Tessellating the Hardy-Ramanujan Taxicab Number, 1729, Bedrock of Integer Sequence A198775.

Here is Trefoil Lattice Labyrinth (32,15). There’s something rather special about it.

ramanujan-3215-for-blogAccording to the celebrated story, the English mathematician G.H.Hardy arrived at the hospital bedside of his Indian protege ( the autodidact mathematical genius) Srinivasa Ramanujan in London taxi number 1729, which apparently uninteresting number Ramanujan immediately pronounced to be the smallest number than can be expressed as the sum of two positive cubes in two different ways, 1729 = 9³ + 10³ = 12³ + 1³  ( the nearest possible miss to a case of  x³  + y³  = z³, declared impossible in the even more celebrated “Last Theorem” of Pierre de Fermat). Several other elegant attributes of 1729 are outlined by  Wikipedia, but their compilers have, at the time of writing, missed one more property: 1729 is also the LOWEST  number which can be represented by a Loeschian quadratic form a² + ab + b²  in FOUR different ways with a and b positive integers. (a,b) can be (25,23), (32,15), (37,8) or (40,3).  I personally discovered this to my amazement when looking up 1729 in my list of Tessellatable Numbers up to 2100.

If you’ve read other posts on this blog you will realise that my interest in numbers pairs is that they form the separation parameters for setting out tessellations on the square or triangular lattices. Here’s what I mean for the triangular lattice case where (a,b) is (4,3).

settingouttriangularlatticepointsThe area of the red triangle, set out according to separation parameters (4,3) measured parallel to the red axes at 60° to each other = 4² + 4×3 + 3² = 37 triangles (for a non-rigorous geometrical derivation of how this comes to be the case see a previous post  on this blog). 37 is also the area of each of the two “supertiles”, the yellow and the blue, shown. Together they make up the repeat unit or fundamental domain of the Trefoil Lattice Labyrinth (4,3) tessellation of the infinite plane. 37 is what I mean by a “tessellatable number”, of which 1729 is another example and is the area of each supertile of tessellations set out using any of the number pairs (25,23), (32,15), (37,8) and (40,3) as the separation parameters.

Had the property, that 1729 is the lowest number that is a Loeschian number in four different ways, been noticed before? My researches took me first to the wikipedia article on the Hardy-Ramanujan Number  and next to the exciting and welcoming George Green Library at Nottingham University, in order to consult the (expensive) second edition of Richard K Guy’s delightfully down-to-earth (at the BEGINNING of each topic) Unsolved Problems in Number Theory. E-mails to Professor Guy at Calgary University, and to Professors Roger Heath-Brown of Oxford University Mathematical Institute and Rainer Schulze-Pillot of Saarland University  were all answered helpfully and promptly.Thank you to all three for taking the trouble to answer a query coming out of the  blue, and congratulations to Richard Guy on his recent 100th birthday. Professor Schulze-Pillot referred me to a source I should have checked in any case, The Online Encyclopedia of Integer Sequences, and there was 1729, the first member of a sequence posted only in the year 2011 (ah well):

A198775 Numbers having exactly four representations by the quadratic form x^2+xy+y^2 with 0<=x<=y

How does this remarkable extra property of the Hardy-Ramanujan Number arise? Let’s see what arises if we multiply some separation parameter pairs together. The number pair (2,1) is PRIME; its only factors are itself and unity, which on the triangular lattice is (1,0). Note that the triangle area it specifies is 2² + 2×1 + 1² = 7. If you multiply it by itself (“square it”), (2,1) x (2,1) you get (5,3) with a corresponding triangle and supertile area of 5² + 5×3 + 3² = 7 x 7 = 49.  If you multiply itself by its mirror reflection (think of the triangle  in the above figure reflected across either axis), (2,1) x (1,2) you get (0,7) also with triangle area 7 x 7 = 49.

