There is a large variety of flowers whose petals are symmetrically arranged around the centre.
Let us look, for example, at the periwinkle shown on the left, and fix our attention on one of its petals: we can pass from this one to any other by rotating the flower about its centre in such a way that each petal reaches one of the five possible positions.

The periwinkle has only these five rotations as symmetries.

On the other hand, this other flower and also the one drawn here on the right has five reflections as well, reflections in each of the lines through the centre and the tip of a petal.

The pentagonal symmetry type is very common in the organic world: flowers apart, we can also find it in starfish and sea urchins. Apples and pears, too, have pentagonal symmetry: to see this, H.S.M. Coxeter suggests cutting an apple in the way you would cut an orange to squeeze out its juice, and look at its core.
Nature offers many examples of shapes that display a similar symmetry based on other numbers. For example, there are six rotations that transform a snow crystal into itself (and six reflections too), while there are three rotations (and three reflections) that transform a slice of banana or a courgette flower into itself.

In art there are extraordinary examples of buildings and patterns that are symmetrically arranged about a centre: the circular Pantheon in Rome, the octagonal Baptistry of the Cathedral in Florence, rose windows of Romanesque and Gothic churches, some mazes, Aztec calendars, and Indian mandalas.
A design is called a rosette if its symmetries are either only rotations or an equal number of rotations and reflections. In fact, it is not possible for a design to have different numbers of rotational and reflection symmetries. It appears that this conclusion was reached by Leonardo da Vinci when he was trying to determine all the possible symmetries of a building with a central plan.

From the mathematical point of view, the simplest rosettes are the regular polygons, that is those polygons with equal edges and equal angles. The snow crystals, for example, are nothing but magnificent elaborations of a single fundamental theme, the regular hexagon.

Would you like to make a paper model of a rosette?
Take a sheet of paper, fold it so that all the creases pass through the same point, and cut out a profile: by unfolding the paper you will have a rosette.
Note a rosette made in this way will always have both rotational and reflection symmetries.
Can you make one with only rotational symmetries?

by Paola Cereda
Transalated by Elisabetta Beltrami and Peter Cromwell

Many everyday objects have bilateral symmetry: furnishings, lamps, road signs, flags, musical instruments and some of our clothes (but not individual gloves, or shoes). So do many animals and leaves; even the human body in its external form has approximate bilateral symmetry (though it would be different if we considered the internal organs too).


What does the phrase "this butterfly has bilateral symmetry" or "this butterfly has a plane of symmetry" mean?

Imagine cutting something in half along a plane and putting a mirror in place of this plane. If the reflected image together with the original half reconstructs the whole object then the plane is a mirror plane or plane of reflection symmetry.

In practice, it can be rather hard to cut a fork in half, not to mention a butterfly or a human being (even if the imagination of Italo Calvino tried it in the novel "The Cloven Viscount"). It is easier to do analogous experiments with planar figures, in which case mirror symmetry is no longer defined by a plane, but by a straight line.

To check whether a particular straight line is a line of reflection symmetry (mirror line) of a figure, we can place a mirror along the line and hold it perpendicular to the plane of the figure. For the line to be a mirror line, the reflected image together with half the original must reconstruct the whole object. As you can see from the two figures below, this need not happen, even if the straight line divides the figure into equal parts.

What happens when we use two mirrors?
Suppose, for example that we place a "p" between two parallel mirrors: then you can see a series of images which repeat along a strip, alternating images of "p" with images of "q".

If we make first a reflection with respect to a straight line, and then another reflection with respect to another line parallel to the first one, the combined effect is equivalent to that of a traslation, in the direction perpendicular to the two mirror lines, and of a distance twice the distance between the two mirrors.

A reflection exchanges the right with the left: in fact the mirror image of a "p", which has a tail on the left-hand side, is a "q", with a tail on the right. A translation, instead, corrisponds to two reflections and so ... the tail of the "p" comes back to its place!

What happens if the mirror lines are not parallel but meet?




Placing an object between two connected mirrors (as shown in the figure, a cat with its tail on the right), you can see some images of the object, and, among these, those with the tail on the left alternate with those with the tail on the right. The images are no longer positioned along a strip, but around a centre, instead. In effect, if we make first a reflection with respect to a straight line, and then another reflection with respect to another line which has a point in common with the first one, the effect obtained is equivalent to a rotation about this point, by an angle twice the angle between the mirror lines.

How many images are seen?

If the angle between the mirrors is a submultiple of 180 degrees, as in the figure here (where the angle is 60 degrees) then the 360 degree angle is divided by the mirrors - both real and virtual ones - into an even number of angles (six in the figure) each of which contains an image: half of them (three in the figure) contain an image obtained from the original object by rotation (and so the cat has its tail on the right), while the others are obtained from the original object by reflection (and so the cat has its tail on the left).

Imagine you are walking in a wood, far from any man-made structure. Among the large variety and complexity of shapes around you, you will see some repeated forms. Regularities and symmetries start to appear. Even with the continuous change of nature, some things seem to be preserved, some directions of growth, or designs, seem to be favoured.

From any wood you can see the sun as round as the full moon, rain falling in approximately spherical drops, or snow falling in hexagonal crystals.
The tree-trunks and mushrooms have rotational symmetry about a central axis. Flowers often have rotational symmetry of pentagonal, square or hexagonal type... Many insects, mammals and leaves have bilateral symmetry. Millepedes, blades of grass, or the leaves of ferns display the beginnings of translational symmetry..

