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Translations of Encyclopedia about Mathematics

 

Display, Rotation and Symmetry

In our daily lives, we run into the most varied forms of display, symmetry and rotation. For example, in a pond, which reflects the surrounding environment, or a mirror in a bathroom.

But there are also some natural examples of symmetry, such as a butterfly or a cat whose sides are perfect reflections of each other.

With our example of a mirror, it must be clear how symmetry can be created if we observe the reflection we see in the mirror.

With such reflections, we find that the points reflected can be perceived as imaginary points located the same distance away from the point of reflection but on the opposite side of the plain of reflection.

In this manner, the points lying one side of a plain of reflection are carried over to the other side of the axis of symmetry, maintaining the same distance from the plain such that the line connecting the points with the plain of reflection make a right angle.

In a three-dimensional space, axial symmetry corresponds to the planar symmetry of a plain, where points, shapes and objects are transferred across the plain of symmetry, as applies in a two-dimensional plain. All distances are maintained and the lines connecting the individual points with their reflection create a right angle with the axis of symmetry.

With rotation, a body rotates a certain angle around a rotation point.

If there is a point of symmetry across which some shape is reflected on the other side, we refer to this as point symmetry, which applies also to the shape itself.

If there is an axis across which some shape is reflected to the other side, we refer to this as axis symmetry, which applies also to the shape itself.

If there is an angle of rotation and centre of rotation across which some shape is displayed on some other shape, we refer to this as angular displacement, which applies even if the angle is zero and the shape displayed is the shape itself.

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Quadrangle

Quadrangles have four vertexes or cusps, of which no more than two can lie on one line or bisector, which would make it a triangle or an abscissa. Furthermore, none of the connecting lines of the individual points can cross.

Quadrangles whose abscissae intersect at another point are called overturned quadrangles. However, it applies that quadrangles have no other intersecting points, even in this case. As with triangles, the sum of their internal angles is constant and equals 360°.

By rule, quadrangles are differentiated according to symmetry. Quadrangles which show symmetry are called parallelograms.

Quadrangles whose axial symmetries are also their diagonals are called equiangular deltoids (shaped like kites).

Quadrangles whose axial symmetries are located in the middle are called equiangular trapezoids and their middles run along axial symmetries through the centre of both sides.

If a quadrangle is axially symmetric according to both diagonals, it is punctually symmetric as well and is called a diamond.
If a quadrangle is axially symmetric according to both axes leading through the centre of each side, it is called a rectangle.

From this definition, it ensues that all rectangle fulfil conditions which apply to parallelograms. In this case, all rectangles are also parallelograms, although this condition does not apply in the reverse situation (a parallelogram does not necessarily have to be rectangular). For diamonds and deltoids, it applies that each diamond is a deltoid but not every deltoid is a diamond. These relations can be explained graphically using arrows pointing from a rectangle to a parallelogram and from a diamond to a deltoid.

Quadrangles which are axially symmetric according to the diagonal and according the centre of the sides are called a square. Squares have four sides of equal length and internal angles of 90°. A square fulfils the conditions which apply to rectangles.

Quadrangles are evident in our lives even more than are triangles, as all books, windows, notebooks, writing pads, postcards, discs, CD cases and so on have these shapes. Most quadrangles we find in our daily lives are in the shape of squares and rectangles.

Circumference of Quadrangles

With quadrangles, their circumference is also the sum of the length of their individual abscissae. However, with the standard quadrangle, there is a simple procedure to determine this. For example, with isosceles quadrangles, isosceles deltoids and rectangles, there are only two differently sized sides. We can label each differently sized side as a and b, in which case:

O = a + b + a + b = 2a + 2b

With squares and diamonds, the process is even simpler in that all sides have the same length, for which it applies that the length of the circumference of these shapes is equal to:

O = a + a + a + a = 4a

Surface Area of Quadrangles

The surface area of quadrangles, in particular with squares or rectangles, is the product of the length of its sides:

P = a.b

With squares, it applies that all sides have the same length, for which reason its surface area is:

P = a . a = aČ

With all other quadrangles, we can calculate their surface areas by breaking them down into simpler shapes, such as into triangles.

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