multiple or large factor of the required number). Hence the various birthyear tessellations to be found in early posts of this blog. The post Tessellatable Numbers up to 2100 tells you which “small” numbers work. 1955 will not, but 2011 corresponds to Trefoil Lattice Labyrinth (39,10) (I know it’s a Trefoil rather than a Honeycomb because a-b is not zero or divisible by 3), This is a monster, in which each supertile will contain 39² + 39×10 + 10² = 2011 (it works!) equilateral triangles. 2011 is a (real) prime number and, in this case, (39,10) is a Loeschian prime, having no number pair factors. It can also be written more conventionally (alas) as Eisenstein prime (49,10). I think I’ll leave tessellating (39,10) to another day.

The Lattice Labyrinths workbook  is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

And finally, to appease search engines, here are the opening lines in American:

So far as Apple’s logo is concerned, it can be any color so long as it’s black, to paraphrase Henry Ford. One color makes it difficult to shade the tiles of a tessellation, so I’ve had to borrow some other colors approximated from their earlier rainbow version.

(Had the USA gone the whole hog in the (strictly imperfectible) attempt to re-write American English phonetically, the two (collections of) dialects might by now be well on the way to becoming mutually incomprehensible languages, like English and Glaswegian.)