?Maths : The challenge for the sixth crown , it can be

Can a plane figure pave a large area without it being possible to pave the entire plan? This problem called Heesch landed since 1968.

Mathematicians interested in tilings of plane geometric shapes as simple as possible. They rank considering their symmetry, design algorithms that discover new, study forms allowing only aperiodic patterns, etc. Alongside these classic problems, some not address what happens on the entire plan, but what is happening locally around a point or a pad: these are the problems of entourages. We present two. The first is solved today. The second, despite steady progress, remains mysterious.

Let's start with a basic conundrum in appearance. A flat shape we give a pad P. Can we totally surround a crown composed of copies of P, leaving no space between pavers used? What is the minimum number of copies of P needed for such surroundings?

A square paved surrounds without difficulty six identical squares (below, a), and is the minimum; best, all rectangle (! not square) is surrounded by four copies of itself (b); more preferably, some pavers surround with only three copies of themselves (c).

Avoid any misunderstanding about the word crown. The ring around a block P must have a certain non-zero thickness e: the minimum distance between a point external to the blocks and a central point of the pad P must always be greater than e. Pavers around a block P (or a set of blocks E) are considered a crown for P (or E) if the removal of a single pad of the crown is contacted outside the box paved with the central pad (or E).

To find out if it is possible to surround a pad P by a ring of two P cobblestones, take a pen and paper and try! (See figure below if you can not do.)

The pentagon Heesch: a single crown

The second issue that will occupy around us originates a small German book published in 1968 when the mathematician Heinrich Heesch (1906-1995) formulated an unexpected remark. He presented a pentagon with angles 90 °, 150 °, 90 °, 150 ° and 60 ° and with the strange property: a copy of the pentagon is surrounded Heesch perfectly without overlapping or space left empty by six seven or eight copies of itself forming a crown, and yet it is impossible to carry out a second ring around the first. One checks without difficulty ...

Jean-Paul Delahaye

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