Thank you to today’s visitor to this blog from Brazil and best wishes to the inspiring neuro-psychologist from Rio de Janeiro who visited the Celebrating Cromford festival on Saturday 21st. June this year.
Not only Brazil, but now also Argentina inspire me to construct a Lattice Labyrinth as a Consolation Tessellation following their defeats by the German football team, with its mysterious knack of retaining possession when they have the ball and of disrupting that of any opposing team on the few occasions they manage to get a kick in.
Amazingly, I can propose the same paving or tiling design for both Argentina and Brazil. Brazil declared independence on the 7th. of September, 1822 and Argentina on 9th. July, 1816 so the pair of numbers honouring each country is the same: nine and seven. I suppose that strictly, I should choose (9,7) as the parameter pair for Argentina (following the British and European not the American convention) and (7,9) for Brazil.
Both pairs of separation parameters specify a Serpentine Lattice Labyrinth, one being the mirror image of the other. So they CAN be sort-of the same, but also reflections of each other – perhaps rather appropriate for the two giants of South America, speaking their closely related but distinct Iberian tongues. N.B. You can always swap tessellations if you want to be dominated by the USA. Otherwise get busy installing them in your iconic public spaces – Michael and I will soon post artist’s impressions of the results. First of all here is the Consolation Tessellation for Argentina, in as near as I can get to the colours of your national flag and football shirt.
and here is the suggested Consolation Tessellation for Brazil, also in approximate National Flag colours.
There may be other ways of interpreting and celebrating by tessellating the Independence of these two countries, each so prodigiously wealthy in its ranges of culture, climate and terrain. Neither 1816 nor 1822 are the sum of two squares, so they can’t be the areas of a lattice labyrinth supertile or fundamental domain/repeat unit on the square lattice. Neither is 1822 a Loeschian number , so it doesn’t inspire a tessellation on the triangular lattice either. I need to find some sheets of paper to check on the Loeschianity or otherwise of 1816.
With the help of the online factoring site I’ve checked for sum of two squares by the Fermat/Conway rule (no prime factor, to whatever power it is present, can be of the form 4n-1). Are there any mathematicians out there who know of a quick way by which a human rather than a computer can check for Loeschianity?