Inspired by the Month of June, Square Lattice Labyrinths with Even Separation Parameters

After many blog posts inspired by dates, including the Queen’s Birthday, Christmas Day, the National Days of Argentina, Brazil and Taiwan and the date of my Hip Op, it’s time for some basic tuition – though June, the sixth month, is still an inspiration, leading me to revisit, Lattice Labyrinths on the square lattice when both separation parameters are even – a situation I’ve been ignoring, as the Chinese and Serpentine families on the square lattice are completely soluble and the families of tessellations on the triangular lattice are so pretty.

When the separation parameters (a,b) are one even or zero, one odd, we get a Chinese Lattice Labyrinth such as (5,0) or (7,0). The area of the supertile a²+b² is always of the form 1+4n which corresponds to a tetragonally symmetrical supertile having a single central square and four identically-shaped arms. (A reminder: The separation parameters tell you how far across (a) and up (b) the lattice you have to count to come to a point identical in its environment to the one you started from; they tell you the scale on which the pattern repeats itself.)

Chinese(5,0),(7,0)The missing links graphs are simple to discover, the tessellation graphs use every lattice link not used in the missing links graphs (see my workbook).

If both a and b are even, a²+b²  is of form 4+4n, so if a tessellation of tetragonally symmetrical supertiles is to exist, each must have at a block of at least four square tiles at the centre of each supertile. Finding such tessellations presents a big problem to me, as the missing-links graph algorithm as used above is of little help. Lower-order cases can be found by trial and error – below I illustrate case (6,4) – two different constructions.

Windmill (6,4)I call the family of which (6,4) is an example the Windmill lattice Labyrinths, a name suggested by the shape of the lowest-order member of the family.  I constructed the left-hand, yellow and green tessellation by trial and error. Its tessellation graph is drawn in black. When one “reverse-engineers” this tessellation by constructing its  missing-links graph , shown in red, one discovers that the result is the tessellation graph of another version of Windmill Lattice Labyrinth (6,4), shown on the right in yellow and blue.

This property, that a Windmill Labyrinths is “complementary” to another Windmill Labyrinth is shared by all members of the family. Some members are complementary to themselves, some are complementary to a mirror image of themselves, the rest to a differently-shaped tessellation, as in the case illustrated above.

Despite the above beautiful property, I find the Windmill family unsatisfactory because the lattice point at the centre of each supertile is unconnected by/to the tessellation graph. We CAN connect these points without upsetting the symmetry of the tessellation as a whole by constructing a cross at each supertile centre to divide each supertile into four identically-shaped arms. This does however mean that these tiles meet in edges of the tessellation graph rather than at vertex (lattice) points, corner to corner. This feels like a generalisation too far. However, this is not the end of the story for (even,even) separation parameters.

In the last few years a Japanese academic team (see reference at the end of the post) have been working their way through all the low-order polyominoes that tessellate, by an ingenious method quite different from mine and soon becoming impractical within the times of the order of the apparent age of the Universe for higher-order cases. Working through their results and correlating them with my lattice Labyrinth Families, I found one that didn’t fit! Here it is.

Japanese (4,2)Well! Both separation parameters are even, the full symmetry of tetrad and dyad axes is there, the supertiles meet corner to corner AND all lattice points (corners of the squares of the square lattice) are connected in the tessellation graph. the above tessellation is indeed the lowest-order member of yet another infinite family, which I had to call the Japanese Lattice Labyrinths in honour of its source – and to balance nomenclaturely with the Chinese Lattice Labyrinth family.

(R, the repeat unit (or fundamental domain) is the number of squares in the pattern that are repeated to form the tessellation, S is the area of each supertile. In this case four supertiles in the four possible different orientations make up the repeat unit)

To return to June, month 6, here is another member of the Japanese family, (6,4).

