ROTATIONAL SYMMETRY IN A PLANE

When we say that a figure has rotational symmetry we mean that it is possible to rotate it about a center in such a way that the whole figure is unchanged but the parts of it have been rotated. The number of ways in which this can be done is called the order of rotational symmetry, and it is usual to count the identity operation (rotation through 0 degree or 360 degree, and so on) as one of the symmetries. Any figure has the trivial rotational symmetry of order one about any point. The letter N, the parallelogram and many other familiar figures, have rotational symmetry of order two.

THE TRAPEZIUM

There is one more special type of quadrilateral which frequently occurs in practical work, the trapezium; this is a quadrilateral with just one pair of opposite sides parallel.

In the side figure, this is a trapezium ABCD with an axis of symmetry. Such a figure is called an isosceles trapezium.

TRIANGLES AND QUADRILATERALS (Part 1)

Thales of Miletus, knew that the measurements of a triangle were fixed when the length of its base and the sizes of its basebase angles were known. He used this fact to determine the distance of a ship at sea. Pythagoras of Samos, who settled eventually in Crotona, develoved  the theory of congruent triangles. It was probably in his school at Crotona that the congruence theorems were first analysed and related to one another in some kind of logical order.

During the life time of Plato (429-348 B.C.) Athens became the chief center of mathematical studies. It held supremacy for about 150 years, when the first university in the world, at Alexandria, displaced it. Plato was born near Athens of rich and noble parents. He studied in Egypt, Cyrille, and Italy, returning to Athens about 380 B.C. About a mile outside Athens was a beautiful walled garden called Academia, after its owner Academos. Here Plato formed a school of students which came to be known as the Academy. Plato was a philosopher who believed in geometry as one of the foundations of a liberal education. The inscription over the entrance to the academy ran 'Let none ignorant of geometry enter my door'. Plato did great service to mathematics in making its foundations logical and secure. We still tend to regards the compasses and ruler as the chief mathematical instruments largely because he would allow no other instruments to be used.