What is Geometry

“Let no one who is unacquainted with Geometry enter here” – Inscription on the door of Plato’s Academy

Geometry is an important branch of Mathematics.

Mathematics developed by our attempts to recognize patterns in the environment around us. This also resulted in a language structure to describe these patterns. Mathematics is both a process of thinking and development of a pattern language.

Humans recognized patterns around them in 2 ways – one through the idea of numbers and the other through shapes. Study of numbers developed into Arithmetic and that of shapes developed into Geometry. Algebra was later developed as a discipline which integrated Arithmetic and Geometry into a universal language for math.

Need for Geometry

As human civilizations developed, they needed to cultivate, trade, build & measure. For living and worshipping they built houses, temples & monuments. For trading & travel they built vehicles, ships & waterways. For agriculture they studied seasonal patterns, dug fields, tanks, wells & canals. As kingdoms developed, kings needed ways of measuring sizes of fields to estimate agricultural production for levying taxes.

Humans obviously took the shapes of objects around them as models for these constructions. They modified these shapes into abstract ideal shapes with perfect surfaces, straight lines, replicable curves etc. Spheres possibly developed from the Sun & the Moon, cylinders from trees, cones from hills, cubes from stone formations and lines from rays of the Sun.

For constructing structures similar to these shapes, they had to understand properties like length, radius etc and arrive at procedures for working out perimeter, area, volume etc.

All ancient civilizations like Egypt, Sumeria, Mesopotamia, China, Greece & India developed such procedures for constructing these shapes. For example, it was well known in all these civilizations that a rope triangle with sides 3, 4 & 5 units, when stretched at the 3 corners will form a shape with which perpendicular lines can be drawn. The Vedic Sulba Sutras contained directions for constructing different kinds of alters required for various yagnas.

From such needs the science of Geometry developed. Geometry literally means “To measure Earth”. The geometry developed by most civilizations consisted of procedures for drawing shapes & calculation of properties associated with these shapes.

It was the Greeks who developed geometry into a science and the developed the idea of a “logically arrived at” proof from certain basic or known assumptions. Many Greek philosophers & geometers studied 2 D figures for at least 2 centuries before Euclid. Thales of Miletus, who lived in the 6th century BCE, is considered the first of the Greek mathematicians.

Greek mathmaticians discovered several properties & relations related to the lines & angles in these figures and formulae for their perimeter and area. Two such results were that the sum of the angles of a triangle is two right angles and that a triangle with sides 3, 4 & 5 will be a Right Angle triangle.

Thales is supposed to have astonished Egyptians by finding the height of a pyramid by using a geometric principle; that of property of similar triangles. He did it by using the relation between the length of a stick and the length of the shadow cast by it & correlating them with the length of the shadow cast by the pyramid. Egyptians had mastered practical geometry and could build amazing structures, but they had not worked out the concepts underlying their methods.

Euclid (around 300 BC) collected all such geometrical proofs, which were known as Theorems, which had been developed in the 3 centuries before his time and compiled them into one encyclopaedic book called “Elements”. In “Elements”, Euclid also developed the “axiomatic method” which has been widely accepted and used in scientific disciplines. We will see details of this method in a later chapter.

Geometry & Logical Thinking

Geometry was also presented as a deductive science. Starting from a few simple assumptions, Euclidean Geometry builds a structure of deductions of increasing complexity. Study of Geometry also stressed the aspect of proof.

It was the first example of an axiomatic deductive system, which became a model for many disciplines. Study of Geometry is expected to provide training in deductive thinking.

The School Geometry Curriculum

Robert Simson (1687–1768), a math professor at the University of Glasgow, adapted Euclid's Elements into English so that Geometry can be taught as a subject in school. It has provided the framework for the current school geometry curriculum.

Learning Geometry in Primary School

Many concepts in geometry are difficult to understand through the medium of language as the vocabulary of geometry consists of may unfamiliar words. Some examples are words like isosceles & hypotenuse which are almost never used outside a geometry class.

However geometrical ideas are easy to experience through the movement of parts of the body.

When we stand with our legs apart, the hips and the legs form a triangle. When we stand straight or hold our arms sideways, we can experience a straight line.

An angle drawn on a sheet of paper is static, though the concept is a dynamic one. We can form a right angle with out forearms and elbow. By "rotating" the elbow we can experience a right angle reducing to an acute angle or increasing into an obtuse angle.

We can form innumerable angles using all parts of our body.

At its basics, a dance is nothing but the continuous formation of several geometric shapes.

These approaches would help the body & brain form muscle memories of geometrical shapes and improve understanding.

Figures & Shapes

In the study of geometry, it is useful to understand the difference between a figure & a shape.

A figure is any collection of points on a plane or space. The world is full of figures. The profile of a rock or shrub is a figure. Geometry studies only a small subset of figures which can be made sense in terms of certain properties. These were called geometrical shapes and include point, line, triangles etc. We can say that the geometers abstracted “geometrical shapes” out of the many “figures” that they see around them.

Chapter 22.2 gives a framework of the different topics that we would be studying in geometry.