Japanese(6,4,)Unlike the Windmill Labyrinths, the Japanese family can be constructed via a straightforward missing-links graph (shown here in red), though spotting general construction rules that cover all cases is not easy. However, my attempts to construct a Japanese Lattice Labyrinth for another “June” case, (6,2) met with persistent failure. It can’t be done! here is the best I can do:

Scarthin (6,2)I decided that Scarthin had to get an entry into my terminology; it’s where I live and work (and well worth a visit to the bookshop, Scarthin Books, the pub, The Boat Inn or just to relax on The Prom, overlooking the millpond, The Dam). The Scarthin Lattice Labyrinth family is undoubtedly also infinite in membership; its lowest-order “basic tessellation” is case (4,0), which you might like to try constructing. Alas, this family, like the Windmills has a block of four squares in each supertile with the central lattice point not connected to the tessellation graph.

We can characterise which (even,even) cases are drawable as Japanese Labyrinths and which as Scarthin labyrinths algebraically, but it’s perhaps easier to get the point from this little sample table, below.

Separation Parameters, Japanese, Scarthin rev2

              At present I’ve forgotten my clear and simple explanation of why the (even,even) separation parameter cases fall into these two distinct families. Analogously to Arthur Sullivan’s musical “lost chord”; this is my “lost proof”. Seated one day at the laptop, I was fluent and free of fear…..

If the constructions above have left you baffled, befuddled and bewildered, then you could always purchase a copy of the explanatory workbook and be admitted to hours of recreational mathematics and artistic designing. The Lattice Labyrinths workbook  is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

The Japanese research paper I referred to above is:

[5] Fukuda, Hiroshi et al, Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry, Symmetry 2011, 3, pp.828-851

 

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Tessellations for Her Majesty Queen Elizabeth the Second’s Ninetieth Birthday. Long may you reign.

I am old enough to remember the celebration of  the Coronation of Queen Elizabeth II, to whom we refer always as simply “The Queen”. I can even remember the date – 2nd. June 1953. In the village of Spratton, Northamptonshire, each of  the pubs hosted a party. The Fir Tree Inn (long since closed) put on a barn dance, which sounded magic to me, but we, the village schoolmaster’s family, were too respectable for that; we went to the party in the village hall (alcohol-free? I wonder). I can just about remember watching the procession and service on a tiny black-and-white television screen. Afterwards we had the classic village hall tea – triangular sandwiches, probably including salmon and cucumber (Mmmm!), butterfly buns. iced cake and jelly or trifle. There was a children’s obstacle race, which included the terrifying task of blowing up a balloon until it burst. In 1977, for the Silver Jubilee of the Queen’s actual accession (not of the  Coronation) I closed the young, but already full-time Scarthin Bookshop and joined in Cromford, Derbyshire’s big Street Party, the tables stretching the length of North Street. The Golden Jubilee in 2002 seemed to pass without a village celebration, or perhaps I was too busy to notice, but by 2012 I was heavily involved in the Celebrating Cromford organisation and helped put on a Diamond Jubilee Street Party on Scarthin Promenade, next to the Boat Inn. We mostly dressed up as queens or kings and there was music and an excellent children’s magician entertainer. The rain stopped and the sun came out just for that afternoon. Here’s a picture or two.

JubileeSreetPartyScarthin 027JubileeSreetPartyScarthin 017

JubileeSreetPartyScarthin 008

Anyway, now to the serious business of some Loyal Lattice Labyrinth Tessellations. The 21st. of April leads to the number pair (21,4), which specifies Trefoil Labyrinth (21,4). here it is.

Trefoil (21,4) ER2 Birthday

I think this has turned out very appropriately for the Queen, it’s a self-disciplined tessellation, tightly controlled, resisting temptations to go out on limbs.

