In this section, we first consider spiral tilings of cyclic symmetry shaped by the logarithmic spiral. Then, based on the cyclic spiral tilings, we introduce several simple conformal mappings to construct derived spiral tilings.
Cyclic spiral tilings
The logarithmic spiral is a spiral whose polar equation is given by
$$\begin{aligned} \rho (\theta ) = e^{k \theta }, \end{aligned}$$
(2)
where \(\theta \in \mathbb {R}\) and \(k \in \mathbb {R}^+\). The k parameter determines the rate of growth. The spiral has the property that the angle \(\alpha \) between the tangent and radial line at any point of the logarithmic spiral is constant. Moreover, the angle \(\alpha \) and the parameter k are related to each other by the relationship \(k = \cot \alpha \). Examples of logarithmic spirals for various values of k are presented in Fig. 2.
Assume \(M \ge 2\) and \(N \ge 2\) are integers, where \(M = m N\) for some \(m \in \mathbb {N}\), i.e., N|M. Moreover, assume that \(k \in \mathbb {R}^+\). In Fig. 3, let the origin and the positive x-axis of the Cartesian coordinate system coincide with the pole and the polar axis of the polar coordinate system. Denote by \(F_k(M, N)\) the polar region bounded by lines
$$\begin{aligned} \rho _1: \theta= & {} \frac{2 \pi }{M}, \end{aligned}$$
(3)
$$\begin{aligned} \rho _2: \theta= & {} 0 \end{aligned}$$
(4)
and spiral arcs
$$\begin{aligned} \rho _3(\theta )= & {} e^{k \theta } \text { with } \theta \in \left[ 0,\frac{2 \pi }{M} \right] , \end{aligned}$$
(5)
$$\begin{aligned} \rho _4(\theta )= & {} e^{k (\theta + \frac{2 \pi }{N})} \text { with } \theta \in \left[ 0, \frac{2 \pi }{M} \right] . \end{aligned}$$
(6)
Let \(\mathbb {C}\) be the complex plane, and \(\arg (z) \in [0, 2 \pi ]\) be the main argument of \(z \in \mathbb {C}\). Then, in the Cartesian coordinate system, the \(F_k(M, N)\) region can be expressed as
$$\begin{aligned} \begin{aligned}&F_k(M, N) = \{z \in \mathbb {C} | 0 \le \arg (z) \le \frac{2 \pi }{M} \text { and }\\&\quad e^{k\arg (z)} \le |z| \le e^{k\left( \arg (z) + \frac{2 \pi }{N}\right) }\}. \end{aligned} \end{aligned}$$
(7)
Define two transformations \(\varphi \) and \(\psi \) from \(\mathbb {C}\) to \(\mathbb {C}\) as
$$\begin{aligned} \varphi (z)&= z \cdot e^{\frac{2 \pi }{N}i}, \end{aligned}$$
(8)
$$\begin{aligned} \psi (z)&= z \cdot e^{-\frac{2 \pi }{M}(k+i)}, \end{aligned}$$
(9)
where \(i = \sqrt{-1}\). According to the Euler’s formula \(e^{i\theta } = \cos \theta + i \sin \theta \), it is clear that \(\varphi \) is a counterclockwise rotation of \(\frac{2 \pi }{N}\) about the origin. Note that \(\psi \) can be rewritten as
$$\begin{aligned} \psi (z) = z \cdot e^{-\frac{2 \pi }{M} k} \cdot e^{-\frac{2 \pi }{M}i}. \end{aligned}$$
(10)
Now, we clearly see that the effect of \(\psi \) is equivalent to first a scaling transformation with a scale factor \(e^{-\frac{2 \pi }{M} k} < 1\) (i.e., reduction transformation) followed by a clockwise rotation of \(\frac{2 \pi }{M}\) about the origin. Consequently, essentially \(\psi \) is a clockwise spiral reduction.
Consider an infinite Abelian symmetry group \(G_k(M, N)\) generated by \(\varphi \) and \(\psi \), namely \(G_k(M, N) = \langle \varphi , \psi \rangle \). Next, we consider the action of the tile \(F_k(M, N)\) under \(G_k(M, N)\).
