It’s fun to derive both theorems using complex numbers displayed on the Argand diagram. If you have not clicked off at the mention of *complex* and *Argand*, hello again. Firstly, here are the presentation slides as a *Powerpoint**,*** ** Trithagoras.Rev.

Possibly easier to understand than slides without words there follows a rather long-winded explanation; far more conscientious than could be included in the presentation at the MathsJam 2017 conference at the likeable conference centre Yarnfield Park, where the five minutes maximum duration was humorously but rigorously enforced.But first let’s look at a perhaps unfamiliar way of generating PYTHAGOREAN triples -the three integral sides of right-angled triangles.

The complex number** 8 + 3i**, for instance, can be represented as a point with coordinates **(8,3)** on the Argand diagram. It also represents a vector connecting this point with the origin with length (the *modulus* of the complex number) **√(8² + 3²)**. You can see from the diagram below how this can specify the figure of a square of area **(8² + 3²)**.

Note that, in general, complex numbers **(a,b)** where **a** and **b** are integral are known as *Gaussian integers* (see the diagram below).

The next figure below illustrates the procedure for multiplying two complex numbers, in this case **(3,2)** and **(4,1)**, to get their product, **(10,11)** and how that is equivalent to multiplying together two square figures, of areas **(3² + 2²)** and **(4² + 1²)** to produce a product square of area **(10² + 11²),** which equals **(3² + 2²)** **x** **(4² + 1²)**. I bet you will not resist checking that this multiplication works.

Now suppose we take a complex number **(a,b)**, that is **a + bi**, and multiply it by itself, in other words square it. The result is the complex number **((a² – b²), 2ab)**, specifying a product square figure of area** (a² – b²)² + (2ab)²**. In the figure below I have chosen to illustrate the specific case **(2,1)** the smallest Gaussian integer **(a,b)** for which **a ≠ b ≠ 0**.

Now suppose we switch the real and imaginary coordinates to get the number** (b,a)**. on the Argand diagram the means we have equivalent to reflected **(a,b)** and its vector about the diagonal line at 45 degrees to the real (x) axis. Suppose we multiply **(a,b)** by **(b,a)**; you could call this reflective squaring. This time the product is **(0, (a² + b²) **which having a zero real part is represented by a point lying on the imaginary (y) axis of the Argand diagram. This has to be the case as, when you multiply two complex numbers you multiply their moduli but ADD their *arguments*, that is the angles their vectors make with the real (x) axis. As the vectors of **(a,b)** and **(b,a)** are symmetrical about a 45 degree angle, their product must have an argument of 90 degrees. (Thanks to John Read of Nottingham MathsJam for pointing out what I hadn’t noticed). For **(2,1)**, the reflective product is **(0,5)**. As the modulus (vector length) of **(2,1)** is obviously the same of that of **(1,2)**, the product square of the reflective squaring must have the same area as that of the squaring, so **3² + 4² **must equal** 5²**, and so, of course, it does. We have found the lowest order Pythagorean triple **3,4,5**.

So many words – below are just two pictures that tell it all.

And here is the familiar figure, linking the triples to their corresponding Pythagorean triangles.

All the Pythagorean triples can (with some complication in a relatively few special cases) be found in this way. Before moving on to Trithagoras, it’s interesting to see what happens if we multiply two DIFFERENT complex numbers, and then reflection multiply them.

You can call the result *Pythagorean quads*,** a,b;c,d** specifying complex numbers **(a,b)** and **(c,d)** such that **(a² + b²)** = **(c² + d²). **The figure below shows how this relates to Pythagorean triangles; we now have not one right-angled triangle with squares constructed on its sides but two right-angles triangles which share a common hypotenuse, no longer itself of integral length.

If we then repeat the process multiplying and reflection both** (a,b)** and** (c,d)** by another number **(e,f)** one can generate Pythagoraen sextuplets. For instance, taking **(-1,8)** and **(4,7)**, illustrated in the figures above (we can interchange **a** and **b**, using use **(8,1)** instead of** (-1,8)** in order to avoid negative values) and multiply, reflection multiply them by **(2,1)** we find that:

**(2,1) x (4,7) = (1,18) (2,1) x (7,4) = (10,15) (2,1) x (8,1) = (15,10) (1,2) x (8,1) = (6,17) all pairs of squares sum to 325**

So, **(10,15)** being essentially the same as** (15,10)**,we now have three pairs of squares, with the same sum. Carrying on this way, we can eventually find the smallest number, **1105**, which is the sum of FOUR different pairs of squares. It is no coincidence that **1105** has factors **5**,**13** and **17** – the hypotenuses of the first three *primitive* (excluding multiples of lower-order cases) Pythagorean triangles.

We follow the same procedure for equilateral triangles as we have just employed for squares. First of all how we construct an equilateral triangle on the Argand diagram.

As you can see, I’ve set out the unit triangular lattice on the Argand Diagram. The points shown are the equivalent on the triangular lattice of the Gaussian integers that make up the unit square lattice. They are known as Eisenstein integers. They are integral with respect to the axes at** 60º** but the real and imaginary parts of an Eisenstein integer** (a,b)** are **(a + b/2)** and **b.√3/2**. This affects the rule for multiplying two Eisenstein integers (see below).

The above diagram also show how the vector represented by an Eisenstein integer specifies an equilateral triangle. Measured in terms of the number of the unit **(1,0)** triangle, the area of this triangle can be shown to be** a² + ab + b²**. Numbers of this form are called *Loeschian numbers*. In the diagram to the right you can see that the area of the red triangle equals half the area of the three parallelograms plus the area of the small triangle **= 3ab + (a-b)²**. QED

(((Experts need to ignore the insignificant annoyance that Eisenstein used axes at 120º; I don’t because to do so would introduce inelegant and vexatious negatives. Inelegant because, using Eisenstein’s axes, the area of a triangle works out as a² – ab + b²; vexatious because thus the modulus of a point on the lattice (and the corresponding triangle area does not increase monotonically as a and b increase, an increase in b can cause the modulus, and triangle area to decrease ))).