It’s also possible to dedicate a Lattice Labyrinth to all those reaching ninety years of age. 90 factorises to powers of primes: 2 x 5 x 3². We can immediately tell that 90 cannot be a Loeschian number of the form a² + ab + b² because such numbers and their prime power factors cannot be of the form 3n+2, which both 2 and 5 are. This rules out a tessellation on the triangular lattice with tile area 90. However, this number CAN be the sum of two squares because none of it’s prime power factors are of the form 4n+3 so it can correspond to a tessellation on the square lattice. We can quickly spot that 90 = 9² + 3², yielding number pair (9,3) which as both are odd corresponds to Serpentine Lattice Labyrinth (9,3) and here it is, not in Union Jack colours this time. For once I’ve included some of the mechanics – the axes of symmetry and the missing-links graph employed in the construction.

Serpentine (9,3) 90th. BirthdayEach supertile contains 45 squares, but they come in two sets, orientated at 90° to each other, so the repeat unit (or fundamental domain) is 90 as desired.

If the construction above has aroused your curiosity, then you could always purchase a copy of the explanatory workbook and be admitted to hours of recreational mathematics and artistic designing. The Lattice Labyrinths workbook  is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

Time to go off to choir.

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Cameron’s Folly – The British IN or OUT of the European Union Referendum – Diatribe, Debate and Tessellation

The question of whether or not Britain should be a member of the EU is too complicated and subtle even for ME, let alone the average British elector to decide upon. It is JUST the sort of question we elect a parliament of professional legislators, able to commission a wide range of expert advice, to determine. But, ironically, these elected savants have determined  to leave it up to us, the ill-informed electorate, to decide, subject as we are to the winds of contingent events and the plausible posturing of high-vis rogues on both sides.  I and many of my acquaintances despair of the wisdom, even of the goodwill, of our government. Actually, this is my permanent state of mind.

Well, if we must have this stupid referendum, at least we should follow the usual constitution of societies and public bodies, for instance that internationally insignificant Derbyshire village body, the Cromford Community Association and require a two-thirds to one-third majority of members (and a simple majority of the Council ) before taking the momentous step of changing our long-established Constitution. What is the UK to do if (as is quite likely) the vote is 51% to 49% in either direction. Should such a divided nation LEAVE despite a tiny majority in favour of STAYING? Should a two-thirds/one third majority be required for us to stay IN, taking OUT to be the historical default situation?

Afterword; as you may know, the vote was 52% to 48% – little different to my “forecast” and so close I’m not bothering to confirm which way it went, though a statistician would confirm that such a split of some 34 million votes means there is a “highly significant” deviation from a 50:50 coin-toss hypothesis. One reaction to this result can be enjoyed on my twitter page: https://twitter.com/davescarthin/status/747816380930596864

Anyway, in an attempt to hear some substantial arguments, rather than the national media’s head-count of moguls, Scarthin Books of Cromford are sponsoring an IN/OUT debate on Saturday 9th. April at the aforementioned Cromford Community Centre. The contestants are Edward Spalton and Brian Mackenzie, both experienced in international enterprise and trade,both committed to their views beyond the economic arguments, both worthy of respect. Here is the poster, and also a parking/access plan as our industrious  village which, with its shops, hotels, stone quarries, engineering and service businesses,historic mills and housing, restaurants and pubs, is a HUB and often congested.

EU Debate

ParkingPlanEUDebate

Of course, Britain will probably be OK either OUT or IN the EU. We’ll all try to make the best of whatever situation we find ourselves in. The result of the referendum will be subject to huge random (i.e. inexplicable and unpredictable) factors; we might just as well have tossed a coin – HEADS WE’RE IN, TAILS WE’RE OUT – you could say that tossing a coin is exactly what we ARE doing.

One afternoon in the the Chemistry Laboratory at Northampton Grammar School, a teacher stuck a pin into a list of Cambridge Colleges and thus randomly allocated a College to each of us five Science Boys. The 100% success record of Alan Bennett’s History Boys was not achieved; the two most promising  did not get in (they shone at Imperial College, London, instead), three of us did – to Pembroke, St Catherine’s and St. John’s. Had I not found myself at John’s, I would not, on my way to Cavendish Laboratory lectures, have passed and become addicted to David’s bookstall  – and Scarthin Books, my children and this blog would all vanish “in a puff of smoke”. But surely another life, not empty of colour and creation, would have come about in place of the life I’ve actually been living. So will it be with Britain, in or out of the EU.

