A chiral aperiodic monotile
- Author:
- Szilassi Lajos
The Einstein (ein Stein) problem
Which are the plane geometric shapes whose not overlapping copies can be used to cover the plane in a single layer without gaps?
Is there a finite number of shapes that are suitable for aperiodic tessellation, i.e. shapes that cannot be transferred into themselves by a shift?
The question has a serious history and present in the history of mathematics.
We now turn to the article reported here.
Surely it would be easier, more exciting, to cover a piece of the plane with tangible copies of the basic shape.
Instead, we give readers the Geogebra applet below to experiment with such fits.
In the applet below, N and M are dynamically movable points.
- N is the interior point of the starting element, this is used to select the vertex that we want to insert into the vertex selected by M on the edge of the already picked element.
- If N is placed at the selected vertex of the initial element, and M is at one of the vertices of the already deposited shapes, the slider controlling the rotation of the element to be deposited is displayed, with the possibility of fixing the element that was set at the selected position.
- While using the applet, it is possible that the "ghost in hand" has been placed in the wrong place, and there is the possibility to undo the hasty fixation. It may even be that we find out much later that the stoning cannot be continued somewhere, so we can retrieve several items.
- If we use the Editing option to suspend the work, we can see which stoning was considered resumable by the authors of this article. Of course, this does not mean that this is the only way to go. For example, the layout given in the original article is slightly different from the layout shown in the applet below.
Readers familiar with the use of GeoGebra spreadsheets should note that by downloading this program and using it in offline mode, you will have the opportunity to
- resume the tesellation work that has been temporarily suspended;
- colour the elements you have laid down as you wish;
- extend the programme, which is capped here at 100 elements;
- analyse the mathematical relationships between the elements deposited, show that this chiral element is indeed suitable for aperiodic coverage of the entire plane, at the same time it is not capable of covering the plane in a periodic way.