Various Orbifolds

Mathematics can give beauty not only to our minds, but also to our eyes. Today I suggest you admire the beautiful pictures that result from combining geometry, group theory and cellular automata.

Any serious discussion of geometry will sooner or later come down to discussions of the topological properties of our space, and then to what other spaces can be studied and used to model all sorts of phenomena in our world. In popular mathematical materials in this regard, multidimensional Euclidean spaces will certainly be mentioned, as well as the classic set of the simplest smooth two-dimensional manifolds: sphere, cylinder, the Mobius strip, Thor, Klein bottleAnd projective plane. We've mentioned them in this blog as well, for example in the article Keeping Our Distance Like Topologists.

The listed spaces are encountered more often than others, since they are best suited for analysis and the construction of geometry on them. This, in turn, is due to the fact that smooth manifolds, in a certain sense, are very similar to the space we are accustomed to. At least, for any point of such a manifold, one can find an open neighborhood that is topologically indistinguishable from the neighborhood of a point in Euclidean space. Thus, it will be very difficult for a small inhabitant of a smooth manifold to distinguish it from the Euclidean world by conducting local observations. It is precisely because the surface of the globe has the topology of a sphere (a smooth compact manifold) that it is so difficult to believe that it is not flat if, moving along it, one examines the neighborhood no further than a few kilometers.

However, we are surrounded by objects for which manifolds are no longer sufficient to describe them. For example, conical surface resembles a cylinder and even easily unfolds into a plane, but it has a special feature – a sharp top. In this blog we have encountered some more extravagant spaces: for example, semicubewhich, being developable, has the topology of a projective plane and is flat almost everywhere except for seven singular points. Finally, considering the space of all conceivable triangles, we constructed a map of this space, and then isolated a quotient space on it, “factoring out” (factorizing) the symmetry group of a regular triangle.

Mathematicians could not pass by such spaces without formalizing them and naming them with a special word: orbifold (orbifold)In Russian it sounds awkward, but it accurately conveys the meaning of its English equivalent, obtained by combining words. orbit (orbit) and manifold (manifold).

Orbit is called the set of all points in space obtained by the action of some transformation on a selected point. If the transformation we are talking about is an element symmetry groups some diversity then the orbits of some one open region, it is called fundamental, can cover everything entirely. In this case, the covering will be seamless, that is, no two points from the fundamental region will be mapped into one, and almost no point of the manifold will be left without its prototype from the fundamental region. The word “almost” here has a precise meaning and means that there are exceptions, but they form a set of zero measure on the manifold: individual special points.

As an example, consider the surface of a cone. It can be turned into a flat angle, remembering that its two sides must be glued together to form a continuous surface. If the resulting flat angle divides 360° into an integer number of parts, then its copies can be used to tile the entire plane by applying a 360°/n rotation transformation to the angle.

Thus, the surface of a cone is not a smooth manifold, but an orbifold constructed using rotational symmetry, with a single singular point representing the vertex of the cone. In addition to the vertex, around any point on the cone one can construct a neighborhood analogous to the neighborhood of a point on the plane.

For orbifolds, a notation system based on the listing of singular points and their types is adopted. I will not clutter this article with an explanation of it, but for reference I will cite the names of the examples encountered, enclosing them in brackets. In particular, a cone whose development divides the plane into identical parts, will have the designation .

Good An orbifold is a space that, under the action of the symmetry group defining it, covers some smooth manifold, while allowing the existence of a finite set of singular points of zero measure. The conical surface described above is a good orbifold.

I am not giving a precise definition here, but my blog is not a textbook, so those who are thirsty for precision, I invite you to consult an encyclopedia with a light heart. The main reason why I even started talking about these spaces is that these abstractions surround us, literally everywhere, and can look very beautiful. Having learned to recognize them, you can broaden your horizons and develop a special quality of thinking, which is called the intellect of a naturalist, which makes life richer.

Orbifolds are most obviously found wherever there is symmetry – in repeating patterns on wallpaper and tiles, in the texture of fabrics and in kaleidoscopes. And invisibly, they help describe the dynamics of force and quantum fields on regular structures: crystals, polymers and metamaterials. Before turning to cellular automata, I will give a few more simple “everyday” examples of good orbifolds.

