The Origins of Geometry

Geometry is a discipline of mathematics that studies the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues. It has its origins in the classical Greek mathematics of Euclid, who presented geometry as an abstract discipline derived from certain assumptions about physical space. In the 19th century, geometric ideas were reinvented by Bernhard Riemann in his theory of manifolds, which allowed for a far deeper analysis of geometry. These days, geometry is used in many different fields, from art to engineering.

Euclidean Geometry

In 300 BC, the Greek mathematician Euclid wrote a treatise entitled "The Elements". In this work, he collected together all of the known geometrical knowledge of the time and presented it in a logical and coherent manner. This work was tremendously influential in shaping Western mathematics and sciences for centuries to come.

One of the most important parts of "The Elements" is Euclid's presentation of five postulates (or axioms), which he uses as the basis for all of his geometry. These postulates are:

1) A straight line segment can be drawn between any two points.

2) Any straight line segment can be extended indefinitely in a straight line.

3) Given any straight line segment, a circle can be drawn with the segment as its radius and one endpoint as its center.

4) All right angles are congruent (i.e., have equal measure).

5) If two lines intersect at a point then the alternate interior angles formed are congruent (i.e., have equal measure).

From these simple statements, Euclid is able to derive all sorts of complicated results about the nature of geometry.

Non-Euclidean Geometry

In the 19th century, mathematicians began to explore what would happen if one or more of Euclid's postulates were not true. This led to the development of non-Euclidean geometry, which studies geometries that don't conform to Euclid's assumptions. The most famous examples are spherical geometry (which takes place on the surface of a sphere) and hyperbolic geometry (which takes place on a saddle-shaped surface). Both of these geometries violate one or more of Euclid's postulates; for example, in spherical geometry there are no parallel lines because any two lines on a sphere eventually intersect! Non-Euclidean geometry has found applications in everything from astrophysics to video game design.

As you can see, geometry has come a long way since its humble beginnings in ancient Greece! Today it exists in many different forms, from Euclidean to non-Euclidean, and has applications across multiple fields. Hopefully this brief overview has given you a better understanding of where geometry came from and how it evolved into the rich subject it is today.

FAQ

Where was geometry started?

Geometry was started in ancient Greece by Euclid, who wrote the Elements around 300 BC. Other Greek mathematicians such as Archimedes and Pythagoras also made important contributions to the subject. Geometry was later developed by Arab mathematicians in the Middle Ages and by European mathematicians in the Renaissance. Today, geometry is an important part of mathematics and is used in fields such as architecture, engineering, and physics.

What is the origin of geometric shapes?

The shapes that we see in geometry are based on the shapes that exist in nature. Euclid’s Elements contains a proof that the five regular solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) are the only possible convex regular solids. These shapes are found in many places in the natural world, from crystals to molecules to planetary orbits. The study of geometry can help us to understand the world around us better.