What is Geometry? An Overview

Geometry is the area of mathematics that deals with the characteristics of space and the shapes of particular things as well as the interactions between them in space. It is one of the oldest fields of mathematics and gets its name from Greek terms that mean "Earth measuring." It developed in reaction to real-world issues like those in surveying. It was eventually realized that geometry need not be restricted to the study of rigid three-dimensional objects and flat surfaces (plane geometry and solid geometry), but rather that even the most abstract ideas and concepts could be represented and developed in geometric terms.

Major Branches of Geometry

Euclidean Geometry

A type of geometry that was appropriate for describing the relationships between the lengths, areas, and volumes of physical things that evolved in a number of ancient societies. On the basis of ten axioms, or postulates, this geometry was formalized in Euclid's Elements around 300 BCE, from which several hundred theorems were proven using deductive reasoning. For many years, The Elements served as the model for the axiomatic-deductive approach.

Analytic Geometry

René Descartes (1596–1650), a French mathematician, developed analytical geometry by introducing rectangular coordinates, which made it possible to locate points and represent lines and curves using algebraic equations. A contemporary extension of the subject to multidimensional and non-Euclidean spaces in algebraic geometry.

Projective Geometry

The concept of projective geometry was developed by the French mathematician Girard Desargues (1591–1661) to address the characteristics of geometric forms that remain unchanged when their image, or "shadow," is projected onto another surface.

Differential Geometry

The discipline of differential geometry was founded by the German mathematician Carl Friedrich Gauss (1777–1855) in conjunction with real-world issues in surveying and geodesy. He identified the fundamental characteristics of curves and surfaces using differential calculus. He demonstrated, for instance, that a cylinder's intrinsic curvature is the same as that of a plane when cut along its axis and flattened, but not a sphere because the latter cannot be flattened without distortion.

Non-Euclidean Geometries

Euclid's parallel postulate, which in its modern form reads, "given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line," was replaced by various alternatives starting in the 19th century. They aimed to demonstrate that the alternatives were illogical. Instead, they found that there are regular non-Euclidean geometries.

Topology

The properties of geometric objects that hold true after continuous deformation—shrinking, stretching, and folding, but not tearing—are the focus of topology, the newest and most complex branch of geometry. Since the introduction of broadly applicable techniques in 1911 by the Dutch mathematician L.E.J. Brouwer (1881-1966), topology has continued to advance.

Practical and Empirical

Geometry has its roots in issues that people face on a daily basis. According to the conventional myth, which was preserved in Herodotus' History (5th century BCE), the Egyptians invented surveying to restore property values following the annual flood of the Nile. Similar to this, the desire to understand solid number volumes arose from the necessity to calculate tribute, store grain and oil, and construct structures like dams and pyramids. Even the three difficult geometrical puzzles of antiquity—how to double a cube, trisecting an angle, and square a circle, all of which will be discussed later—were likely inspired by practical considerations in pre-Greek societies of the Mediterranean, from religious ritual, timekeeping, and construction, respectively. The theory of conic sections, the main topic of later Greek geometry, owed its significance.

Ancient Geometry

The writing in Euclid's Elements was so thorough and precise that it utterly destroyed the work of his forebears. The majority of what is known about Greek geometry prior to him comes from passages cited by Plato, Aristotle, and other mathematicians and writers. The general philosophy and some of the results of Pythagoras and his disciples (c. 580–c. 500 BCE) are among the priceless treasures they kept. All things are numbers or owe their relationships to them, the Pythagoreans came to believe. The doctrine gave mathematics the top priority when it came to exploring and comprehending the universe. A similar viewpoint was developed by Plato, and philosophers who were influenced by Pythagoras or Plato frequently wrote ecstatically about geometry serving as the basis for understanding the cosmos. Thus, the sublime and ancient geometry became associated.

Ancient surveyors and builders needed to be able to quickly create perfect angles on the spot. It appears that the Egyptians used a rope to set out their construction requirements, which gave them the nickname "rope pullers" in Greece. One way they could have used a rope to build right triangles was to mark a looped rope with knots, requiring that the rope form a right triangle when held at the knots and pulled tightly. Take a rope that is 12 units long, tie a knot of 3 units from one end and another 5 units from the other end, and then tie the ends together to form a loop to execute the trick. 

Modern historians have uncovered puzzles in Babylonian clay tablets (c. 1700–1500 BCE) whose solutions show that the Pythagorean theorem and several unique triads were understood more than a thousand years before Euclid. However, it is highly improbable that a right triangle formed at random will have all of its sides measurable by the same unit, i.e., every side is a whole-number multiple of some standard unit of measurement. The Pythagoreans were shocked to learn this fact, which led to the development of the idea and theory of incommensurability.