Plane Geometry

What is Plane Geometry ?

The idea of Plane Geometry could have emerged when geometers tried to represent 3D shapes on a flat surface (either sand or parchment) for studying them in detail.

In two dimensions, a sphere becomes a circle, cone a triangle and a cube a square and a cuboid a rectangle. To describe these 2D figures, Face became Surface, Edge became Line and Corner became Point. In a closed figure any two lines which meet at a point are also called sides.

The flat surface on which these figures were drawn is known as a Plane. A Plane was defined as a surface on which a Straight Edge or a ruler could be kept in any orientation without leaving any gaps between the straight edge and the surface. In a plane, if we mark any two points and join them by a straight line, the line will also be on the plane completely.

We can contrast a plane with a curved surface like that of a cylinder. On this we can mark 2 points so that the line joining them would be curved and not straight.

The top of a table or the face of a cube is examples of planes.

Elements of Plane Geometry

All shapes drawn on a plane can be thought of as consisting of straight lines & curved lines. Plane Geometry studies the properties when several lines intersected (or did not) and made several patterns on a plane. It also studies the properties of closed figures and angles which are formed by intersecting lines.

Use of Deductive Logic & the Axiomatic System

Euclid gathered all such previous results, including some of his own.

Apart from just compiling all the theorems of Plane Geometry, Euclid also organised them in a particular order. He arranged them in such a manner that all the results needed to prove any particular theorem were all proved in theorems which appeared before that theorem. He used ideas of logic (what today is called deductive logic) developed by Plato & Aristotle.

This provided a logical structure by which the entire discipline of “plane geometry” could be developed independently from a few simple & unproved assumptions.

But what about the very first theorem? What proofs does it depend on?

For this, he assumed right at the start, certain "postulates or axioms" and "common notions" regarding points, lines & angles, which he said were not provable but can be "seen" as true. These axioms were obviously based on our real life experiences but abstracted into ideal shapes.

He then achieved a system where each result followed from results which had been proved earlier in the sequence.

Today such a system, which starts with a few “unproved” assumptions, is called an axiomatic system. Development of Plane Geometry in Elements was the first example of an axiomatic deductive system.

Power of the Axiomatic System

For 2000 years, mathematicians & philosophers were not satisfied that the Fifth Postulate of the Elements was not a fundamental postulate. But they could not put their finger on the exact problem.

It was in the 19th century, they realised that the Fifth Postulate, as stated by Euclid, described the behaviour of straight lines on a plane.

When they altered the Fifth Postulate they came up with other geometries which were equally valid! This was the intuition behind Spherical Geometry & Hyperbolic Geometry.

Through this process, the strength & flexibility of the Axiomatic System was realised.

The Elements

He established the discipline of Plane Geometry on a firm foundation. His book "Elements" is the most published textbook after the invention of the printing press. For almost 2000 years it was the standard book on geometry. It has been translated into all the major languages of the world. It established the idea of "proof" as one of the foundations of the scientific method.

Proclus, author of "Eudemian Summary", the main source of our information about very early Greek Geometry, says that once Ptolemy asked Euclid whether there was any shorter way to a knowledge of Geometry, other than by a study of the Elements, Euclid is supposed to have answered "There is no royal road to Geometry".

Conic Sections

Apollonius (1stcentury AD) also showed that some of the 2D figures like circle, ellipse and parabola can also be got by shearing a cone with a plane surface at different angles. His method was called Conic Sections.

"Greek" is a Cultural Idea

We must remember that many of the "Greek" mathematicians who contributed to the development of geometrical ideas, may not be from the area that we today call Greece.

Alexander the Great, who lived in the 4thcentury BC expanded the Greek rule to many countries around the Mediterranean Sea. Hence many of them lived in modern Turkey and Egypt!

Geometry in the Present School Math Curriculum

One of the purposes of math education itself is to train students in logical thinking. Study of Euclid's Geometry is one of the best introductions to logical thinking which is accessible even to Primary School students.

It is unfortunate that the current geometry curriculum in schools has been rid of these logical processes and reduced to a set of recipes to be memorized.

Coloured Version of Elements

In 1847 mathematician Oliver Byrne employed a color scheme for the figures and diagrams in his most unusual edition of Euclid's Elements.

The author says that “The Elements of Euclid in which coloured diagrams and symbols are used instead of letters for the greater ease of learners.”