Let
$$\begin{aligned} A = \bigcup ^{+\infty }_{p = -\infty } \psi ^{p}(F_k(M, N)) \end{aligned}$$
(11)
be the region tiled by \(F_k(M, N)\). Note that geometrically \(\psi ^{-1}(z) = z \cdot e^{\frac{2 \pi }{M}(k + i)}\) is a counterclockwise spiral expansion (with the scale factor \(e^{\frac{2 \pi }{M} k} > 1\)). Thus, A consists of tiles obtained by a continuously spirally reducing and expanding tile \(F_k(M, N)\). Therefore, geometrically A is a spiral arm. On the other hand, by the definition of A and \(F_k(M, N)\), we see that \(\varphi ^q(A)\) (\(q = 0, 1, \ldots , N-1\)) perfectly cover \(\mathbb {C}\) without gaps or overlaps. More precisely,
$$\begin{aligned} \mathbb {C} = \bigcup ^{N-1}_{q = 0} \varphi ^{q}(A) = \bigcup ^{N-1}_{q = 0} \bigcup ^{+\infty }_{p = -\infty } \varphi ^q(\psi ^{p}(F_k(M, N))). \end{aligned}$$
(12)
Equation (12) gives a concise approach for constructing spiral tilings of cyclic symmetry, see Fig. 4 for an example presenting a spiral tiling of the \(G_{0.15}(9, 3)\) symmetry.
Given \(k \in \mathbb {R}^+\), integers \(M\ge 2\) and \(N\ge 2\) satisfying N|M, by the method described above, one can construct a corresponding cyclic spiral tiling of \(G_k(M, N)\) symmetry. For convenience, we call the tiling as \(G_k(M, N)\) tiling. It is clear that \(F_k(M, N)\) is a fundamental region associated with the \(G_k(M, N)\) tiling, i.e., it is a connected set whose transformed copies under the action of a symmetry group cover the entire space without overlapping, except at boundaries.
The spiral tiling \(G_k(M, N)\) is defined by three parameters k, N, and M. These parameters give an easy way to change the structure of the tiling. In Fig. 5, we present examples of various tilings generated with the proposed construction method and varying values of the parameters. The k parameter is responsible for the rate of growth of the spirals, see Fig. 5a, b in which we fixed \(M = 12\), \(N = 6\), and changed k. The next parameter, N, is responsible for the number of the spiral arms that form the tiling, see Fig. 5c, d in which we fixed \(k = 0.3\), \(M = 15\), and changed N. Finally, the M parameter is responsible for the division along the spiral arm, see Fig. 5e, f in which we fixed \(k = 0.25\), \(N = 4\), and changed M.
Because \(F_k(M, N)\) is the fundamental region, so each \(z \in \mathbb {C}^* {\setminus } F_k(M, N)\) is a transformed copy of some \(z_0 \in F_k(M, N)\) under the action of the symmetry group \(G_k(M, N)\) (see equation (12)). To generate Escher-like spiral tessellations, we need a fast algorithm that transforms the given \(z \in \mathbb {C}^* {\setminus } F_k(M, N)\) to a corresponding point \(z_0 \in F_k(M, N)\). The algorithm can be divided into the following four steps:
-
1.
Let \(n_1 = \left\lfloor \frac{\arg (z)}{\frac{2 \pi }{M}} \right\rfloor \in \mathbb {Z}^+\), where \(\lfloor \chi \rfloor = \max \{ q \in \mathbb {Z} | q \le \chi \}\), and \(z_1 = \psi ^{-n_1}(z)\). We have \(0 \le \arg (z_1) \le \frac{2 \pi }{M}\).
-
2.
Suppose \(e^{2 n_2 k \pi } \le z_1 \cdot e^{-\arg (z_1)(k + i)} \le e^{2(n_2 + 1)k \pi }\) for certain \(n_2 \in \mathbb {Z}\). (Note that \(z_1 \cdot e^{-\arg (z_1)(k + i)}\) is a real number.) Let \(n_2 = \left\lfloor \frac{\ln |z_1| - k \arg (z_1)}{2k \pi } \right\rfloor \) and \(z_2 = z_1 \cdot e^{-\arg (z_1)(k + i)} \cdot e^{-2 n_2 k \pi }\). Then, \(z_2 \in [1, e^{2k \pi }]\) is a real number.
-
3.
Let \(n_3 = \left\lfloor z_2 \cdot e^{-\frac{2k \pi }{N}} \right\rfloor \in \mathbb {Z}^+\) and \(z_3 = z_2 \cdot e^{-\frac{2 n_3 k \pi }{N}}\). Then, \(z_3 \in [1, e^{\frac{2k \pi }{N}}]\).