The general expression for multiplying number pairs (a,b) and (c,d)  on the triangular lattice to get (m,n) is: (m,n) = (ac-bd)(ad+bc+bd). This expression is derived in the aforementioned previous post, where the numbers pairs are treated as specifying complex numbers on the Argand diagram, the Loeschian a² + ab + b² being the norm of (a,b) referred to axes at 60° (the imaginary part of the number is still that measured along a y axis at right angles to the real x axis). Apologies if I’m confusing you as well as myself.

The figure below illustrates the tessellations corresponding to (2,1) x (2,1) = (5,3) and   (2,1) x (1,2) = (0,7)


I call 3,5 and 7 a Trithagorean triple by analogy with Pythagorean triples such as 3,4,5. For the derivation and geometric implication of Trithagorean triples see that same previous post yet again. Below I repeat from that post an illustration of the geometric meaning of the lowest-order Trithagorean (or ????????an) triple, compared and contrasted with the ubiquitous lowest-order Pythagorean triple.


The lovely feature of the Trithagorean case is that the sum of the areas of the pink, yellow AND grey triangles equals the area of the pale blue triangle.

The lowest order prime number pairs (known as Eisenstein primes on the complex plain, but conventionally expressed with reference to axes at 120º rather than 60º to each other) are (1,1), (2,0), (2,1), (3,1) and (3,2). Multiplying cases where one parameter is zero such as (2,0) or the parameters are equal such as (1,1) do not yield Trithagorean triples when you “square” and “mirror-square” them. We have  seen that:

(2,1) x (2,1) = (5,3)    and  (2,1) x (1,2) = (0,7).  Trithagorean triple  3,5,7.     Likewise  (3,1) x (3,1) = (8,7)    and  (3,1) x (1,3) = (0,13). Trithagorean triple 7,8,13    and     (3,2) x (3,2) = (5,16)  and (3,2) x  (2,3) = (0,19). Trithagorean triple 5,16,19.

You may not be surprised, by now, to find that 7 x 13 x 19 = 1729. To see why 1729 can be represented by four different number pairs we need to do some more multiplying. To arrive at a number pair with Loeschian number 1729 we need to multiply each of the above three prime number pairs together. There are eight different ways of multiplying all three together if we include their mirror-image pairs, but four ways are enough to generate the four different representations of 1729 (the other four generate the mirror pairs, with the numbers in each pair interchanged). I list them below, with the negative integers that turn up converted into positive integers by the rule “replace – a by a and b by b-a”. You can check algebraically that (-a)² + (-a).b + b² = a² + a(b-a) +(b-a)² or check this geometrically when you plot the separation parameters on the triangular lattice.

(2,1) x (3,1)=(5,6);  (5,6) x (2,3)=(-8,45) = (8,37)

(1,2) x (1,3)=(-5,11);  (-5,11) x (3,2)=(-37,45) = (37,8)

(2,1) x (1,3)=(-1,10);  (-1,10) x (2,3)=(-32,47) = (32,15)

(3,1) x (3,2)=(7,11); (7,11) x (1,2)=(-15,47) = (15,32)

(2,1) x (3,2)=(4,9);  (4,9) x (3,1) = (3,40)

(1,2) x (1,3)=(-5,11);  (-5,11) x (2,3 )=(-43,40)=(43,-3) = (40,3)

(2,1) x (1,3)=(-1,10);  (-1,10) x (3,2)=(-23,48) = (23,25)

(3,1) x (2,3)=(3,14); (3,14) x (1,2)=(-25,48) = (25,23)

I’ve chosen to show the geometry of the multiplications in red because they fit conveniently onto the compact graph below.


I employed  significant chunks of the Christmas 2016 to new Year 2017 holiday very enjoyably finding Trefoil Lattice Labyrinth tessellations for each of the four different pairs of separation parameters, each containing 1729 triangles within the supertile. Here they are, in the flag colours of India, starting with the tessellation with which I opened the post:

ramanujan-3215-for-blogramanujan-2523-for-blogramanujan-403-for-blogramanujan-378-for-blogJust one sibling seems to be of a quiet and retiring disposition. Ramanujan is taken to be the epitome of the autodidact, but I think that long months of work in solitude has been a necessary part of the (self-) education of all great and many of us far-from-great thinkers.