WHAT ABOUT A TESSELLATION ?! The EU referendum takes place on June 23rd., so here is Trefoil Lattice Labyrinth (23,6), first of all the usual pretty pattern of just six supertiles, each comprised of 23² + 23×6 + 6² = 703 equilateral triangles. I’ve produced it in the EU flag colours. You could call it the Brussels Labyrinth, Brussel Labyrint, Le Labyrinthe de Bruxelles.

Trefoil (23,6) EU 6 finalBUT, there are 28 nations in the EU. Here they are in all their waning and waxing variety fully interlocked and symmetrically disposed, three nines, about one central supertile (I wonder who that can be?). If the UK leaves, the symmetry will certainly be destroyed

Trefoil (23,6) EU 28

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

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A Complete Hip Replacement Tessellation for Thornbury Hospital, Sheffield & the NHS, from a Grateful Hipster

For the technical brilliance of my Complete Hip Replacement operation at Thornbury Hospital, Sheffield I am more than grateful, but I cannot personally endorse the craftsmanship as, in addition to the epidural total-nether-number (pronounced “nummer”), I opted to be “put to sleep” for the duration. The duration seems to me to have been negative as I woke up in the recovery suite BEFORE the  general anaesthetic had even been administered! At any rate, that’s what I remember. I was referred to this BMI hospital by the NHS, Britain’s pioneering, legendary and embattled National Health Service, and the interval between my (belatedly) requesting to be referred for the procedure by my GP (General Practice) doctor and the actual operation was just three months.

thornburyHospitalFor much of January and February I considered myself to be on HIP ROW, hoping for much longer stays of operation, but the hip in question adopted a different  OP-WISH strategy so in the end I was EAGER for the PROCEDURE. Thank you to the consultant anaesthetist, whose name alas I didn’t gather, and, of course, to Prof. Ian Stockley who radiates reassurance and, I think, joy in his work and who has become Hip Replacement Surgeon by Appointment to Scarthin, the hamlet within Cromford where I live and work. After the operation, my three nights’ individual-room bed and board was simply luxurious. So thanks also to Mappin Ward and to the smiling nurses, most with the Sheffield and South Yorkshire accents that reminded me of my Conisborough roots. By day the sun lit up the valley sides, by night the owls hooted in the trees under the moon.

The sometime Birmingham University Wayfarer Chris Ford tells me he has reached HR50, the age at which half his friends have had a hip replacement, said by Prof. Stockley to be the second-best of all “routine” operations, after cataract-removal. I think he is citing both the high success rate and the life-transforming effect.

Incidentally, HR50 ≅ PhD50 + 45

OP DAY was Monday 22nd. February, 2016.The number pair (22,2), taken to be  separation parameters on the triangular lattice, corresponds to a  Lattice Labyrinth in the Trefoil family because 22 – 2 = 20 is not divisible by 3. For (a,b), if a-b IS divisible by 3, a Honeycomb Labyrinth (and a Diamond or Dart Labyrinth) results.

As a thank you for my “free” operation, this time I won’t hide the stages by which the tessellation is constructed. First of all we set out  superlattice points on the triangular grid, each at separation  22 lattice points  across and 2 lattice points at 60º up and then repeating the construction symmetrically to produce the superlattice array. At each of the superlattice points there is a hexad axis of 6-fold rotational symmetry (see Rotational Symmetry simply set out, an early blog post). It helps to find our way around if we also mark in the triad 3-fold and dyad 2-fold axes. Crucially this completes the very same symmetry group as displayed by the triangular lattice itself.

