Rotational symmetry is a type of symmetry studied in mathematics. Apart from that, you will also learn about folding symmetry which is similar to rotational symmetry. After studying the symmetry material, students will usually be asked to practice how to apply rotational and folding symmetries to a flat shape. An example is a square shape that has fold and rotate symmetry.
Definition of Symmetry
The term symmetry refers to a transformation that is applied to a plane as the medium. A flat shape can be said to be symmetrical if the shape can cover each other when folded or rotated.
Symmetry Types
Following are the types of symmetry
1. Rotate Symmetry
A plane shape is said to have rotational symmetry if the plane shape can be rotated less than one full rotation and can return to its original position exactly. Plane shapes that have rotational symmetry, including squares, rectangles, equilateral triangles, regular pentagons, regular hexagons, and rhombuses. The flat shapes that do not have rotational symmetry are isosceles triangles and trapezoids.
2. Fold Symmetry
A plane shape can be said to have folding symmetry if the plane shape can be folded into two parts, so that it can produce two shapes that are the same and congruent. In addition, the fold will produce a fold line or axis of symmetry that divides the plane into two equal parts. The number of fold symmetries of a flat shape is equal to the number of axes of symmetry that will be produced.
Examples of flat shapes that have folding symmetry are squares, rectangles, equilateral triangles, isosceles triangles, regular pentagons, regular hexagons, isosceles trapezoids, circles, kites, and rhombuses. Meanwhile, a flat shape that does not have fold symmetry, namely a parallelogram.
Steps to Determine the Amount of Rotational Symmetry in Mathematics
A flat shape is said to have rotational symmetry if it has a central point, which, when rotated for less than one turn, can return to its original shape. So, rotational symmetry in a plane shape is the number of shadows that can be produced in less than 1 rotation.
Each plane shape has a different number of rotational symmetries. Here are 4 steps to determine the amount:
1. Determine the Rotation Center Point
First, determine the center point of the plane’s rotation, which is obtained from the intersection of the symmetry axes of the plane.
2. Trace the Shape
Second, trace the shape of the flat shape onto a blank white paper. The trace will later serve as a base.
3. Name the Angle
Third, name or give a symbol in each corner. For example, on a square: A, B, C, D.
4. Compute Rotational Symmetry
Finally, rotate the square 360 degrees clockwise. That way, you can count how many times the square fits exactly on the base, which is the square we traced earlier.
After doing the 4 steps above, we finally find 4 rotational symmetries in the square.
Total Rotational Symmetry in Various Planes
Rotational symmetry is the rotation of a plane shape determined by the center point of rotation and the rotation angle and direction of rotation, whose rotation is determined by a central point P with a certain rotation direction (Marini, 2013:30). Based on this understanding, a plane shape will know the number of rotational symmetries if the clockwise rotation can be determined by the center point.
According to Winarni (2012: 63) rotation or so-called rotational symmetry is a rotation determined by a point P with an angle magnitude and a clockwise rotation direction. Thus the rotational symmetry is determined by the center point through a clockwise rotation.
Furthermore, Zuliana (2017: 153) concluded that rotational symmetry is included in the scope of geometry related to transformation which is the object of study in mathematics learning. Based on this understanding, rotational symmetry material is in the study of mathematical objects as students’ understanding of the mathematics learning process in the scope of geometry, so that students can know more clearly about rotational symmetry material.
Based on the description above, it can be concluded that rotational symmetry is an object of mathematical study within the scope of plane geometry defined by a central point P with a certain magnitude and direction of rotation.
1. Square
Haryono (2014: 251) says that a square shape is a flat figure that is bounded by 4 equal sides. The properties of a square shape are that it has 4 equal sides, all four angles are equal right angles.
Here’s an example of rotational symmetry on a square shape:
Astuti (2009: 159) concludes that the first round of quadrilateral ABCD is 90º causing angle A to occupy D, B to occupy A, C to occupy B, and D to occupy A. The second rotation of 180º results in angle A to occupy C, B to occupy D, C occupies A, and D occupies B.
The third rotation of 270º causes angle A to occupy B, B to occupy C, C to occupy D, and D to occupy A. The fourth rotation of 360º results in angle A to occupy A, B to occupy B, C to occupy C, and D to occupy D. So, a flat shape a quadrilateral has rotational symmetry of the fourth degree or has 4 rotational symmetries.
The characteristics and properties of a square flat wake, among others:
- Has sides that are the same length.
- It has two diagonals that are the same length (both of which intersect and form a perpendicular and divide it into two equal parts).
- It has four right angles that are equal in size, which is 90 degrees.
- Has four fold axes of symmetry.
- Has four corner points.
- Has four axes of rotational symmetry.
A square is a special case of a rhombus (equal sides, opposite angles equal), a kite (two pairs of equal sides abut), a trapezoid (a pair of opposite sides are parallel), a parallelogram (all opposite sides are parallel), a quadrilateral or a tetragon ( four-sided polygons), and rectangles (equal opposite sides, right angles) and thus have all the properties of all these shapes, namely:
- The diagonals of the square bisect each other and meet at 90°.
- The diagonal of a square bisects its corners.
- The opposite sides of the square are parallel and the same length.
- The four square corners are equal. (360° / 4 = 90° respectively, so every corner of the square is a right angle.)
- All four sides of the square are equal.
- The diagonals of the squares are the same.
- Squares are the n = 2 cases of the n-hypercubes and n-orthoplexes families.
