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