Trefoil (22,2) Hip Op SuperlatticeNow for the creative bit – which means I have no hard-and-fast rule for this procedure – constructing a valid missing-links graph, so named because it uses up ALL the lattice links that will NOT form part of the tessellation graph. First of all, around each hexad axis I draw in the biggest nest of hexagons that I can without the outermost hexagons overlapping. I then know from experience with separation parameters which share a common factor (2 in this case) that the dyad axis lattice points are potentially hard to reach in a symmetrical fashion, but also from experience I know that a paddle wheel construction, also shown in red, centred on each heaxad axis, will do the trick.

The missing-links graph elements have to pass through every lattice point twice, except for the hexad-axis superlattice points, so as to leave just one route through each lattice point unused. The green constructions made up of triangles and closed triangularish figures symmetrical about each triad axis complete this job. This particular green assemblage is the most elegant I’ve found, there may be other possibilities for this part, or even for the whole, of the missing links graph. here it is.

Trefoil (22,2) Hip Op ML GraphNow for the automatic, but entertaining job of tracing the graph made up of just those lattice links NOT used in the missing links graph. To shorten the task we can take advantage of the symmetry which means we only need to trace the graph, by way of the only links left unused, from one superlattice hexad axis point to the next. We can then take advantage of the rotational symmetry and of the repetition at (22,2) separations to construct as little or as much as we like of the infinite tessellation graph. Here is enough to demarcate just six trigonally symmetric supertiles, three in each of the two orientations at 180º to each other – the equivalent of the equilateral triangles that make up the basic lattice itself.

Trefoil (22,2) Hip Op Tess GraphLike magic, isn’t it!  Finally, here it is, coloured in:

Trefoil (22,2) Hip Op Tess Only

AIN’T THAT HIP,MAN!

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

 

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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.

Trefoil (2,24), (10,5) JobsThese 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.)

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A Christmas Card from Dave at Lattice Labyrinths and Scarthin Books

I wish all visitors to this blog a Merry Christmas and a Peaceful New Year.

Christmas Card 2015

The tessellation is Trefoil Lattice Labyrinth (25,12) with Snowdon seen from Crib Goch in the background. The occasion was a classic traverse of The Snowdon Horseshoe in Alpine conditions by a party from Birmingham University Wayfarers Society. At sunset, while we sheltered beneath a little crag below Llewedd, Ekrem Gacic, from Sarajevo, brewed a “Turkish” coffee which we sipped through a sugar lump from a tiny cup.

The cardboard is being unloaded for recycling at the highly-organised, indispensable Peak Waste near Kniveton, Derbyshire. The cycle, just visible (in fact a tandem) was NOT recycled.

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

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Celebrating Anglo-Chinese Relations; 21st October 2015 is Double Ninth Day, Chongyang 重九節

The Hamlet of Scarthin within Cromford, where this blog is based, leads the way in Anglo-Chinese relations, and here are some photographs from the narrative (which also involved the hamlet of Upperwood within Matlock Bath).

IMG_6088R. soon changed into traditional RED.

OliverPiggyBackingWith niece and Ma and Pa … in the country.

ChinaFamilyPlus2015With sister and son … in the city.

RuthShezhenHospitalA student at John Radcliffe Hospital, Oxford visits Beijing University Hospital, Shenzhen

北京大学深圳医院

..and there’s lots more where those come from (in fact I give illustrated talks).

This blog post is being assembled on 21st. October 2015 which this year is the ninth day of the ninth lunar month, Double Ninth Day or Chongyang. Yang means nine but also denotes the masculine as opposed to Yin which is feminine  and ‘chong’ in Chinese means double, hence Chongyang.

(In addition, the state visit of Xi Jinping, President of the People’s Republic, is today in full swing. Can he lend us some of our money back?)

