Author: Gabor Damasdi

In a small house in Portugal, the floor is tiled with quadrilateral tiles of the same shape and size (see figure). Find all four angles of the tile. Give your answer in degrees.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-46.jpg?height=940&width=1829&top_left_y=382&top_left_x=0)
At the center of the mosaic, 6 identical angles with a common vertex form a full angle, so each of them is equal to $360^{\circ}: 6=60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-46.jpg?height=829&width=1169&top_left_y=1538&top_left_x=-1)

A bit to the right, a full angle of $360^{\circ}$ is formed by the already known angle of $60^{\circ}$ and two more equal angles—thus, they each measure $150^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-47.jpg?height=757&width=1189&top_left_y=-1&top_left_x=0)

Similarly, another angle can be calculated: it is equal to $\left(360^{\circ}-150^{\circ}\right): 2=105^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-47.jpg?height=757&width=1189&top_left_y=839&top_left_x=0)

The last angle of the tile can be calculated by knowing the sum of the angles of a quadrilateral, or using another vertex of the mosaic: $\left(360^{\circ}-150^{\circ}-2 \cdot 60^{\circ}\right): 2=45^{\circ}$. Thus, the angles of the tile are $45^{\circ}, 60^{\circ}, 105^{\circ}$, and $150^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-47.jpg?height=777&width=1192&top_left_y=1718&top_left_x=-1)

## Comments.

1) Note also that such a tile is divided by one of its diagonals into an equilateral and an isosceles right triangle.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-48.jpg?height=386&width=252&top_left_y=1&top_left_x=0)

2) A tiling of the plane with polygons without gaps or overlaps is called a tessellation. Tessellations can be either periodic (there are two different directions in which the tessellation overlaps with itself when shifted) or aperiodic (such shifts do not exist). The problem presents a part of an aperiodic tessellation of the plane with such quadrilateral tiles, which was devised by Gábor Damásdi. However, such quadrilaterals can also tessellate the plane periodically (try to figure out how—comment 1 will help). Is there a polygon that can only tessellate the plane aperiodically? This is unknown!

## Answer

$45^{\circ}, 60^{\circ}, 105^{\circ}$, and $150^{\circ}$.