I recommend extracting the next-lowest triangular lattice prime number pairs (the parameters unequal and neither zero) from my table of Tessellatable Numbers up to 2100 and seeing how multiples of their Loeschian number norms correlate with members of the integer sequence A198775 Numbers having exactly four representations by the quadratic form x^2+xy+y^2 with 0<=x<=y of which the Hardy-Ramanujan number is the first member.

Hummm – and what do you get if you multiply the lowest order PYthagorean norms?

To finish with the customary commercial:

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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Tiptoe tentatively into tessellated 2017

HURRAH!  2017 is the first tessellatable year since 2011, either on the square or on the triangular lattice. To start with the last but best image I’ve found:


This shows symmetry axes and six supertiles of Trefoil Lattice Labyrinth (41,7), each supertile comprised of 2017 equilateral triangles. Here’s just one of the supertiles that fit together to potentially cover the infinite plane:

trefoil-417-2017-try-2-single-tileThis unexpectedly wayward shape is in fact generated by an elegantly simple missing-links graph which I arrived at only by a hit-and-miss evolutionary process – a metaphor of last year’s wayward results of evolved democratic processes?

We had better re-cap to sketch out why 2011 and 2017 yield tessellations, but 2012 2013, 2014, 2015 and 2016 do not, but by all means skip or speed-read the next two paragraphs if you like and go straight to some images of tessellations on the square lattice.

To construct a tessellation on the square lattice we set out points by counting a along one axis and b along the other, forming squares of area a²+ b², so for a number, such as 2017, to be the area of the repeat unit (fundamental domain) of a   tessellation on the square lattice it is necessary that it can be partitioned into the sum of the squares of two integers, a and b (one of which may be zero). This is possible only if the number in question is of the form 4m, 4m+1 or 4m+2, never of form 4m+3. You can get a feel for why this is the case by drawing dots in square arrays or sticking matchsticks in a pinboard. Less obvious intuitively, it is also necessary that all the number’s prime factors (raised to whatever power they occur) must also be of one of the allowed forms. (See Conway and Guy’s Book of Numbers (expensive in paper). For instance, if 3 is a factor, or , no tessellation is possible because both these powers of a prime are of form 4m+3, but if is the factor, this presents no problem, as we can see that 9 can be expressed as 2m+1 where m=4.

For tessellations on the triangular lattice we set out points counting e  and f along axes at sixty degrees to one another, and the area of the triangles so constructed will be the Loeschian number  e² + ef + f². There is an analogous rule for the form of Loeschian numbers. All are of form 3n or 3n+1, never of form 3n+2. Again, you can get a non-rigorous feel for this by counting points or matchsticks set out in triangular arrays.   Once again, this rule applies not only to the number itself but also to all its prime factors, raised to a power if it occurs as such. So 2 or cannot be factors of a Loeschian number but can be, as 4 can be expressed as 3n+1, where n=1.

Note that the “tessellatability test” is a member of that beguiling but frustrating family of theorems that tells you whether something exists but includes no method of deriving it. One has confidence that e and f exist (there may be more than one possible pair) but get no further help towards finding them.

2011 is a prime number and  of the form 4m+3 so cannot be the sum of two squares, but it is of form 3n+1 so can be a Loeschian number e²+ef+f² and a Trefoil Lattice Labyrinth can be constructed (I believe – I’ve yet to try) with e=39, f=10.

2012 = 2² x 503 and 503 is of form 4m+3 and also of form 3n+2, so is tessellatable neither on the square nor the triangular lattice. The same applies to the next four years:

2013 = 3 x 11 x 61 and both 3 and 11 are of form 4m+3 while 11 is of form 3n+2.

2014 = 2 x 19 x 53 and 19 is of form 4m+3 while both 2 and 53 are of form 3n+2.

2015 = 5 x 13 x 31 and 31 is of form 4m+3 while 5 is of form 3n+2.

2016 =  25 x 3² x 7 and 7 is of form  4m+3 while 25 = 32 is of form 3n+2.

