Geometry

Geometry is a branch of mathematics that studies the properties of space. It looks at things like distance, shape, size, and how objects are placed in relation to each other. Along with arithmetic, geometry is one of the oldest parts of mathematics. People who study geometry are called geometers.

For a long time, geometry mainly focused on Euclidean geometry. This uses basic ideas like points, lines, planes, distances, angles, surfaces, and curves. This type of geometry helps us understand and describe the world. It is used in many fields, including science, architecture, and art.

In the 19th century, geometry grew in new directions. Scientists discovered that surfaces could be studied on their own. They also found that new types of geometry, called non-Euclidean geometries, were possible. Today, geometry includes many different areas, such as differential geometry, algebraic geometry, and computational geometry. Each has its own special focus.

History

Main article: History of geometry

Geometry began a long time ago in places like Mesopotamia and Egypt around 2000 BC. People used simple geometry to help measure land, build things, and watch the stars. Important early books and tablets include the Egyptian Rhind Papyrus and Moscow Papyrus, and Babylonian clay tablets like Plimpton 322.

Later, Greek thinkers like Thales of Miletus and Pythagoras used geometry to solve problems and find new ways to prove ideas. Around 300 BC, Euclid wrote a book called Elements. This book organized many geometry ideas into a clear system that we still use today. Many different cultures added more ideas to geometry over time.

Main concepts

See also: Euclidean geometry and Axiom

Geometry is a part of math that looks at shapes, sizes, and where points, lines, and surfaces are placed. It is one of the oldest parts of math. It started with Euclid’s book Elements. In this book, he used simple ideas, called axioms, to create more complicated ideas.

Geometry studies many kinds of spaces and shapes. Points are the simplest ideas, but lines and planes are also important. We learn about angles, how to measure length, area, and volume. We also study ideas like symmetry and how shapes can look the same. Geometry helps us understand the world and is used in many places, from art to physics.

Main articles: Congruence (geometry) and Similarity (geometry)

Main article: Compass and straightedge constructions

Main articles: Rotation (geometry) and Orientation (geometry)

For broader coverage of this topic, see Dimension (mathematics)

Main article: Symmetry

Contemporary geometry

Main article: Euclidean geometry

Euclidean geometry is the study of shapes and spaces around us. It helps us learn about points, lines, planes, angles, triangles, circles, and other flat shapes. People use this kind of geometry in many areas like mechanics, astronomy, crystallography, engineering, architecture, and navigation.

Main article: Differential geometry

Differential geometry uses math to study curves and surfaces in detail. It helps scientists learn about the shape of the universe, especially how space can be curved. This type of geometry also looks at how shapes change and move.

Applications

Geometry is useful in many places. In art, geometry helps make designs and patterns. Artists use ideas like proportion and the golden ratio to make their work look nice. Tilings and tessellations are important in art, especially in Islamic art.

In architecture, geometry helps design buildings. Architects use geometry to plan shapes, create balance, and build structures like domes. In physics, geometry helps us learn about the places of stars and planets. It is also important in math areas such as calculus and number theory.

Main article: Mathematics and art

Main articles: Mathematics and architecture and Architectural geometry

Main article: Mathematical physics