I am told that on this day, the tombs of ancestors may be visited and swept and a mountain climbed for protection against the dangers of such doubled masculinity. I recommend Lantau Peak (thanks, Wikipedia) which is a Monroe (3000 ft+, 914.4 m+). Chrysanthemum wine is a traditional drink. But treat this webknow with scepticism.

Lantau_Peak,_Hongkong

Finally, of course a Lattice Labyrinth Tessellation, The two odd numbers, nine and nine generate  Serpentine Lattice Labyrinth (9,9), and here it is. Maybe the snakes could be transmuted into dragons. The colours are of the Chinese flag superimposed on England’s Green and Pleasant Land.

Serpentine (9,9) ChonyangFinal

An inexpensive how-to-do-it  Lattice Labyrinths workbook  is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

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The Tessellated Body – Seahorses embellish a Handbook for Medical Students

Behold the cover of the helpful Handbook for October 2015’s new students at Oxford University Medical School.  Both the front and back covers are shown.

PowerPoint PresentationThis design by Ruth Mitchell employs the  seahorse tessellation , Diamond Lattice Labyrinth (4,1).

Should you wish to work up some graphic designs of your own, here are some links to the inexpensive how-to-do-it  Lattice Labyrinths workbook

BothCoversOctoberwhich is available from the publisher, Tarquin (UK), from  you-know-who, from a good independent bookshop or via Google.

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A Lattice Labyrinth for Cristián: Trefoil (29,3)

Cristian is a correspondent in Chile who helped spread the word of my abortive but world-wide 2012 campaign for an Olympic Fourth Place Medal. Cristián’s special date is 29th. March, so here, as a belated thank you, is Trefoil Lattice Labyrinth (29,3), five elements displayed in the colours of the Chilean flag. I haven’t shown the boundaries of the 29² + 29×3 + 3²  = 937 equilateral triangles that make up each of the six supertiles pictured.

Trefoil (29,3) Cristián Final

It took several false starts and some “cheating” to arrive at this design, but I learned a lot in the process – and it’s been nostalgic to revisit that delightful interview in a London Park with ABC’s Julie Foudy.

In the absence of my friend Dave Jackson, daughter Ruth and several originally South American residents of Wirksworth to do a human job, here is the Google translation of the opening paragraph:

Cristian es un corresponsal en Chile que ayudó a difundir la palabra de mi abortiva sino en todo el mundo de campaña 2012 para una cuarta medalla olímpica Place. Fecha especial de Cristián es 29o. De marzo, por lo que aquí, como una tardía gracias, es trébol Entramado Laberinto (29,3), cinco elementos que se muestran en los colores de la bandera chilena. No he mostrado los límites de las 29² + 29×3 + = 937 triángulos equiláteros que componen cada una de las seis supertiles fotografiados.

I think I detect some wild mistakes in the above.

As always, here are some links to the inexpensive how-to-do-it  Lattice Labyrinths workbook which is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

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Tessellatable Numbers up to 2100, with corresponding Gaussian and Loeschian Number Pairs

The following tables, which should appeal to your inner nerd and might even be USEFUL, pick out which integers N in the range 1 to 2100 are of the form a²+b² and/or of the form e²+ef+f². The first category are the square of the “norm” or “modulus” of Gaussian integers a+ib on the complex plane, the second the square of the norm of Loeschian integers e+ωf on the complex plane, where ω = (1+i√3)/2.

If N is of the form a²+b² it specifies a point (a,b) of the unit square lattice and corresponds to a tessellation on the square lattice. If N is of the form e²+ef+f² it specifies a point (e,f) of the triangular lattice and corresponds to a tessellation on the triangular lattice. This post replaces earlier tables of “tessellatable birth-years”, now deleted from the blog (though perhaps they live on in some limbonic Cloud).

Note that to every Loeschian number e²+ef+f² there corresponds an Eisenstein number g²-gh+h² with the same value of the norm, √N, but described by taking axes on the Argand diagram at 120° rather than 60° to each other. The Eisenstein representation I consider to be a menace; the negative sign in front of the product gh means that N jumps up and down as the value of  h increases. The Loeschian representation means that N always increases with an increase in either g or h.