The above figure shows how, analogously to multiplying squares on the square lattice we can multiply equilateral triangles on the triangular lattice. Remembering that the real and imaginary parts of **(a,b)** are **(a + b/2)** and **b.√3/2**. we can slog through the algebra and find that the rule for multiplying together two Eisenstein integers **(a,b)** and **(c,d)**and the triangles they specify is **(a,b) x (c,d) =** **( ac-bd, ad+bc+bd)**; we have an extra term, **bd**, over and above the Gaussian formula. Now we can see what happens when we square and reflection square a single Eisenstein integer and the corresponding triangle.

We’ve generated in each case two triangles with the same area, but one having a zero coordinate, yielding TRITHAGOREAN TRIPLES, the equivalent for equilateral triangles of Pythagorean triples for squares. Below you see the equivalent of the classic Pythagorean triangle figure for Trithagorean triples.

The beautiful property of these figures is that as for instance **3² + 3×5 + 5² = 7²**, the area of each big blue triangle equals the sum of the areas of the two smaller triangles PLUS THE AREA OF THE GREY *TRITHAGOREAN TRIANGLE* ,the analogy to the Pythagorean triangle that links three squares.

Those lovely figures (and we can carry on indefinitely generating all the higher order triples) are the real crux of this post, but I can’t resist finishing with a remarkable link to the Hardy-Ramanujan taxicab number, **1729**. Just as in the Pythagorean cases, you can generate quads, that is Eisenstein integers **(a,b)** and **(c,d)** such that the corresponding Loeschian numbers and the triangles of which they are the areas are the same, that is, ** a² + ab + b²** = **c² + cd + d²**. Here are two examples,

and the figure below shows how these triangles can be related together. Now we have not a single grey triangle in the middle, but two, sharing their longest side, analogous to the sharing of a hypotenuse in the Pythagorean case.Trithagorean triangle diagram above and analogous to the quads figure in the Pythagorean half of this post.

Once again, we can keep on the process of multiplying and reflection multiplying until, to my amazement, we find that the lowest Loeschian number and triangle area which can be made in FOUR different ways, corresponding to four different Eisenstein integers,** (a,b)**,**(c,d)**, **(e,f)**, **(g,h)**, say, – actual numbers are in the above figure – is the celebrated Hardy-Ramanujan taxicab number **1729**. So we have:

**a² + ab + b²** = **c² + cd + d²** = ** e² + ef + f²** = **g² + gh + h² = 1729.**

Apart from its famous property of being the lowest number that is the sum of two positive cubes in two different ways** 1729 = 7 x 13 x 19**, the product of the three lowest primitive Trithagorean “hypotenuses”, just as in the corresponding Pythagorean case we had **1105 = 5 x 13 x 17**. The correspondence raises the intriguing question as to whether **1105** has at least one other very special property. Ramanujan would have instantly tendered an answer. I would love to hear from a reader what his answer might have been.

No wonder I didn’t get through all that very clearly in five minutes at Yarnfield Park.

PS These days almost all the known properties of almost any number can be found by checking with the On-Line Encyclopedia of Integer Sequences. Entering **1105,1729** into the search field comes up with ten sequences in which these appear consecutively – most of them closely related. In particular they are the second and third lowest *Carmichael numbers* https://oeis.org/A002997 and the first and second lowest numbers which are *pseudoprimes both to base 2 and to base 3*, https://oeis.org/A052155.

The **Lattice Labyrinths** workbook is available from the publisher or you-know-who , or from a good independent bookshop or, if clicking on these makes you anxious, via Google.

..but, writing on Tuesday 5th. September, the question is, “Will it be finished in time?” the snapshot below give some idea of the 99% perspiration involved. Thanks to Kwik Split for supplying Raimondi hexagonal tile dividers – a rare breed.

Furthermore, as I write my good friend Jacob the Joiner, busy with Middleton-by-Wirksworth community endeavours, is still to cut up the 6 x (64 + 72 + 81) = 1302 equilateral triangles needed, from mdf boards (plywood would spall under the saw), two of which are to be seen above, each anointed with seven coats of paint, two of which are “magnetic”. The time is ripe to prototype laser-cutting from metal sheets.

Update of Thursday 7th: the redoubtable and obliging Jacob Butler has nearly finished cutting the triangles. Here he is at work. First the boards are cut into strips, then fed in to the saw again, about five strips at a time at a 60 degree angle. After each cut, each strip is turned over and…hey presto! Jacob’s table saw is an excellently solid and accurate piece of kit, but not surprisingly, however, he recommends I root out a CNC (Computer Numerical Controlled) router next time. The last time he experienced such repetitive tedium was when Harrod’s ordered 100 (was it?) identical wooden toy trains.

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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According 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)**.

The 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:

Just 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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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:

This 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 **3³**, no tessellation is possible because both these powers of a prime are of form **4m+3**, but if **3²** 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 **2³** cannot be factors of a Loeschian number but **2²** 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 =** **2 ^{5}**

**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!)

* 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:

At 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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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.

Thanks 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”.

**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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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.

Thank 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.

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

Here are some pictures of previous Apple Days.

It’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.

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

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.

and 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).

and 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.

Other 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:

]]>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)**.

The 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.

The **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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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.)*

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.

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.

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)**.

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:

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.

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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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.

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.

Each 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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