A digression:

2016 has 36 factors: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. leading me to think that 2016 might be a “highly composite number”, the smallest number to have 36 factors. But no, according to Wikipedia ( donate!)  1260 is the relevant highly composite number, the lowest with 36 factors. One observes that 1260 is an anagram of 2016. The highly composite number that comes first to my mind is 24, the lowest number to have 8 factors. One observes that 42, an anagram of 24, also has 8 factors.

End of digression. At last we come to:

2017, a prime of the form 4m+1 and 3n+1, so we should be able to construct tessellations on both the square and the triangular lattice.  2017 = 44² + 9², corresponding to Chinese Lattice Labyrinth (9,44) (my convention is to put the odd number first) and 2017 = 41² + 41×7 + 7², corresponding to Trefoil Lattice Labyrinth (41,7) (oddness and evenness are not especially significant for the construction, so I put the largest number first), a realisation of which heads this post.

Many of us are anxious that 2017 will be a difficult year, and so it proved when searching for a satisfactory tessellation on the square lattice. The standard missing-links graph for a Chinese lattice Labyrinth, consisting simply of nest of squares, work as always but leads to a very swastika-like (“swastikoid”) supertile. Attempt after attempt to use non-standard missing-links graphs failed – what emerged would have been (if completed) a tessellation of tiles within tiles rather than a monohedral tessellation. Here is an example:

chinese-944-2017-failure-5At this point (a dyad axis) I gave up, as the tessellation graph traced thus far, if copied, rotated through 180 degrees and joined up, would lead directly to the next superlattice point –  far too quickly, leaving most lattice points unvisited. I suppose I’d better show you what transpires if I do keep right on to the end of the road … tho’ the way be long … keep right on round the bend as the rather appropriate old song goes. So here it is:


– very  (left-leaning) swastikoid of course. I’ve used three colours to distinguish the (two) different shapes that make up the tessellation, but two colours would do, yellows being surrounded by blue and vice versa.

Eventually, I arrived at a successful tessellation, but still reminiscent of a shape disgraced in Western eyes, though there is no such problem in South East Asia.


and finally, employing a radically new type of missing-links graph, a pattern that I feel able to recommend as a suitably tortuous representation of a year through which there is no clear path for many communities and nations.


I’ve illustrated all these designs against skies that at this time of year represent a Shepherd’s warning.

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

Happy New Year!




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Thanks for the MathsJam Conference 2016 at Yarnfield Park near Stone, Staffordshire

I’ve only just discovered MathsJam, which may be meeting at a pub near you on the last but one Tuesday of each month, with Tuesday 13th.December 2016 a pre-Christmas exception. Here’s the MathsJam twitter site and a list of cities and other places where MathsJam is probably currently active; my personal centre is Nottingham, meeting at the Crafty Crow, opposite the Castle gateway.

Aberdeen, Antwerp, Auckland, Bangkok, Bath, Baton Rouge, Berlin, Birmingham, Bombay, Brighton, Brisbane, Brunei, Cambridge, Canterbury, Cardiff, Cheltenham, Chicago, Delhi, East Dorset, Edinburgh, Ghent, Guelph, ON, Guildford, Kolkata, Lagos, Leeds, Leuven, Leicester, Lincoln, Lisbon, London, Lund, Manchester, Newcastle, Norwich, Nottingham, Oshawa, Oslo, Oxford, Peterborough, Phoenix, Portsmouth, San Antonio, Sheffield, Stockholm, Swansea, Sydney, Tacoma, Winnipeg  and York

Each month, Katie Steckles co-ordinates and issues a MathsJam SHOUT – a page of problems and puzzles to break any ice at your meeting. Here is a recent example and here is another. (Those two links are to dropbox – they did eventually appear for me without attempts to sign in with mis-remembered details and I did manage to print them off as A4 sheets.) The October sheet occupied me happily for an hour or more on a broken-down train to Nottingham for a meeting I consequently never made.

