## Repaving Red Square for Lenin’s Birthday – вновь прокладывает Красную площадь

Vladimir Ilyich Lenin (Владимир Ильич Ленин) seems to have been a fortunately rare example of an idealist who had the political skill to enable him to try his ideas out in practice. According to John Gray (whose oratorical skills I admire but who’s books are an order of magnitude more numerous and bulkier than the ideas contained in them), Lenin’s attempt in “war communism” to totally abolish money and markets impoverished the Russian proletariat rather than enriching them as he’d expected and led to the starvation of millions of peasants.

Lenin’s  notorious mausoleum lies within Red Square, so despite the Russian Federation’s abandonment of communism , if we are going to re-pave this megalomaniacal (sic) space, Lenin’s birthdate is the obvious inspiration. April 22nd. (by the modern, not Orthodox calendar) gives us separation parameters (22,4), which can be set out on the triangular lattice to give a Honeycomb Lattice Labyrinth.

The basic tessellation of the Honeycomb family of labyrinths is of course the honeycomb tessellation of heaxagons. I think of the hexagon not as a primitive shape in itself but as a supertile made up of six equilateral triangles, set out on the triangular lattice with a separation of (1,1) along axes at 60 degrees to one another.Here is the familiar Honeycomb tessellation, which I call the basic tessellation (1,1) of the Honeycomb Lattice Labyrinth family.

It’s easyish to show that an equilateral triangle set out on the triangular lattice with separation parameters (e,f) has area e^2 + ef + f^2. Here’s a diagram to explain some of what I mean, with e =4 and f= 3 as an example.

For the honeycomb, with separation parameters (1,1), e^2 + ef + f^2 = 1^2 + 1×1 + 1^2 = 3 and the area of the supertile is twice this, 6 triangles. You can see from the above diagram that the hexagon surrounded by the red arrows has twice the area of the black-edged triangle. In general, the supertile of Honeycomb Labyrinth (e,f) has area equal to         2(e^2 + ef + f^2).

Lenin’s birthdate Honeycomb lattice Labyrinth (22,4) is a rather more Byzantine beast, with a supertile area of 2(22^2 + 22×4 + 4^2) = 1176 triangles. Here is just one supertile of the tessellation I have come up with after the usual hours of struggle.

and here, to show that it works is this supertile tessellating with six others, analogously, in fact homeomorphic to, the portion of the basic honeycomb shown above, once more in an approximation to the colours of the Russian Federation flag. Are those forking bits sticking out stylised hammers and sickles?

So now I call on Майкл, the architect in the blogteam, to apply this pattern to the paving of Red Square, which is overlooked by the crazy skyscape  of St. Basil’s Cathedral.

So, finally, here is Honeycomb Labyrinth (22,4), the tessellation is inspired by 22nd. April, the birth date of Vladimir Ilyich Lenin, Владимир Ильич Ленин,  employed to repave Red Square, Красный квадрат, in  colours  matching the terracotta, gold and green of St. Basil’s, Собор Василия Блаженного.

Thanks, Майкл. Another first for Russia. The Sputnik syndrome revisited, they’re well ahead of the West in the manufacture of equilateral triangular paving slabs. Alas, the phlegmatic pedestrians seem oblivious of the exciting mathematical puzzle beneath their feet. Not even the children are rushing to the magical hexagons on which the labyrinthine corridors converge.

P.S. Links to the how-to-do-it workbook: From the publisher, or via you-know-who , better,  from a good independent bookshop or just Google.