On stones, a triangle full of holes

This image represents a “Sierpiński” triangle, drawn in chalk on Place Carnot, in Lyon. We created it at the age of twelve, including five children, in two hours, under the project # streetmath.

The goal of #StreetMath is to create artistic and civic works of sports inspiration. It is an urban exploration laboratory open to all. Our business in Lyon’s yards is ephemeral and disappears with the rain. If you live in Lyon or not far, Come join us !

So what were we drawing that day?

A map where each point is a crossroads?

We are reasonably disappointed with the cheese full of holes – there is almost nothing to eat! On the other hand, mathematically, such cheese can be very attractive.

An example of a holly object is the Sierpiński triangle. It is named after the Polish mathematician Wacław Sierpiński who, thanks to this triangle, solved, in 1915, the following problem: to find a map of the ways in which All The point will be a crossroads. We immediately feel that this card must be very complex…

Here’s how to create a Serpiskie Triangle. Take a full equilateral triangle. Next, draw a triangular hole, the vertices of which are located in the middle of the edges of the initial triangle – this takes up a quarter of its area. Keep removing the triangle in the middle of every triangle that appears… until infinity! The figure obtained in the limit no longer has any space at all, because at each stage of this slimming the total area is multiplied by 0.75 (the remaining three quarters).

And it’s roughly a Sierpińsky map – apart from the three corners of the first triangle where our bike has to rotate 60 degrees. By arranging six Sierpiński triangles together into a hexagon, the angle problem has been solved and our hexagon map is ready!

Sierpiński triangle, line by line

Find a cobblestone place. Grab some chalk, invite passersby to join you, and get started.

The color of one of the cobblestones (preferably in the middle of the edge of the square). Now equals 1, the other empty squares equal 0. This will be the first square on top of Sierpiński’s triangle. In the line below this first square, each piece will be the “sum” of the pieces above it, with a specificity: 1 + 1 = 0! Go down line by line. You’ll see Sierpiński’s triangle appear, in a “polka dot” version!

This algorithm allowed us to draw the Sierpiński triangle of the image in 128 lines. The attraction of this pattern lies in its “self-similarity”: it is made up of three smaller copies of itself, each a triangle with 64 square sides. Thus, the large figure can be separated into three independent parts, on which three groups act simultaneously. A little practical advice: First make crosses on the cobblestones before coloring them. This will make it easier to fix a bug that can spread quickly – and give a stretch seashell drawing.

Moreover, this triangle looks more rectangular than it really is – but that’s just the effect of perspective!

One question remains

You might be wondering why the drawing process that follows the algorithm line by line produces an image of a Holly triangle, similar to a Serpiski triangle? This is a very good question! And that is exactly the purpose of mathematics: to find connections between things that seem to have nothing in common, but which turn out to describe the same thing.

The basic idea here is that the rows of squares whose numbers are Powers of 2 (1, 2, 4, 8, 16, etc.) They are all filled and each forms the base of a perforated triangle. This also means that the lines following these solid lines will have only two squares filled in: one at the beginning and one at the end of the line. It is these two cobblestones that each result in a new triangle with holes, similar to the one shown above.