12. Finite figures. Example: Let F be the line segment determined by two points P and Q.

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1 12. Finite figures We now look at examples of symmetry sets for some finite figures, F, in the plane. By finite we mean any figure that can be contained in some circle of finite radius. Since the symmetry set of any figure has the algebraic structure of a group, this set will be referred to as the symmetry group of the figure. Example: The simplest finite figure is the figure consisting of single point, P: F = {P}. In this, T(F) = F if T is any rotation with center P, or any reflection in a line containing P. So, in this case the symmetry group is an infinite group (it contains an infinite number of symmetries). So a point is a highly symmetric figure. Example: Let F be a circle centered at point C, with radius r. Then T(F) = F if T is any rotation with center C, or any reflection in a line containing C. So again, the symmetry group of F is an infinite group. A point and a circle have the same symmetries. Example: Any figure F will have at least one symmetry in it s symmetry group; the identity isometry I. Clearly, these are the figures that one would say have no symmetry, but what is really meant by that is that the figure has no nontrivial symmetry. So the symmetry groups form a continuum where on one end reside the asymmetric figures and on the other are the highly symmetric figures like circles and points. The next few examples illustrate the middle ground. Example: Let F be the line segment determined by two points P and Q. P Q

2 Let M be the midpoint of this segment. Denote by m reflection through the perpendicular bisector of the segment and r the rotation about center M by 180º. Then both m and r are symmetries of F. There is another symmetry that is less obvious, and that is the reflection through the line containing the segment. This is the reflection, mr, obtained by composing m and r. On the segment it is the identity but it is clearly not the identity isometry. The multiplication table looks like: 1 r m mr 1 1 r m mr r r 1 rm rmr m m mr 1 mmr mr mr mrr mrm 1 This can be simplified since, for example mrr = mr 2 = m, mmr = m 2 r = r, mr = rm, so mrm = mmr = r, etc. The simplified table shows the group properties of closure and the existence of inverses. 1 r m mr 1 1 r m mr r r 1 mr m m m mr 1 r mr mr m r 1 The line segment has both mirror and rotational symmetry. Example: F is the figure The only non-trivial symmetry of this figure is r, the rotation about center M by 180º, where M is the midpoint of the horizontal segment. This gives the relatively simple table: 1 r

3 1 1 r r r 1 Note that this table is embedded in the one above, and corresponds to the rotational symmetry. Example: Let F be an isosceles triangle: Let m be reflection in the perpendicular bisector of the base. Then the only non-trivial symmetry of F is m. This is called bilateral symmetry and the group table looks like: 1 m 1 1 m m m 1 Note that this table is also embedded in the table for the segment, although it s a little harder to see. Example: Let F be an equilateral triangle:

4 ! This time let r be the rotation about the center of the triangle by 360 = 120, and m the reflection through the perpendicular bisector to 3 the base. We will determine the symmetries of this figure and fill in it group table during lecture. Rosette: n. any arrangement, part, object, or formation more or less resembling a rose. An architectural ornament resembling a rose or having a generally circular combination of parts. Typically, rosettes have point and/or reflection symmetries. For a given rosette, the group of symmetries is called a rosette group. It is known that there are only two possible types of rosette groups: C n for a figure with n-fold rotational symmetry, and D n for a figure with both rotational and reflection symmetry. This fact is known as Leonardo s theorem, after Leonardo da Vinci, who first showed that these are the only possibilities. C n is also called a cyclic group, while D n is known as a dihedral group (dihedral indicates the presence of a reflection line symmetry). Each group has a standard form as follows: For the cyclic group with n-fold rotational symmetry C n = {1,r,r 2,...,r n "1 },where r = 360 n.!! For the dihedral group with n-fold rotational symmetry D n = {1,r,r 2,...,r n "1,m,mr,mr 2,...,mr n "1 },where r = 360 n From above, the symmetry group for the figure and m is the reflection. is C 2 and the symmetry group for the equilateral triangle is D 3. Looking at a couple of examples should convince you that the symmetry group of a

5 regular n-gon is the dihedral group D n. However, you shouldn t think that these groups are only found in a mathematical context. If you search the internet on the phrase rosette group, you will find examples of architectural rosettes, among other examples.