-
4.
\(z_0 = z_3 \cdot e^{\arg (z_1)(k + i)} \in F_k(M, N)\).
For convenience, we summarize the algorithm as pseudocode in Algorithm 1.
Derived spiral tilings
A mapping is conformal (angle preserving) at a point if it preserves oriented angles between curves through the point with respect to their orientation. Conformal mappings preserve angles and shapes of infinitesimally small figures, but not necessarily their size. Now, we introduce three conformal mappings with the help of which we can easily construct derived spiral tilings.
Let \(\mathbb {S}^2 = \{ (x, y, z) \in \mathbb {R}^3 | x^2 + y^2 + z^2 = 1 \}\) be the unit sphere in \(\mathbb {R}^3\), and \(P_1 = (0, 0, 1) \in \mathbb {S}^2\) and \(P_2 = (0, 0, -1) \in \mathbb {S}^2\). The stereographic projection of \(\mathbb {S}^2\) to \(\hat{\mathbb {C}} = \mathbb {C} \cup \{ \infty \}\) from \(P_1\) is the map \(\Phi _1 : \mathbb {S}^2 \rightarrow \hat{\mathbb {C}}\) given by the following formula
$$\begin{aligned} \Phi _1(x, y, z) = {\left\{ \begin{array}{ll} \frac{x}{1 - z} + \frac{y}{1 - z} i, &{}\quad \text {if } (x, y, z) \ne P_1, \\ \infty , &{}\quad \text {if } (x, y, z) = P_1, \end{array}\right. } \end{aligned}$$
(13)
where \((x, y, z) \in \mathbb {S}^2\). The inverse map \(\Phi _1^{-1} : \hat{\mathbb {C}} \rightarrow \mathbb {S}^2\) of \(\Phi _1\) is given by the formula
$$\begin{aligned} \Phi _1^{-1}(z) = {\left\{ \begin{array}{ll} \left( \frac{2 a}{a^2 + b^2 + 1}, \frac{2 b}{a^2 + b^2 + 1}, \frac{a^2 + b^2 - 1}{a^2 + b^2 + 1} \right) , &{}\quad \text {if } z \in \mathbb {C}, \\ P_1, &{}\quad \text {if } z = \infty . \end{array}\right. } \end{aligned}$$
(14)
where \(z = a + bi \in \hat{\mathbb {C}}\). Note that \(\Phi _1^{-1}\) maps the origin 0 to \(P_2\) and \(\infty \) to \(P_1\). Figure 6a illustrates a spherical spiral tiling obtained by projecting a cyclic spiral tiling onto \(\mathbb {S}^2\).
Let \(\Phi _2\) and \(\Phi _3\) be, respectively, mappings from \(\mathbb {C}\) to \(\mathbb {C}\) defined as
$$\begin{aligned} \Phi _2(z) = \frac{z - i}{z + i} \end{aligned}$$
(15)
and
$$\begin{aligned} \begin{aligned} \Phi _3(z) = \Phi _3(x + iy)&= \tan (x + iy) = \frac{\sin (x + iy)}{\cos (x + iy)} \\&= \frac{\sin x \cosh y + i \cos x \sinh y}{\cos x \cosh y - i \sin x \sinh y}. \end{aligned} \end{aligned}$$
(16)
It is easy to check that the one-to-one linear rational transformation mapping \(\Phi _2\) maps i to 0 and \(-i\) to \(\infty \), whereas \(\Phi _3\) is a periodic mapping about the real axis with the period \(\pi \). Note that \(\mathbb {C}_j = \{z^* \in \mathbb {C} | z^* = \Phi _j(z),\ z \in \mathbb {C}\}\) for \(j = 2, 3\) are complex planes reshaped by the mappings \(\Phi _j\). Given a group \(G_k(M, N)\), for all \(z^* \in \mathbb {C}_j\), by Algorithm 1, we can transform \(z^*\) into a symmetrically placed point \(z_0 \in F_k(M, N)\). Thus, similar to the construction of spiral tilings shown in Fig. 4, we can produce derived spiral tilings characterized by \(G_k(M, N)\) and \(\Phi _j\) that have two or more spiral whirlpools, see the examples demonstrated in Fig. 6b, c.