The annual MathsJam Conference at Yarnfield Park, Staffordshire, was  a great treat. More than fifty talks were presented over the two days – each limited to just FIVE MINUTES after which escalating audio penalties are applied, and with just ONE MINUTE set-up time. After a batch of six or seven talks there is a coffee break during which the recent lecturers remain available to talk to. Delegates seat themselves at round tables, many strewn with mathematical games and puzzles, which makes for a very sociable time. Under the windows were arrays of free books, craft exhibits, a T-shirt competition, mathematical cakes competition, activities and puzzles competition,a competition for the best competion breaking the competition rules and arching over all a competition for the best competition. Next year we are threatened with a competition for the best best of competitions competition.

I think I’ve got that right. There was lots of laughter, generated by a very un-nerd-like array of stand-up mathematicians of all ages.  After an excellent serve-yourself and seat yourself dinner there was an evening of activities, a quiz and mathematical musical jam session..

Scalene, scalene, scalene, scaleeeen, of triangles you’re my favourite one.                    Scalene, scalene, scalene, scaleeeen, all your angles have a different tan.        and

Hark the herald angels sing                                                                                                           Trigonometry is King

give you a flavour of that. Summaries and slides of the talks at the 2015 conference are already online and those for 2016 will be posted soon. Here are a few pictures from 2016, including the answers to an anagram (or was it an acronym) competition.

wp_20161112_14_07_41_pro wp_20161112_14_09_10_pro  wp_20161112_23_08_19_prowp_20161112_16_19_18_proThanks to the organisers, presenters, moderators, lift-givers, microphone-fitters and all who made MathsJam 2016 go so far as I know without a serious flaw. Finally, of course, here are my tessellations for the two days of the conference, November 12th. and 13th., Chinese Labyrinth (11,12) and Serpentine Labyrinth (13,11) each constructed via non-standard missing-links graphs in a quest for more intricate and less swastikoid forms. In the second example the illustrated missing links graph is itself swastikoid. Finding such graphs that work is helped by some experience and requires some “tweaking”.


serpentine-1311-mathsjamThe 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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Cromford Apple Day 2016, a Lattice Labyrinth for the 15th October 2016

Tomorrow, October 15th. 2016 is the 21st. annual Apple Day on Scarthin Promenade. The first was on October 21st, 1995, but we missed a year when it coincided with one of my many decadal birthdays. Until about five years ago it was run by Scarthin Books, but nowadays the energetic Celebrating Cromford team have taken over most of the work and the range of entertainments and instruction that accompanies the serious business of fruit pressing has grown. It feels almost like a harvest festival and for a long time very year has seemed to be the best ever.

It so happens that one supertile of Chinese Labyrinth (15,10), generated employing a fairly ingenious non-standard missing links graph fits, on my 44 x 44 squares, 2.4m x 2.4m tileyard, so I have been obliged to construct this celebratory tessellation. Here it is, seen whilst under construction.

tessellation-2016-sidelongand here is a fully-frontal:

tessellation-2016Thank you to Eve Booker for the photographs taken in the gloaming. There’s more to be done tomorrow if time – I was called in for dinner.

wp_20161015_12_53_36_proMore was done to modify the design – but not by me.

Here are some pictures of previous Apple Days.




blappleday2011-016-2ajdsc01225-2It’s hard to stop adding memorable  and heart-warming images, and yes, the sun does often shine for us….touch wood, the forecast for tomorrow is a bit iffy.

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 Tessellation in the Wirksworth Festival Art and Architecture Trail, 10,11 September 2016

The 2016 Wirksworth Festival, Derbyshire, UK, begins on Saturday 10th. September with the pioneering and now widely-famed Art and Architecture Trail which takes you from surprise to surprise, in and out of gardens, up and down the ancient hillside town temporarily or permanently colonised by artists from all over the town and from far and near.

