Converting tilings with multiple types of rhombuses to pentagonal tilings

Countless tilings with multiple types of rhombuses exist, and several rotationally symmetric tilings with multiple types of rhombuses are known as well. By using pentagons that satisfy (2) or (3), the values of $\theta$ in (2) and (3) are important when a tiling with multiple types of rhombuses is converted into one with multiple types of pentagons, which is simply referred to as “pentagonal tiling.” The reason is because each pentagon must have the same value of $\theta$ (vertex $B$) to be “$B+D+E=360^{\circ}$” to hold within that pentagonal tiling.

As mentioned in [7], pentagons that satisfy (2) or (3) can be concave, convex, trapezoidal, or parallelogram depending on the parameters. Therefore, when a tiling with multiple types of rhombuses is converted into a pentagonal tiling, the pentagons in the tiling could be solely convex or concave, possess both a concave and convex shape simultaneously, etc. The pentagons that satisfy (2) with two parameters, $\alpha$ and $\theta$, are difficult to treat as an example because it leaves the value of $\alpha$ without a clear guideline after determining the value of $\theta$, which is an important factor in generating a pentagonal tiling. The pentagons that satisfy (3) have two parameters, $n$ and $\theta$, where $n$ is an integer of three or more. Specifically, a pentagon that satisfies (3) is easy to treat as an example, because the shape of the pentagon can be determined only by the value of $\theta$ after the value of $n$ is determined. (Of course, the pentagon that satisfies (3) is also the pentagon that satisfies (2), but here it is distinguished for the sake of explanation.)

Therefore, in this study, we utilize rotationally symmetric tilings with pentagons that satisfy (3) a case example where a tiling with multiple types of rhombuses is converted into a pentagonal tiling. Specifically, we introduce pentagonal tilings based on five-fold or eight-fold rotationally symmetric tilings with two types of rhombuses (i.e., the tilings are formed by two types of pentagons that satisfy (3)), and pentagonal tilings based on a 12-fold rotationally symmetric tiling with three types of rhombuses (i.e., the tilings are formed by three types of pentagons that satisfy (3)).

The properties of the shape of pentagons that satisfy (3) depending on the values of $n$ and $\theta$ are arranged below. If the pentagons that satisfy (3) exist geometrically, then $n>2$, because $C>0^{\circ}$. It is $0^{\circ}<\theta<180^{\circ}$ because the edge $e$ of the pentagon that satisfies (3) exists and does not intersect the edge $b$. As mentioned in Section 4 of [7], the pentagons that satisfy (3) with $n=3$ and $n=6$ correspond to rhombuses with an acute angle of $60^{\circ}$, and these pentagons are opposite to each other. The shapes of the pentagons that satisfy (3) change depending on the values of $n$ and $\theta$ are as follows:

1. Case where the pentagons that satisfy (3) admit $n=4$

   • •

     $0^{\circ}<\theta<90^{\circ}$ : Convex pentagons

   • •

     $\theta=90^{\circ}$ : Parallelograms ($B=180^{\circ}$)

   • •

     $90^{\circ}<\theta<180^{\circ}$ : Concave pentagons ($B>180^{\circ}$)

1. Cases where the pentagons that satisfy (3) admit $n\geq 5$

   • •

     $0^{\circ}<\theta<90^{\circ}-\dfrac{360^{\circ}}{n}$ : Concave pentagons ($E>180^{\circ}$)

   • •

     $\theta=90^{\circ}-\dfrac{360^{\circ}}{n}$ : Trapezoids ($E=180^{\circ}$)

   • •

     $90^{\circ}-\dfrac{360^{\circ}}{n}<\theta<90^{\circ}$ : Convex pentagons

   • •

     $\theta=90^{\circ}$ : Parallelograms ($B=180^{\circ}$)

   • •

     $90^{\circ}<\theta<180^{\circ}$ : Concave pentagons ($B>180^{\circ}$)

1. Cases where the pentagons that satisfy (3) become the polygons with $a=b=c=d=e$

   • •

     Case where $n=4$ and $\theta\approx 41.41^{\circ}$ (Convex pentagon with $E\approx 114.30^{\circ}$)

   • •

     Case where $n=5$ and $\theta\approx 37.95^{\circ}$ (Convex pentagon with $E\approx 149.76^{\circ}$)

   • •

     Case where $n=6$ and $\theta=30^{\circ}$ (Trapezoid of $E=180^{\circ}$)

   • •

     Case where $n=7$ and $\theta\approx 16.41^{\circ}$ (Concave pentagon with $E\approx 213.69^{\circ}$)

If the pentagons that satisfy (3) have $\theta=90^{\circ}$ (i.e., $B=180^{\circ}$), then they are parallelograms (the case of $n=4$ is a rectangle). Such parallelograms correspond to half of the rhombus corresponding to the basic unit of Figures 3.1(a) or 3.1(b).