In the tables, real primes are highlighted in yellow, Gaussian primes in green and Loeschian primes in blue.

There are all sorts of properties and patterns to be descried in these tables. Notice that there is one category of Gaussian and Loeschian primes for which N is NOT a real prime. For instance, N = 49 corresponds to Gaussian prime (7,0). this arise because                     N = 7 ≡ 3 modulo 4 and cannot be an a²+b² (which are always ≡ 0,1 or 2 modulo 4) and so corresponds to no Gaussian integer, and there is therefore no Gaussian integer that can be squared to give (7,0), so (7,0) is a Gaussian prime.

Similarly, N =25 corresponds to Loeschian prime (5,0) as N = 5 ≡ 2 modulo 3 and cannot be an e²+ef+f²  (which are always ≡ 0 or 1 modulo 3)so there is no Loeschian integer that can be squared to give (5,0).

Real PRIMES also cannot be  0 modulo 3 (divisible by 3),                                                 nor can they be  0 or 2 modulo 4 (divisible by 2 or 4 respectively), so all TESSALATABLE prime N are  1 modulo 4, corresponding to a Chinese Lattice Labyrinth or 1 modulo 3, corresponding to a Trefoil Lattice Labyrinth.

N = 121 corresponds to both Gaussian prime (11,0) and Loeschian prime (11,0) as       121 ≡  1 modulo 3 and 121 ≡ 1 modulo 4, that is 121 ≡  1 modulo 12.

My previous posts Eine kleine Mathmusik und Eine nicht so kleine Mathmusik  show how multiplying number pairs gives rise to tessellations of tessellations corresponding to the product(s) of the number pairs (Gaussian or Loeschian integers) and how Pythagorean triples can be generated. These triples, and other cases such as Gaussian  (9,2) and (7,6) both having N = 85 are a prominent feature of the tables. The lowest N corresponding to THREE Gaussian integers is 425, the lowest corresponding to FOUR is 1105. It is no coincidence that 5 is a factor of both these values of N and 13 a factor of the latter. The lowest N corresponding to three Loeschian integers is 931, the lowest corresponding to four is 1729. it is no coincidence that 7 is a factor of both these values of N and 13 a factor of the latter.

No value of N can correspond to more than a single prime Gaussian or Loeschian number. The fact that a real prime number can be an a²+b² in no more than one way was proved by Fermat. That it can be an e²+ef+f² ( or g²-gh+h²) in no more than one way appears also to be the case, though I haven’t come upon a formal proof. Note that this uniqueness extends to cases where N is not itself a prime but is the SQUARE of a prime which is not the square of the norm of a Gaussian or Loeschian integer, as outlined above.

Some real primes turn up in the intriguing and notorious adjacent pairs, such as 5 and 7, 29 and 31,……, 137 and 139 and so on without end. You will notice that these pairs correspond either to a Gaussian and a Loeschian integer respectively or the greater of the pair corresponds to both. This can be explained by simple algebra. Look for the cases where N ≡ 12 modulo 1.

Enough preambling. In the course of enhancing and playing with the  tables I have corrected some six “typos”, errors of transcription. Though I don’t like the admission, it’s “statistically likely” that a similar number of errors remains to be found. So just check the arithmetic before making use of an element of the tables and please comment on the blog post if you do spot an error.

Note that each Loeschian integer correspond to an Eisenstein integer; the number pair will of course be different in the two representations but the point on the complex plain referred to is the same and the value of N is the same. the conversion to Eisenstein (g,h) from Loeschian (e,f) is given by g = e+f, h = f. Another advantage of Loeschians; they are more economical with the natural numbers.

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Diamond (7,4)As always, here are some links to the inexpensive how-to-do-it  Lattice Labyrinths workbook which is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

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