Facebook: https://www.facebook.com/Wirksworth-Festival-86444826559/

Twitter: https://twitter.com/wirksworth_fest

Lattice Labyrinths entry: http://www.wirksworthfestival.co.uk/artists/davie-mitchell-2016

This year the trail includes a half-foot-sized tiled realisation of a version of Chinese Lattice Labyrinth (9,8), chosen to celebrate the patronal (or matronal?) festival of St Mary the Virgin, patron of Wirksworth’s mediaeval St Mary’s Church, which fell as recently as yesterday, as I write, September 8th. Four supertiles of this tessellation have a wingspan of 44 squares, which, if sized two-by-two-inches, can just be fitted onto two 8 foot by 4 foot (approx 2.4 metres by 1.2 metres) plywood boards. Here’s the design, with added doodles, submitted for the approval of the Trail curators.

chinese-98-st-artarchtrail-a4-300dpiand in a rare moment of self-revelation here I am screwing  tile-separators to one of the boards at 54 mm intervals (+ or – 1mm, I hope, or the 47mm x 47mm tiles won’t fit).

dave-screwing-separatorsand here is the highly-skilled, self-moulded philosopher Jacob the Joiner (Jacob Butler), cutting up 1500 or so tiles on his rock-solid and millimetre-precise bench/table saw. It took only about three hours. The tiles are medium density fibreboard which when sawn splinters much less than real wood.

jacob-sawing-3Other curiosities of the installation can be demonstrated if you dip into the Trail (completing it might take the two days) after purchasing a guide and badge at one of  several accessible locations in the centre of Wirksworth. On a damp evening, Friday 9th. September, the completed piece was assembled by myself and son Michael, laid out flat on the lawn behind the Memorial Hall on St.John’s Street, necessarily sheltered by a substantial but obligingly pop-up gazebo.



The whole concept has worked but I’m a bit disappointed by its appearance; tomorrow we’ll perhaps add some  parts of some further supertiles in pale blue (against the red) and charcoal grey (against the yellow) to achieve more contrast with the board.

PS now we’ve made this beast, its parts can be reassembled anywhere and to designs tailored to other dates or significant numbers. All rush at once.

The Lattice Labyrinths workbook  is available from the publisher or you-know-who , or from a good independent bookshop , via Google or for two days only on the lawn behind the Memorial Hall in Wirksworth, where two striking giftwrap designs, “Logistical Nightmare” and “Diamonds are Forever” (oops – a copyright infringement, I take back that title), are also to be had at £1.50 a sheet, 10% to the Festival.

P.P.S. there are less arcane uses for the Tesslab Tile Yard:


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Vive la France; An Addition to the Grandeur of Paris for le Quatorze Juillet, Bastille Day, la Fête Nationale,

Bill Bryson remarks somewhere that a Martian visitor desiring to be taken to our leader would chose Vienna as the likeliest place to find him or her, judging by the magisterial grandeur of its imperial  architecture and the overbearing opulence of the baroque sculptures that strut their muscles and bosoms along the Ringstrasse.  Surely other capitals, in particular Saint Petersburg and  Washinghton D.C.,would be major contenders, scale-model London being a 100:1 outsider.  Whatever the Graeco-Roman grandeur of other capitals, however, I think that the archetype they all seek to emulate is PARIS.

So, on or around Bastille Day, it is time for a typically Gallic histrionic, over-the-top, ruthless and megalomaniac makeover of one of the city’s great public open spaces. The upset will be as nothing to that occasioned when les grandes Places et les Boulevards were originally butchered through the homes, workshops and lives of a teeming city, municipal destruction in scale and senselessness commensurate with  British “slum clearance” of the 1950’s and 1960’s , except that little very impressive came out of the latter.

Le Quatorze Juillet is the number pair (14,7), is of form one even, one odd, so yielding a Lattice Labyrinth tessellation of the Chinese family, specifically of a subfamily where the two numbers share a common factor,in this case 7.

Here is a little region of Chinese Lattice Labyrinth (14,7).

Chinese (14,7) FrenchThe outline of this figure does display that cockiness (suffisance?) which forms part of the French national self-image and can arouse animosity among les étrangers.

Below is the present unfinished state of the re-paving of the Place de la Concorde; batches of slabs in the national colours have to be specially manufactured.

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

Le classeur Lattice Labyrinthes est disponible auprès de l’éditeur ou de vous-savez-qui, ou à partir d’une bonne librairie indépendante ou via Google.


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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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