If the pentagons that satisfy (3) have $E=180^{\circ}$, then “$A=D,\;B=C$.” Thus, they are trapezoids with a line of symmetry and $\theta=90^{\circ}-\frac{360^{\circ}}{n}$ holds. It is when $n\geq 5$ that the pentagons that satisfy (3) become trapezoids, because $\theta>0^{\circ}$.

If the pentagons that satisfy (3) have $a=b=c=d=e=1$, then $1=2\sqrt{1-\sin(2\alpha)\cos\theta}$ where $\alpha=90^{\circ}-\frac{180^{\circ}}{n}$. In this case, $n\leq 7$ for $\theta$ to exist.

By using pentagons that satisfy (2) or (3), when a tiling with multiple types of rhombuses is converted into a pentagonal tiling, there are two possible methods. The first method is to convert each rhombus in the tiling into two pentagons, as illustrated in Figure 4.1(a). The second method is to divide each rhombus in the tiling by its own similar figures and convert each rhombus formed by division into two pentagons. If a tiling with rhombuses is monohedral, the pentagonal tiling generated by conversion of the above methods is the same, but if it is a tiling with multiple types of rhombuses, the pentagonal tiling generated by the conversion differs depending on how the rhombuses are divided.

In the tiling with multiple types of rhombuses, the method of dividing a rhombus into its own similar figures has two rules that must be followed to be able to convert it into a pentagonal tiling. First, it is necessary to divide the base rhombus so that it contains $4\times u^{2}$ for $u=1,2,3,\ldots$ of its own similar rhombuses. Accordingly, when converted to a pentagon, the base rhombus contains $8\times u^{2}$ pentagons. To form a pentagonal tiling as illustrated in Figures 3.2(g), 3.2(h), etc., we combine basic units of the pentagonal pair with the anterior and posterior sides as illustrated in Figures 3.1(a) and 3.1(b). Specifically, when dividing the base rhombus into its own similar figures, the four divisions in Figure 4.1(b) are the standard units of division. Because the standard unit of division contains eight pentagons, with four pentagons each on the anterior and posterior sides, it is considered as “divide a rhombus into eight (congruent) pentagons” or “convert a rhombus into eight (congruent) pentagons.” Therefore, dividing a rhombus by its own similar figures, which can be converted into a pentagonal tiling, is allowed as follows: four divisions with one standard unit of division (refer to Figure 4.1(b)), 16 divisions with four standard units of division arranged in two rows and two columns (refer to Figure 4.1(c)), 36 divisions with nine standard units of division arranged in three rows and three columns, etc. The second rule is that each rhombus in the tiling must have the same number of divisions. For example, in the case where a tiling with two types of rhombuses, $H$ and $R$, is converted into a pentagonal tiling, if $H$ is divided into 16 by its own similar figures, $R$ is also divided into 16 by its own similar figures (i.e., unifying all rhombuses $H$ and $R$ in the tiling into 16 divisions).

In this subsection, we present pentagonal tilings based on the five-fold rotationally symmetric tiling with rhombuses of acute angles $72^{\circ}$ and $36^{\circ}$ discovered by Penrose, as illustrated in Figure 5.1 [1]. For this conversion, two types of pentagons corresponding to $n=5,10$ in (3) are utilized. When the pentagons that satisfy (3) are degenerated into trapezoids, the trapezoids are axisymmetric; hence, they have no distinction between their anterior and posterior sides. A shape of two trapezoids based on rhombuses, as illustrated in Figures 3.1(a) and 3.1(b), has two-fold rotational symmetry but no axis of reflection symmetry (refer to Figure A-3, etc.). Specifically, the shape has a distinction between the anterior and posterior sides. Therefore, in the figures of tilings in this study, the trapezoids that correspond to the posterior side arrangement are given an asterisk mark “*,” indicating that they are for the posterior side.

