Pavimenta: A Photo Tiling Tool

Year: 2011 Authors: António Salgueiro

Core claim

A simple interface can apply wallpaper-group symmetries to user images in real time to create beautiful tilings.

Topics

wallpaper groups, plane tilings, image symmetry, real-time transformation, ornamental design

Domains

group theory, Euclidean geometry, orbifolds, linear algebra, crystallographic symmetry, digital art, pattern design, photographic tiling

Methods

fundamental regions, isometry group selection, rotation translation reflection, interactive image tiling, real-time rendering

Media

digital images, photographs, tiling interface, computer-generated patterns

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2011: Mathematics, Music, Art, Architecture, Culture

Pavimenta: A Photo Tiling Tool

António Salgueiro Departamento de Matemática Universidade de Coimbra ams@mat.uc.pt

Abstract

It is well known that there are seventeen wallpaper groups, that generate distinct tilings of the plane. Some of the famous Escher drawings are based on these groups. The Pavimenta program [2] uses these seventeen groups to generate tilings of the plane using fundamental regions taken out of any image. Its ease of use makes it straightforward to create surprisingly beautiful tilings.

1 The Seventeen Wallpaper Groups

Plane tilings have been used throughout the centuries to create artistic geometric figures. All seventeen plane isometry groups have appeared in ornamental creations, although only in the late 19th century Fedorov and Schoenflies have proved that there are only seventeen of these groups. A simple proof comes from the fact that the quotient of the plane by a plane isometry group is an euclidean 2-orbifold, which has null Euler characteristic. The Euler characteristic $\chi (\mathcal{O})$ of an orbifold with underlying surface $S$, $k$ cone points of orders $n_1,\dots ,n_k$ and $t$ corner points of orders $m_1,\dots ,m_t$ is

Since the only surfaces with nonnegative Euler characteristic are the sphere (with $\chi (S)=2$), the projective plane and the disk (with $\chi (P)=\chi (D)=1$), the torus, the annulus, the Klein bottle and the Möbius band (with $\chi (T)=\chi (A)=\chi (K)=\chi (M)=0$), a straightforward calculation shows that there are only 17 possibilities for $\{S,\{n_i\},\{m_j\}\}$.

These plane symmetry groups have been described by different notations, some of which are given in the table below. The Montesinos notation [1], indicates the underlying surface of the quotient orbifold, the orders of the cone points, and the orders on corner points, in the form $S{n}_1\dots n_k(m_1\dots m_t)$.

2 The Pavimenta Program

There are several programs that generate tilings of the plane using the seventeen plane isometry groups. It is possible, using these programs, to make a drawing on the plane and see automatically how it is reproduced by the isometry group selected. The Pavimenta program, besides being capable of doing this, lets the user choose any image and see the effect of the isometry group on it. The interface is simple, with three areas on the screen.

The upper right area has seventeen buttons corresponding to the plane isometry groups. Clicking each one of these buttons changes the image on the main region. The lower right area shows the fundamental region associated with the selected isometry group. This fundamental region is either a triangle or a quadrilateral, according to the isometry group selected. Since the image chosen by the user is a rectangle, the

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fundamental region contains only a subset of this image. The user can draw in this fundamental region, whether it contains an image or is blank, and see immediately the result of the drawing on the main area. Right clicking on the fundamental region lets the user select another image.

Figure 1: The seventeen plane symmetry groups. Figure 2: The Pavimenta interface.

The main area shows the image generated by the selected isometry group on the current image. The program uses standard linear algebra to rotate, translate and reflect the fundamental region to the various

Pavimenta: A Photo Tiling Tool

positions on the screen. These calculations are done for every change made on the fundamental region, in real time.

Although it would be possible to construct these images with a standard photo editing program which is capable of applying linear transformations to pictures, Pavimenta generates them with a single click, which makes it extremely easy to create beautiful and surprising images. Below are some examples of tilings produced with Pavimenta and the images that originated them.

Figure 3: A tiling of an anemona by $D(333)$ .

Figure 4: A tiling of marbles by $D(2222)$ .

Figure 5: A tiling of palm trees by D4(2).

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Figure 6: Since this portuguese azulejo is already highly symmetrical, all tilings from $T$ to $D4(2)$ produce the same image.

Figure 7: A tiling of eyes by $D(632)$ .

Figure 8: A tiling of tulips by $S(333)$ .

References

[1] Montesinos, J. M., Classical Tesselations and Three Manifolds, Universitext, Springer, Berlin (1987). [2] Salgueiro, A., Pavimenta, executable available at http://www.mat.uc.pt/~ams/Pavimenta, (2011).