11. Pantheism – Joanie Lemercier

Joanie Lemercier (FR/BE)

Pantheism

Texts by William Hautekiet

1. Tiling of a plane

The tiling or tessellation of a plane is a set of shapes or tiles that fill a plane with no overlaps or gaps. These tiles are often polygons, i.e. triangles, quadrilaterals, hexagons, etc. By definition, the plane goes on infinitely, which means that the set of tiles can cover the surface infinitely.

You are most certainly familiar with examples of surface tilings, such as square floor tiling, mosaics, or the hexagons in a honeycomb.

But, do you know what type of polygons can fill a tiling of a plane, and thus cover it entirely through infinite repetition?

Reminder: a polygon is a flat and closed geometrical figure figure that is made up of several vertices, and straight line segments, or sides, that are joined at a vertex. Some common polygons are triangles, rhombi, squares, parallelograms, pentagons, hexagons, trapezoids, etc.

Let’s get back to our question and focus for the time being on regular polygons, which have equal side lengths and equal angles.

The answer is simple, but rather surprising. Only three types of polygons can tile a plane: a triangle, a square and a hexagon!

To tessellate a plane, the interior angle measure of a regular polygon must be a multiple of 360°. Use the following formula:  to find the interior angle measure of a regular polygon with n sides.

Tessellation of a plane can only work with equilateral triangles and their three 60° angles (360°=6×60°), with squares and their four 90° angles (360°=4×90°) or with regular hexagons and their six 120° angles (360°=3×120°).

Being that the interior angle of a pentagon is 108°, which is not a multiple of 360°, it is impossible for regular pentagons alone to tessellate a plane.




Regular tiling and pentagonal tessellation attempts

There are evidently many more possibilities if regularity is no longer required; it becomes possible to use polygons with unequal sides and angles.
Here are some examples of tessellations with irregular polygons.

Different irregular pentagonal tessellations

To construct a tessellation, the base pattern cannot be changed, but it can be repeated by using three types of geometric transformations: translation, rotation and symmetry.

So far, we have focused on periodic tiling.These tilings are constructed by infinitely reproducing an unchanged pattern, or bounded region of interest in the plane, through translations in at least two directions.

Play with the images below to verify the tiling periodicity!

But, is there such a thing as non-periodic tilings, with other characteristics?

The answer is yes. A slight change in the first pentagon tiling provides ample proof. It shows that a simple translation is no longer enough to tile the plane with the same pattern. These pentagons can tile the plane both periodically and non-periodically.

Periodic and non-periodic tiling

Mathematician Hao Wang believed it was always possible to construct a periodic tiling from an existing non-periodic tiling by using the same polygons. His student, Robert Berger, proved him wrong: he found a set of 20,426 different polygons that could cover a plane only in a non-periodic fashion.

In 1974, Roger Penrose found two polygons that cover a plane non-periodically. These two polygons – two quadrilateral “kite” and “dart” shapes – need to placed in such a way as to create red and green continuous curves.

The following animation shows that it is not possible to use translation in Penrose tiling without it changing.

Pavage de Penrose (animation)

In the following section, we will construct a Penrose tiling and show that it cannot be periodic. We will also find out how all this is related to the Fibonacci sequence and the golden ratio!
Finally, we will discuss the triangulation of a polygon, or how it is divided into triangles.

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