Looking Glasses and Kaleidoscopes

Looking at a clean, well-washed mirror, we do not see the boundary between our world and its reflection. The mirror, reflecting the half-space (fundamental region) inhabited by us, complements it into a new space – an orbifold (R^3)

except that all geometric and optical phenomena in it must have mirror symmetry. We will see that the resulting space is filled with paired objects, resting and moving exactly the same way relative to the plane of symmetry.

gets to know its orbit.

Photographer caught in an orbiform

Picture drawn in the Inspirit program. Here the crystallographic symmetry group p4m is used, forming an orbifold on the plane .

describes the space of the world of triangles, in which each point has many copies created by the symmetry group of a regular triangle, which coincides with the permutation group of three elements.

An example of an orbifold representing the factorization of an open disk by the symmetry group of a regular triangle.

An example of an orbifold representing the factorization of an open disk by the symmetry group of a regular triangle.

The examples above demonstrate the main feature of good orbifolds: they are, except for singular points, locally indistinguishable from the underlying manifold. This means that the same geometric and analytical constructions can be made in them as in a “normal” space, but at the same time they will have one or another symmetry group sewn into them. It is this property of orbifolds that makes them especially suitable for the work of cellular automata.Cellular automataLovers of mathematics and generative art are familiar with the concept of a cellular automaton: an ensemble of elementary information carriers in which local rules of information exchange determine the global dynamics of the entire system. Cellular automata can model systems considered by mathematical physics, field theory, catastrophe theory, synergetics, and other disciplines.

The most famous example of a cellular automaton is the game “

In most machines, the state of each cell is determined entirely by the state of its neighbors. On a flat square grid, this might be the eight neighbors adjacent to the edges and diagonals.

Example of a neighborhood of a black cell (green cells).

• Example of a neighborhood of a black cell (green cells).

• Example of a neighborhood of a black cell (green cells).The rules of interaction of cells with neighbors and the number of possible states of cells determine a specific automaton. To obtain beautiful images, I chose several of them:the above mentioned game “Life”;

• machine gun “Rock Paper Scissors “, in which each cell plays this famous game with its neighbors, or its five-state analogue, as described in the TV series The Big Bang Theory; machines simulating autocatalytic chemical reactions (mixing machine

Gerhard-Schuster, and the finite-difference reaction-diffusion model

Gray-Scott ), in which the states of the cell form a continuum. In all these automata, information about all eight of a cell's neighbors is required to calculate its state. This implies either the infinity of the cell space (in which case neighbors always exist) or its compactness (closed and bounded). It is possible to use rigidly defined constant boundaries, “zeroing” all border cells. But most often, periodic boundary conditions are used, embedding the automaton in the topology of a flat torus. This already turns the cell space into a torus, in which there are no singular points and the boundaries exist globally, but are not “felt” by individual cells at the level of their neighborhoods. Using more complex orbifolds instead of a torus allows one to generate complex and very effective ornaments using cellular automata. In this case, all singular points are hidden “under the lattice” (fall between cells) and therefore do not violate the continuity of the finite automaton's coverage of arbitrarily large areas of the Euclidean plane. In this case, calculations are performed only in a small fundamental area, which makes the calculation of images very effective.For this article I will restrict myself to orbifolds constructed by factorization of the planein groups ornaments

. It has long been known that any flat ornament that has a regular pattern and covers a two-dimensional Euclidean space must have some of

17 symmetry groups Each of these groups forms its own orbifold on the Euclidean plane.As I have already said, the “seamless” covering of the entire plane by good orbifolds allows one to correctly define on a small and finite fundamental region a full-fledged cellular automaton in which all cells without exception “do not feel” the finiteness of this region. Especially for this article (and for my math circle) I wrote a simpleonline program

of this repository

.

I will give examples of my program's work on 12 suitable groups of ornaments. In the upper left corner is shown the structure of the orbifold, its fundamental region and special points, which give the formal name to the orbifold (the color of the figures and lines, designating the points, corresponds to the numbers in the notation). Once again, I would like to draw your attention to the absence of any sharp boundaries, visible mirror planes or breaks in the resulting patterns. Cellular automata behave exactly as if they were defined uniformly on the entire infinite plane, but the initial and all subsequent configurations were defined with the observance of the given symmetry and would be infinitely repeating. ***