Aperiodic Tiling Co-office
Aperiodic tiling
The Penrose tiles are an aperiodic set of tiles, since they admit only non-periodic tilings of the plane: Any of the infinitely many tilings by the Penrose tiles is non-periodic. More informally, many refer to the 'Penrose tilings' as being 'aperiodic tilings', but this is not well-defined mathematically.The informal term aperiodic tiling loosely refers to both an aperiodic set of tiles, and to the tilings which such sets admit. Properly speaking, aperiodicity is a property of the set of tiles themselves; any given finite tiling is either periodic or non-periodic. Further confusing the matter is that a given aperiodic set of tiles typically admits infinitely many distinct tilings. One proposed formal definition is that a tiling of the plane is aperiodic if and only if it consists of copies of a finite set of tiles, that themselves only admit non-periodic tilings. A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings which remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well (for example, randomly arranged tilings using a 2×2 square and 2×1 rectangle will typically be non-periodic). An aperiodic set of tiles, however, admits only non-periodic tilings.
The various Penrose tiles are the best-known examples of an aperiodic set of tiles. Few methods for constructing aperiodic tilings are known. This is perhaps natural: the underlying undecidability of the Domino problem implies that there exist aperiodic sets of tiles for which there can be no proof that they are aperiodic. Quasicrystals — physical materials with the apparent structure of the Penrose tilings — were discovered in 1984 by Dan Shechtman et al.;[5] however, the specific local structure of these materials is still poorly understood. Wikipedia
Some constructions of aperiodic tiling
Aperiodic hierarchical tilings
Substitutions
Ammann Tiling
In geometry, an Ammann–Beenker tiling is a nonperiodic tiling generated by an aperiodic set of prototiles. Because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns. The Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings, most notably: They are nonperiodic, which means that they lack any translational symmetry. Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, if one looks only at finite patches. They are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction; the diffractogram reveals both the underlying eightfold symmetry and the long-range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called “deflation” or “inflation.” Wikipedia
Iteration 1
Iteration 2
Iteration 3
空间现有两种尺寸桌子，用来丰富使用者的选择。 Ammann Tiling的概念用 在这里，同时满足两种桌子所需要的使用面积，创造一种集合办公的 空间Co-office。 Ammann Tiling本身是一种分形图形，可以根据空间需求无限延伸。两种尺寸的拼图实际上限定了办公桌的空间范围，给整个大空间一个划分的 规则，同时又允许办公桌在所在区域内任意移动。 这种空间的利用形式使Co-office成为一个集合的办公环境，人们可以租赁其中的桌子。实际上出租的是桌子所在的一个小的空间，这里允许和周 围产生多种联系，带来的即是整个办公室里多样的办公方式。