When combining two types of pentagons corresponding to $n=5,10$ in (3), their shapes in a tiling change depending on the value of $\theta$ are as follows:

• •

  $0^{\circ}<\theta<18^{\circ}$: $n=5,10$ are both concave pentagons with $E>180^{\circ}$

• •

  $\theta=18^{\circ}$: $n=5$ is a trapezoid ($E=180^{\circ}$) and $n=10$ is a concave pentagon with $E>180^{\circ}$

• •

  $18^{\circ}<\theta<54^{\circ}$: $n=5$ is a convex pentagon and $n=10$ is a concave pentagon with $E>180^{\circ}$

• •

  $\theta=54^{\circ}$: $n=5$ is a convex pentagon and $n=10$ is a trapezoid ($E=180^{\circ}$)

• •

  $54^{\circ}<\theta<90^{\circ}$: $n=5,10$ are both convex pentagons

• •

  $\theta=90^{\circ}$: $n=5,10$ are both parallelograms ($B=180^{\circ}$)

• •

  $90^{\circ}<\theta<180^{\circ}$: $n=5,10$ are both concave pentagons with $B>180^{\circ}$

In this subsection, we present pentagonal tilings based on the eight-fold rotationally symmetric tiling with rhombuses of acute angles $90^{\circ}$ and $45^{\circ}$ discovered by Ammann and Beenker, as illustrated in Figure 5.2 [1]. Of course, a rhombus with $90^{\circ}$ angles is a square. For this conversion, two types of pentagons corresponding to $n=4,8$ in (3) are utilized. As described in Subsection 5.1, the trapezoids corresponding to the posterior side in the tiling are given an asterisk mark. The pentagons corresponding to $n=4$ in (3) have a line of symmetry connecting the vertex $B$ to the midpoint of the edge $e$, i.e., there is no distinction between its anterior and posterior sides. A shape of square converted into two pentagons corresponding to $n=4$ in (3), as illustrated in Figures 3.1(a) and 3.1(b), has $D_{2}$ symmetry. In other words, because the shape has no distinction between its anterior and posterior sides (refer to Figure A-15, etc.), a unit comprising two anterior pentagons can be freely replaced with one comprising two posterior pentagons. Therefore, in the figures of tilings in this study, the pentagons with $n=4$ in (3) that correspond to the posterior side are not given an asterisk mark.

When combining two types of pentagons corresponding to $n=4,8$ in (3), their shapes in a tiling change depending on the values of $\theta$ are as follows:

• •

  $0^{\circ}<\theta<45^{\circ}$: $n=4$ is a convex pentagon and $n=8$ is a concave pentagon with $E>180^{\circ}$

• •

  $\theta=45^{\circ}$: $n=4$ is a convex pentagon and $n=8$ is a trapezoid ($E=180^{\circ}$)

• •

  $45^{\circ}<\theta<90^{\circ}$: $n=4,8$ are both convex pentagons

• •

  $\theta=90^{\circ}$: $n=4,8$ are both parallelograms ($B=180^{\circ}$)

• •

  $90^{\circ}<\theta<180^{\circ}$: $n=4,8$ are both concave pentagons with $B>180^{\circ}$

In this subsection, we present pentagonal tilings based on the 12-fold rotationally symmetric tiling with rhombuses with of acute angles $90^{\circ}$, $60^{\circ}$, and $30^{\circ}$ discovered by Socolar, as illustrated in Figure 5.3 [1, 4]. Of course, a rhombus with $90^{\circ}$ angles is a square. For this conversion, three types of pentagons corresponding to $n=4,6,12$ in (3) are utilized. As described in Subsections 5.1 and 5.2, in the figures of tilings in this study, the trapezoid corresponding to the posterior side is given an asterisk mark, and the pentagons with $n=4$ in (3) are not given an asterisk mark.

When combining three types of pentagons corresponding to $n=4,6,12$ in (3), their shapes in a tiling change depending on the values of $\theta$ are as follows:

• •

  $0^{\circ}<\theta<30^{\circ}$: $n=4$ is a convex pentagon and $n=6,12$ are concave pentagons with $E>180^{\circ}$

• •

  $\theta=30^{\circ}$: $n=4$ is a convex pentagon, $n=6$ is a trapezoid ($E=180^{\circ}$), and $n=12$ is a concave pentagon with $E>180^{\circ}$

• •

  $30^{\circ}<\theta<60^{\circ}$: $n=4,6$ are convex pentagons and $n=12$ is a concave pentagon with $E>180^{\circ}$

• •

  $\theta=60^{\circ}$: $n=4,6$ are convex pentagons and $n=12$ is a trapezoid ($E=180^{\circ}$)

• •

  $60^{\circ}<\theta<90^{\circ}$: $n=4,6,12$ are all convex pentagons

• •

  $\theta=90^{\circ}$: $n=4,6,12$ are all parallelograms ($B=180^{\circ}$)

• •

  $90^{\circ}<\theta<180^{\circ}$: $n=4,6,12$ are all concave pentagons with $B>180^{\circ}$