These symmetries are hardly ever perfect: the leaves of a four-leaved clover are not exactly equal, so its symmetry is not as precise as that of a square. However, we will not be too bothered about this and say coolly that a star fish has pentagonal symmetry even if one of its five arms is folded or broken.

Having awakened our curiosity, we can make some hypotheses about how and why nature is organised in symmetric shapes, hypotheses linked to criteria such as functionality, replacement, locomotion, or economy.

A system evolves interacting with its environment and, if the external conditions that impose on it a particular effect have some symmetry, then the result may display, completely or partially, the symmetry of the cause that produced it. The Earth is not perfectly spherical because its rotation causes it to bulge out around the equator and so become flattened at the poles. However, the rotational symmetry about its axis and reflection symmetry with respect to the equator is preserved.

Crystals are a notable example of regularity in the inorganic world. The crystals of salt, iron oxide and pyrite are respectively of cubic, octahedral, and dodecahedral shape. It is likely that the pyrite crystal suggested the dodecahedral shape to the ancient Greeks, who found it in Sicily, where it is abundant.

Symmetrical structures can have advantages for living organisms. They are stable, compact, homogeneous, and interchangeable. As in a production line in a factory, they reproduce themselves more easily and quickly. A spherical body disperses less heat because a smaller surface area is needed to separate it from outside.

As living organisms evolved, they lost the high degree of symmetry that is characteristic of lower orders. Many unicellular organisms, like the simplest ones suspended in water, are as symmetrical as possible, that is they have a spherical shape.
More complex sea organisms, creatures like starfish or jellyfish, or plants that are fixed to the ground where they grow, are conditioned by the direction of gravity: their upper and lower parts are different, but they still have rotational symmetry about a (vertical) axis.

In higher organisms that move by themselves, like fish or mammals or birds, we can also distinguish the front (where the eyes are) from the back. But these creatures often preserve bilateral symmetry. This is favoured by linear motion and acts against helical motion (like a screw).

Even the human body has substantially bilateral symmetry. The embryo initially develops a repetitive translational structure. Such symmetry is then lost when the organs such as the heart or liver, of which we have only one copy, develop and position themselves off the central axis, or when the intestine, as it needs to lengthen, folds itself up.

Some crustaceans break their bilateral symmetry by developing one claw more than the other. If the largest claw is lost, a new one is generated in its place, while the other grows and becomes the largest one. The two possible shapes are reflections of each other.

Translational symmetry is a property that applies only to unbounded figures. But a structure that grows in only one direction by repeatedly adding the same unit, like an ear of wheat, or an earthworm, is potentially symmetric with respect to translation: as time passes, new segments equal to the previous ones grow.

The need to fill all available space with the same unit gives rise to numerous symmetrical situations, for example the hexagonal structure of honeycomb, the tetrahedral symmetry of a foam of soap bubbles, the pigment of the retina in our eyes, or the polyhedral shape of some vegetable cells like those in the heart of a tree trunk or in the structure of some diatoms.

The 48 symmetries of the cube form a family with interesting properties. It is a very closed family: the result of the composition of two or more elements of the family is still a member of the family.

The identity transformation has an exceptional role in it. It is the neutral element of the family, in the sense that any symmetry of the cube composed with the identity remains itself.

In general the order in which you compose the transformations affects the final result.
For example, in the two figures on the left we see where the reference pyramid moves if we first reflect the cube with respect to the plane p and then we rotate it of 1/4 of turn about the line r. In the other two figures on the rigth, we see where the pyramid moves if we reverse the order of composition.

Any symmetry of the cube has a mischievous twin with which it has a privileged relationship. The mischievous twin undoes the symmetry by moving the reference pyramid back to the start. It is called the inverse of the symmetry. When a symmetry is combined with its inverse, independent of the order, the result is the identity. For example, the rotation of 1/4 of turn about the line r has as its inverse the rotation of 3/4 of turn about r; any reflection is its own inverse; the composition represented in the figures above, reflection in the plane p followed by a rotation of 1/4 of turn about r, has as its inverse the composition of a rotation of 3/4 of turn about r followed by reflection in the plane p -- in this order! This is the same as when you reverse the operation "put on socks then put on shoes", first you take off shoes then you take off socks, not vice versa!

Any symmetry of the cube composed with itself a certain number of times gives the identity. For example, the reference pyramid goes back to its initial position if we perform four times a rotation of 3/4 turn about a line joining the centres of two opposite faces, or if we repeat twice a reflection in a mirror plane.

In describing the characteristics of the family of symmetries of the cube we have given an example of one of the most important concepts in mathematics: the structure known as a group.
The symmetries of the cube, with the operation given by performing first one and then the second (composition), constitute a group. Another example of group is given by the set of (positive and negative) integers, with the operation of sum.
This family too is closed (the sum of two integers is still an integer); the number 0 has the same role as the identity transformation has for the family of symmetries of a cube (the sum of any number and zero is equal to the starting number); and for any number, its relative with which it has a privileged relationship is the number of opposite sign (any number summed with its opposite is zero).

The group structure is found in many other contexts: for example, the set of vectors in the plane with the combination rule given by the parallelogram rule for the sum of two vectors is a group.

Not only the symmetries of a cube, but also the symmetries of any plane figure or of any solid object form a group with the operation of composition, just like in the case of the cube.
If the starting shape is completely asymmetric, the symmetry group is the group containing a single element: the identity.

The symmetry group of the plane figure on the left has 6 elements. (Which ones? What is the inverse of each element?)

Rosettes are plane figures whose symmetry group contains a finite number of elements. There are also some figures whose symmetry group contains infinitely many elements, for example a circle or a straight line.