WELCOME TO THE WORLD OF GEOMETRY

WHAT IS GEOMETRY???

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

A Brief History of Geometry

Geometry began with a practical need to measure shapes. The word geometry means to “measure the earth” and is the science of shape and size of things. It is believed that geometry first became important when an Egyptian pharaoh wanted to tax farmers who raised crops along the Nile River. To compute the correct amount of tax the pharaoh’s agents had to be able to measure the amount of land being cultivated.

Around 2900 BC the first Egyptian pyramid was constructed. Knowledge of geometry was essential for building pyramids, which consisted of a square base and triangular faces. The earliest record of a formula for calculating the area of a triangle dates back to 2000 BC. The Egyptians (5000–500 BC) and the Babylonians (4000–500 BC) developed practical geometry to solve everyday problems, but there is no evidence that they logically deduced geometric facts from basic principles.

It was the early Greeks (600 BC–400 AD) that developed the principles of modern geometry beginning with Thales of Miletus (624–547 BC). Thales is credited with bringing the science of geometry from Egypt to Greece. Thales studied similar triangles and wrote the proof that corresponding sides of similar triangles are in proportion.

The next great Greek geometer was Pythagoras (569–475 BC). Pythagoras is regarded as the first pure mathematician to logically deduce geometric facts from basic principles. Pythagoras founded a brotherhood called the Pythagoreans, who pursued knowledge in mathematics, science, and philosophy. Some people regard the Pythagorean School as the birthplace of reason and logical thought. The most famous and useful contribution of the Pythagoreans was the Pythagorean Theorem. The theory states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

Euclid of Alexandria (325–265 BC) was one of the greatest of all the Greek geometers and is considered by many to be the “father of modern geometry”. Euclid is best known for his 13-book treatise The Elements. The Elements is one of the most important works in history and had a profound impact on the development of Western civilization.

Euclid began The Elements with just a few basics, 23 definitions, 5 postulates, and 5 common notions or general axioms. An axiom is a statement that is accepted as true. From these basics, he proved his first proposition. Once proof was established for his first proposition, it could then be used as part of the proof of a second proposition, then a third, and on it went. This process is known as the axiomatic approach. Euclid’s Elements form the basis of the modern geometry that is still taught in schools today.

Archimedes of Syracuse (287–212 BC) is regarded as the greatest of the Greek mathematicians and was also the inventor of many mechanical devices including the screw, the pulley, and the lever. The Archimedean screw – a device for raising water from a low level to a higher one – is an invention that is still in use today. Archimedes works include his treatise Measurement of a Circle, which was an analysis of circular area, and his masterpiece On the Sphere and the Cylinder in which he determined the volumes and surface areas of spheres and cylinders.

There were no major developments in geometry until the appearance of Rene Descartes (1596–1650). In his famous treatise Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, Descartes combined algebra and geometry to create analytic geometry. Analytic geometry, also known as coordinate geometry, involves placing a geometric figure into a coordinate system to illustrate proofs and to obtain information using algebraic equations.

The next great development in geometry came with the development of non-Euclidean geometry. Carl Friedrich Gauss (1777–1855) who along with Archimedes and Newton is considered to be one of the three greatest mathematicians of all time, invented non-Euclidian geometry prior to the independent work of Janos Bolyai (1802–1860) and Nikolai Lobachevsky (1792-1856). Non-Euclidian geometry generally refers to any geometry not based on the postulates of Euclid, including geometries for which the parallel postulate is not satisfied. The parallel postulate states that through a given point not on a line, there is one and only one line parallel to that line. Non-Euclidian geometry provides the mathematical foundation for Einstein’s Theory of Relativity.

The most recent development in geometry is fractal geometry. Fractal geometry was developed and popularized by Benoit Mandelbrot in his 1982 book The Fractal Geometry of Nature. A fractal is a geometric shape, which is self-similar (invariance under a change of scale) and has fractional (fractal) dimensions. Similar to chaos theory, which is the study of non-linear systems; fractals are highly sensitive to initial conditions where a small change in the initial conditions of a system can lead to dramatically different outputs for that system.