Symmetry is often thought of as a purely geometric phenomenon and it is usually defined in terms of rigid motions of a figure to itself but there are other types of symmetry. Formally the term "topological symmetry" refers to a group of transformations of a figure to itself that can be extended to a group of homeomorphisms of the space in which the figure is embedded. To avoid technicalities I will introduce the concept of topological symmetry with three examples. We will restrict ourselves to examples that only involve a finite number of homeomorphisms. In two dimensions this restriction provides a strong constraint on the types of symmetry that can occur. But in higher dimensions this restriction is not so strong and it allows for some rather exotic types of symmetry to occur. We will start with a rather mundane example and work our way up to more interesting cases.

Our first example of topological symmetry is shown in the picture to the left. It depicts a geometric figure made up of six line segments connected at a single point in the center. Geometrically speaking the figure has the same symmetry as an equilateral triangle. We can rotate it by one third of a turn, two thirds of a turn, and reflect it about three different lines in the plane. When we include the identity map we get six symmetries altogether. But there are more symmetries hidden within this figure.

The intrinsic structure of the figure provides one kind of constraint on how it can be mapped to itself. The vertex in the center must be mapped to itself and each line segment must be mapped to itself or to some other line segment. The fact that the figure is embedded in the plane provides another kind of constraint. The cyclic order of the line segments radiating away from the central vertex must either be preserved or reversed.

Another more subtle constraint is that, as a consequence of restricting ourselves to finite groups of symmetries, a transformation which maps a line segment to itself must map every point in the line segment to itself. Otherwise repeatedly applying the transformation would generate an infinite number of transformations. Therefore any transformation which maps all six line segments to itself must be the identity map.

There are six ways we can map the six line segments so that their cyclic order is preserved and six ways we can map them so their cyclic order is reversed. If two transformations permuted the line segments in the same way then combining one of these transformations with the inverse of the other would produce a transformation that maps every line segment back to itself and therefore must be the identity map. This implies that the two transformations were actually the same. So the only finite groups of homeomorphisms of the plane to itself that can be symmetries of the figure have at most twelve members.

These twelve symmetries are very much like the Euclidean symmetries found in a regular hexagon. Indeed if we stretched the three short line segments radially outward so that they had the same length as the three long line segments then the line segments would form the diagonals of a regular hexagon and the whole figure would have the Euclidean symmetry of a regular hexagon.

Even though this figure has more topological symmetries than Euclidean symmetries these topological symmetries did not give the figure a new type of symmetry. The topological symmetry was the same type as the Euclidean symmetry of a different figure. This is generally true in two dimensions. Figures in the Euclidean plane with a finite number of topological symmetries can be continuously deformed without changing their topology so that their topological symmetries become Euclidean symmetries given by rigid rotations and/or reflections of plane.

Our second example of topological symmetry is the fractal to the left which is the attractor for a discrete dynamical system obtained by numerically integrating the Volterra-Lotka equations . The Volterra-Lotka equations were one of the original mathematical models of predator/prey relationships and they are well known among population biologists. In this mathematical model the population sizes of the predator and prey species vary over time in a precisely periodic fashion. If we plot the population size of the prey versus the population size of the predator then over time these points will trace out a simple closed curve in the plane.

However in making this fractal the time step of the numerical method has been deliberately made too large for the purpose of approximating the solutions to the Volterra-Lotka equations. Making the time step too large in the numerical integration of a differential equation often results in a map that has a fractal attractor. This is a convenient way to custom design fractals.

To help understand how this works consider using a small time step in the numerical method and imagine slowly increasing its size. For small enough values of the time step most numerical methods do a good job of approximating the periodic solutions of the Volterra-Lotka equations. As the numerical method is iterated it produces a discrete set of points which tends to closely follow a periodic solution of the Volterra-Lotka equations.

As the time step increases resonances can occur in the numerical method. In other words an attractor appears which consists of just a finite set of points. In this particular case the number of points in the attractor was seven. The numerical method maps each point of the attractor to another point of the attractor and in every seven iterations each point of the attractor is visited. In this case we say that the periodicity of the resonance is seven. Many different starting values tend to converge towards the seven point attractor.

The periodicity of the resonances can be controlled by appropriately choosing the numerical method. In this case increasing the time step a little more resulted in a period doubling bifurcation. A pair of points emerged from each of the seven points in the original attractor. This produced a fourteen point attractor while the original seven points formed a repellor. Further increasing the time step resulted an infinite sequence of period doubling bifurcations that produced a fractal attractor for the system.

This fractal can not be rotated or reflected back into itself. Aside from the identity map there is no rigid motion or congruence of the Euclidean plane which can map this fractal into itself. However the period seven resonance has effectively given us an attractor which can be decomposed into seven topologically equivalent pieces. These seven pieces emerged from the seven points of the original period seven resonance through the period doubling cascade. Each of these seven pieces individually resembles maple syrup being poured out from a bottle. Geometrically these pieces have different shapes but their complicated topological structures are the same. The numerical method maps each piece of this attractor to another piece of this attractor and in every seven iterations each of these pieces is visited.

Although the symmetry of this fractal is in sense hidden the type of symmetry that it displays is not really new. We can stretch and compress some regions of the plane to turn this fractal into a figure which can be mapped back into itself with a rigid rotation by one seventh of a turn.

If we relax the requirement that the group of topological symmetries has to extend to a group of homeomorphisms of the entire Euclidean space then we can find many interesting examples of topological symmetry in three dimensions. Our third example is an object often referred to as "Klein's quartic surface" hown to the left. This is a highly symmetrical figure made up of twentyfour regular heptagons with three heptagons meeting at each vertex. It is like the five Platonic solids (regular polyhedra) but it has to be distorted to fit inside three dimensional Euclidean space, E³. Klein's quartic can be embedded in a higher dimensional space in which it is possible to rigidly rotate the figure by a seventh of a turn about the center of each heptagon. But in E³ these symmetries involve stretching and compressing different parts of the figure.

Altogether the group of topological symmetries for Klein's quartic has 336 members. It is very different from any geometric symmetry group in E³. For instance any polyhedron in E³ can have at most one axis in which it can be rotated by a seventh of a turn. But for each curved heptagon in an embedding of Klein's quartic in E³ there is a homeomorphism which maps the figure to itself, the chosen heptagon to itself, and the heptagon's edges to adjacent edges. The homeomorphism fixes exactly one point in the interior of the heptagon and on "average" the other points in the heptagon are rotated about the fixed point by one seventh of a turn. In this sense we can rotate the figure by one seventh of a turn about every one of its faces. These homeomorphisms can be extended to all of E³ except for a subset with zero volume.

The picture of Klein's quartic embedded in E³ is from "Patterns on the Genus-3 Klein Quartic" which has many other nice pictures of Klein's quartic. This article is available online. A whole book has been written on this object: "The Eightfold Way: The Beauty of Klein's Quartic". This book is also available online. Below are a list of links to these and other websites with pictures and text which explain the topological symmetry of Klein's quartic in greater detail.