What is the picks set

The fabulous sentence of the mathematician Georg Pick

A single photo of Georg Pick is circulating on the Internet. It shows a young man with short hair and a hint of mustache. In addition, like fashion back then, a mighty tie and a daring look.

Pick was a little younger than Sigmund Freud, a little older than Arthur Schnitzler. He was born in Vienna-Leopoldstadt and studied mathematics at the University of Vienna - apparently with success, because he soon took up a position at the venerable Prague University, where he became professor and dean. His colleague was none other than the famous Ernst Mach. For a year, around 1910, Einstein was also part of Pick's faculty.

After a pleasant career, Georg Pick retired in 1929 and returned to his hometown. Less than ten years later, he had to save himself across the border to Prague. Hitler caught up with him there too. The eighty-year-old Pick was deported and killed in Theresienstadt in 1942. From Leopoldstadt to Theresienstadt - such a journey through life was not an individual fate.

Documents about Pick's life are rare. But he has left his mark on mathematics through demanding contributions in many areas, especially complex analysis. His most famous discovery, however, is a doctrine that requires no mathematical training at all and can be fun even for twelve-year-olds - proof included!

How big is a polygon

Even in elementary school you learn what a polygon is. You put the pen on a sheet of paper, draw a straight line, then without stopping the next, then the next and so on, without ever touching the already drawn line, until you finally return to the starting point. Then the polygon is ready. It consists of corner points and lines that connect them.

That makes triangles, octagons, dodecagons ... they may or may not be regular, it doesn't matter. And how big is the polygon? No problem. Even the most complicated polygon can be broken down into simpler ones: triangles, rectangles, trapezoids, for which the area can easily be calculated. You measure heights and widths, multiply, halve here and there and finally add everything up. Child's play.

Pick's polygons are drawn on squared paper. The drawing sheet is covered with a grid of thin lines, and the corner points of the polygons lie on the grid - where two grid lines intersect. Such polygons are called lattice polygons.

And for these grid polygons Pick has proven his theorem, namely that the area can be obtained by a simple recipe. Count all grid points on the edge (this includes the corner points, but maybe also other grid points on the lines) and divide this number by 2. Then count all grid points inside and add them to them. Then subtract 1. And done. The area has been found, and that by simply counting without measuring.

If you get the number 31, it means: The area is 31 times the size of the unit square (i.e. the smallest diamond on the squared paper). Exactly 31, by the way: not 31.04. It can only come out as an integer or an integer plus 1/2.

While you often have to be satisfied with approximate values ​​when measuring, Pick only counts: The grid points on the inside count completely, those on the edge count half. Such an edge point is, so to speak, only half of the polygon, it is neither completely inside nor completely outside.

Review of picks recipe

It's easy to check Pick's recipe with a few examples. With the uniform diamond - a single small square on our squared paper - there is no inner point and 4 corners at all. Each of these counts half, results in 2. Subtract one from this, results in area 1. Fits.

If you divide the small square even further, by a diagonal, you get two triangles. Such a triangle has 3 edge points, each one counts half. You subtract 1 and you have the area: 1/2. It never gets smaller.

Now you can try out the formula on larger rectangles parallel to the axes. The result: Pick's method always delivers the length times the width as it should be. The same applies to right-angled, axially parallel triangles, which you get when you split a rectangle by means of a diagonal.

And once you get that far, you will already be on the trail of the proof of Pick's theorem: After all, all grid polygons can be broken down into triangles. However, be careful! These do not necessarily have to be axially parallel right-angled triangles. So you still need a good idea. It is withheld from the inclined reading public. One does not reveal the resolution in the detective novel either.

Does the set of pick have applications? How you take it. It's kind of like asking if a good joke has any uses. A joke can provide deep insight and yet be of no practical use. But while the question is already in the room: Yes, Pick's theorem has remarkable applications - in mathematics!

Here is an example: One of the deepest findings is Euler's formula for polyhedra, i.e. for bodies in space that are bounded by polygons. The cube, for example, is a polyhedron. It has 6 faces, 12 edges and 8 corners. Euler's formula applies to all polyhedra that are convex (i.e. not hollow-cheeked or perforated).

It says: If the number of corners and surfaces is added, this results in the number of edges plus 2. The formula for the cube is correct: 6 + 8 = 12 + 2. With pyramids too. With rhombs also and with parallelepipeds, with prisms and icosahedra, dodecahedra and always! Regular solids played a central role in Greek mathematics - and yet the formula was only found in modern times: corners + surfaces = edges + 2.

Euler's formula for polyhedra does not seem to have anything to do with Pick's theorem: the polyhedra are three-dimensional, the polygons two-dimensional. With Pick the corners are on a grid, with Euler there is no grid. And yet the polyhedron formula follows amazingly simply from Pick's theorem.

A final example: Let us choose any natural number, say 5. The sequence of Farey F5 contains all fifths between 0 and 1 plus all quarters, all thirds and all halves (there is only one half, namely 1/2). Some fractions occur several times, so you take the shortened one: not 2/4, but 1/2. Let's order these fractions according to their size and add 0 and 1 as endpoints. We also write them as fractions, i.e. 0/1 and 1/1. So that gives

F5 = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1)

The main theorem for Farey sequences says: If you take two consecutive fractions, say 3/4 and 4/5, and multiply the numerator of the smaller by the denominator of the larger (i.e. 3 by 5), the product of the denominator of the smaller with the numerator of the larger (i.e. 4 times 4) is always 1 larger. Always at exactly 1. This applies not only to F5, but also to F23 and F276990 and to all Farey episodes in general.

That is a curious result, but the curious thing is that it follows directly from Pick's theorem. How can that be? In Farey, fractions between 0 and 1 are strung like swallows on a telephone wire. What does this have to do with bars? With polygons? With area? In fact, it has a lot to do with it - and again there is something magical about the trick of proof (which is easy when you know it). That's why Pick's fabulous theorem is considered a little mathematical jewel. His evidence is so insightful and beautiful that it almost looks cheerful - which makes the tragic end of its discoverer even sadder.

It's a shame that Pick's sentence doesn't appear in school material. The hours are so tight that there is no time for the fun aspects of math. This means that children are missing out on something, and so are adults. Many people enjoy thinking. They enjoy sudokus and many other games that require acumen. And often do not suspect what the ancient Greeks knew: that mathematics is the game of the gods. (Karl Sigmund, March 14, 2020)

Karl Sigmund is professor emeritus for mathematics at the University of Vienna.