History of geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is the study of shapes and spaces. It is one of the two main parts of math from long ago, the other being the study of numbers (arithmetic).

Classic geometry was about making shapes and solving problems using simple tools like a compass and straightedge (compass and straightedge constructions). The field changed a lot because of Euclid. He made math more exact and logical. He used the axiomatic method, a way to start with simple ideas and build knowledge from them. This way of thinking is still used today. His famous book, The Elements, was very important and was read by many people in Europe for a long time.

Today, geometry has grown into many new and abstract forms. It now works closely with calculus and algebra, creating new parts of math that look very different from older geometry. Now, geometry helps us understand the world in many unexpected ways (see Areas of mathematics and Algebraic geometry).

Early geometry

The earliest known geometry began with ancient people like those in the ancient Indus Valley and Babylonia around 3000 BC. Early geometry helped with practical tasks such as surveying, construction, astronomy, and crafts. People learned about lengths, angles, areas, and volumes, and some of their ideas were very advanced.

The ancient Egyptians and Babylonians knew about the Pythagorean theorem long before Pythagoras. They also had ways to estimate the area of a circle and find the volume of shapes like pyramids. In Vedic India, geometry was used to build altars, and texts like the Śulba Sūtras showed knowledge of the Pythagorean theorem and special number triples.

Greek geometry

See also: Greek mathematics

Thales of Miletus, who lived long ago, was one of the first people to use careful thinking to study shapes. He was the first known person to use steps to solve math problems. Pythagoras, who lived after Thales, studied math and music. He and his students found important ideas about shapes, like the relationships between the sides of triangles, that we still learn about today.

Plato, another famous thinker, said that geometry should only use a compass and a straightedge to draw lines and circles. This idea helped mathematicians see what could be made with just these tools. Later, Euclid wrote a well-known book called The Elements of Geometry. This book organized geometry into clear rules that many people still use. Euclid showed how geometry could be built step by step from simple ideas. Other important mathematicians like Archimedes discovered new things about shapes and how objects behave in water.

Classical Indian geometry

See also: Indian mathematics

Classical Indian geometry had many interesting problems and answers. Ancient Indian mathematicians studied the sizes of different shapes, including some that don’t have smooth edges. They also found ways to work out areas and amounts to fill up space, showing great skill in math.

One important mathematician, Brahmagupta, wrote about useful math questions. He learned special rules for four-sided shapes where you can join opposite corners with a straight line. He made a formula to find the space inside these shapes, which was an important discovery in geometry. His work helped other mathematicians learn more about these shapes.

Chinese geometry

See also: Chinese mathematics

The earliest known Chinese book about geometry is the Mo Jing, written by followers of the philosopher Mozi around 330 BC. This book described simple ideas about points, lines, and shapes. It explained that a point is the smallest part of a line and gave rules for measuring lengths and spaces.

Later, during the Han dynasty, Chinese mathematicians kept developing geometry. The book The Nine Chapters on the Mathematical Art included many geometry problems, such as finding the areas of squares and circles and the volumes of different 3D shapes. It also had early proofs of the Pythagorean theorem. Famous mathematicians like Zhang Heng and Zu Chongzhi found better ways to calculate pi, an important number in geometry.

Islamic Golden Age

See also: Islamic mathematics

During the Islamic Golden Age, mathematicians made big steps in geometry. Thābit ibn Qurra found new ways to show why the Pythagorean theorem works for all triangles. Researchers found that girih tiles, used in pretty Islamic art, made patterns like fractal designs.

Renaissance

Greek ideas about geometry reached Europe through books written in Arabic. This happened in the 10th century and became very important in the 12th century when these ideas were translated into Latin. These new ideas added to the work of Euclid and helped create many new geometry rules.

In the 14th and 15th centuries, during the Renaissance, artists and builders made big steps in showing depth and space in their paintings and buildings. Filippo Brunelleschi showed how to use geometry to make paintings look deep, a way that artists like Masolino da Panicale and Donatello started to use. This helped paintings show one clear scene instead of many small scenes. Later, Leon Battista Alberti and Piero della Francesca wrote books about these geometry methods, helping others learn them too. These ideas traveled from Florence to artists across Europe.

Modern geometry

The 17th century

In the early 1600s, geometry changed a lot. Two big changes happened. First, analytic geometry was created by René Descartes and Pierre de Fermat. This new way of thinking about geometry used coordinates and equations. This helped make calculus and physics more exact. The second big change was the study of projective geometry by Girard Desargues. This type of geometry looks at how points line up with each other.

Later in the 1600s, Isaac Newton and Gottfried Wilhelm Leibniz both worked on calculus. Calculus helps solve problems, like finding the slope of curved lines and the area under those curves. Even though calculus is not a part of geometry, it helps solve geometry problems.

The 18th and 19th centuries

Non-Euclidean geometry

Main article: Non-Euclidean geometry § History

For a very long time, people tried to prove something called Euclid’s Fifth Postulate, known as the “Parallel Postulate”. This rule was hard to prove using only the first four rules Euclid had written. In the early 1800s, three mathematicians—Gauss, Johann Bolyai, and Lobachevsky—decided to create a new kind of geometry where this rule wasn’t true. They succeeded, and this new geometry is called non-Euclidean geometry. Later, Bernhard Riemann used calculus to study the shapes of smooth surfaces. This work later helped form the base for Einstein’s theory of relativity.

Introduction of mathematical rigor

As mathematicians worked on the Parallel Postulate, they realized how hard it was to separate logical thinking from what we naturally think space looks like. To fix this, David Hilbert created a new set of rules in 1894 called Hilbert's axioms. These rules were complete and didn’t rely on pictures or natural feelings about space.

Analysis situs, or topology

In the middle of the 1700s, mathematicians noticed that some ideas repeated when they looked at numbers, flat shapes, and solid shapes. This led to the idea of a metric space, where they could study these ideas in a more general way. This area of study was called analysis situs, and later topology. Instead of focusing on exact shapes and angles, topology looks at ideas like connectedness and boundaries.

Geometry of more than 3 dimensions

In the 1800s, Ludwig Schläfli expanded geometry to more than three dimensions. He found all the special shapes, called Platonic solids, that can exist in four dimensions and discovered there are three such shapes in even higher dimensions.

In 1878, William Kingdon Clifford created geometric algebra, which brought together different math ideas and showed their geometric meaning, especially in four dimensions.

The 20th century

In the 1900s, algebraic geometry grew, studying curves and surfaces using finite fields and also real or complex numbers. Finite geometry found uses in coding theory and cryptography. With computers, new areas like computational geometry and digital geometry appeared, focusing on using computers to solve geometry problems and work with geometric data.

Timeline

Main article: Timeline of geometry

Geometry started a long time ago as a way to understand spaces and shapes. It began in ancient times, especially in places like Greece. People there wanted to measure and describe the world around them. Early geometry used simple tools like compasses and straightedges to draw and solve problems about shapes and sizes. This knowledge helped people build structures, plan farms, and navigate the land. Over many years, geometry grew and changed, becoming an important part of mathematics.