- The box has the symbol Schläfli {4}. The truncated square, t {4}, is an octagon, {8}. Alternate squares, h {4}, are digons, {2}.
2. Rectangle
A rectangle (English: rectangle ) is a two-dimensional flat shape formed by two pairs of sides, each of which is the same length and parallel to its partner, and has four angles, all of which are right angles.
A rectangle is a flat shape that has 2 rotational symmetries (Sugiono, 2009:162). The characteristics and properties of a rectangular flat wake, among others, are as follows.
- Has four sides (where the two sides are opposite each other in length and parallel).
- It has four right angles that are equal in size, which is 90 degrees.
- It has two diagonals (crosses) that intersect into two equal parts.
- Has two fold axes of symmetry.
- Has two axes of rotational symmetry.
- It has rectangular sides that are perpendicular to each other.
3. Triangle
In geometry, an equilateral triangle is a triangle in which all three sides are the same length. In Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are 60° each. They are regular polygons, and therefore can also be called regular triangles.
Soenarjo, (2008:253) concludes that an equilateral triangle occupies its frame 3 times in a full rotation and has 3 rotational symmetries. If the triangle in figure (a) first rotates 120º, it will result in position A occupying B, B occupying C, and C occupying A. If the triangle in image (b) rotates second by 270º, it will result in position A occupying C, B occupying A , and C occupies B. If the triangle in figure (c) rotates 360º, it will result in position A returning to A, B returning to B, and C returning to C.
In the isosceles triangle above, it has one axis of rotational symmetry or is said to have no degree of rotational symmetry. Because the triangle only occupies the frame once with a 360º rotation.
If the side lengths of an equilateral triangle are expressed by a , using the Pythagorean theorem we can determine that:
- The area is .
- Circumference is .
- The radius of the outer circle is .
- The radius of the inner circle is or .
- The geometric center of a triangle is the center of the inner and outer circles.
- And the altitude (height) of each side .
Denoting the radius of the outer circle as R , using trigonometry we can determine that:
- The area of the triangle is .
Some of these equations have a simple relationship to the altitude (“h”) of each angle on the opposite side:
- Area .
- The height of the center of each side, or apothem, .
- The radius of the outer circle of the three vertices is .
- The radius of the inner circle is .
In an equilateral triangle, the altitudes, bisectors of the angles, perpendicular bisectors and the median for each side coincide.
4. Parallelogram
A parallelogram or parallelogram (English: parallelogram ) is a two-dimensional plane shape formed by two pairs of edges, each of which is the same length and parallel to its partner, and has two pairs of angles, each of which is equal to the angle opposite it. A parallelogram is a derivative of a quadrilateral which has special characteristics. A parallelogram with four edges of the same length is called a rhombus.
A parallelogram can be formed by combining two triangles of the same type and size (congruent triangles). The characteristics of a parallelogram are that the opposite sides are the same length and parallel, the opposite angles are the same length, the sum of the adjacent angles is 180º, the diagonals bisect each other, the parallelogram has unequal lengths, does not have a symmetrical axis, the parallelogram can fit into its frame in 2 ways. The rotational symmetry of a parallelogram is 2.
5. Trapezoid
A trapezoid is a two-dimensional plane shape formed by four edges, two of which are parallel to each other but not the same length. A trapezoid is a type of quadrilateral that has special characteristics.
Trapezoid consists of 3 types, namely:
- An arbitrary trapezoid, that is a trapezoid whose four sides are not the same length. This trapezoid has no fold symmetry and no rotational symmetry.
- An isosceles trapezoid, that is a trapezoid that has a pair of ribs that are the same length, besides having a pair of parallel ribs. This trapezoid has 1 fold symmetry and no rotational symmetry.
- A right-angled trapezoid is a trapezoid in which two of the four angles are right angles. The parallel sides are perpendicular to the height of the trapezoid. This trapezoid has no fold symmetry and no rotational symmetry.
A trapezoid is a quadrilateral with one pair of parallel sides. The trapezoid has elements consisting of the base side, the top side, and the legs of the trapezoid (Haryono, 2014: 260). The trapezoid will only return to its frame when it is rotated 360º (one full turn). So, the trapezoid is said to have no rotational symmetry, because according to its axis of symmetry it only has one degree of rotational symmetry.
6. Rhombus
A rhombus has two axes of symmetry, its diagonals are axes of symmetry, it has 2 fold symmetries, it has 2 rotational symmetries, the rhombus is attached to its frame in 4 ways. Rotational Symmetry in Rhombus, as follows (Haryono, 2014:261).
In a flat shape, the rhombus has 2 rotational symmetries, the first rotation of the rhombus is rotated clockwise with a magnitude of 180º, that is, C occupies A, D occupies B. The second rotation is 360º, namely, A occupies C, B occupies D, so it returns to position start as before playing.
7. Kites
Haryono, (2014: 262) a kite has 4 opposite angles that are equal, has 2 diagonals that are perpendicular to each other, the kite can occupy its frame in 2 ways, and has 1 axis of symmetry. Because the kite’s flat shape occupies its frame with a magnitude of 360º Rotational Symmetry on the Kite’s flat shape, as follows:
8. Circle
The circle is a unique flat shape with a value of Phi (π). The shape of a circle has the following properties, namely a circle including a closed curve, the number of degrees of a circle is 360º, the circle has one center point, the axes of symmetry are an infinite circle because it is rotated at any angle at the corner point P.