Euclidean geometry

Euclidean geometry is a mathematical system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One important idea is the parallel postulate, which tells us how parallel lines behave on a Euclidean plane. Euclid was the first to organize these ideas into a logical system where every result is proved using axioms and earlier proven theorems.

The Elements begins with plane geometry, which is still taught in secondary school as the first example of an axiomatic system and the first steps in learning about mathematical proofs. It also covers solid geometry in three dimensions. Many ideas from algebra and number theory appear in the Elements, but they are explained using geometry.

For over two thousand years, people thought Euclidean geometry was the only kind of geometry possible because its ideas seemed obviously true. But today we know about two other types of geometry called non-Euclidean geometries: hyperbolic and elliptic geometry. Because of Albert Einstein's theory of general relativity, we now understand that space itself isn’t perfectly Euclidean; it only seems that way over short distances or where gravity is weak.

Euclidean geometry is a type of synthetic geometry, meaning it starts with simple ideas about points and lines and builds more complex ideas from there. This is different from analytic geometry, a method started by René Descartes that uses coordinates and algebraic formulas to describe geometry.

The Elements

Main article: Euclid's Elements

The Elements is a book about geometry written by Euclid. It brings together many ideas that were already known and organizes them in a clear way. The book has 13 parts, or "books," covering different areas of geometry.

The first four books and the sixth book talk about flat shapes, like triangles and squares. They prove many facts, such as the Pythagorean theorem, which tells us about the relationship between the sides of a right-angled triangle. Books five and seven to ten discuss numbers, looking at them as lengths and areas. They talk about prime numbers and numbers that cannot be written as simple fractions, called irrational numbers. The last three books, eleven to thirteen, discuss solid shapes, like cones and cylinders, and also build special shapes called the platonic solids.

Axioms

Euclidean geometry is a system where we start with a few simple rules, called axioms, and build up more complex ideas from them. Euclid listed five basic rules, or postulates, for geometry. These include ideas like being able to draw a straight line between any two points, or drawing a circle with any center and radius. One important rule is the parallel postulate, which talks about what happens when a line crosses two other lines.

Parallel postulate

Main article: Parallel postulate

The parallel postulate was seen as tricky even in ancient times. It says that if a line crossing two other lines makes the angles on one side add up to less than two right angles, then those two lines will eventually meet up on that side. This idea is important because it helps us understand how parallel lines behave.

Methods of proof

In Euclidean geometry, proofs often show how to construct shapes using just a compass and a straightedge. This makes the geometry very hands-on and practical. Euclid used different ways to prove his ideas, including proofs by contradiction, where he shows that the opposite of what he wants to prove leads to a problem.

Notation and terminology

Points in geometry are usually given names using capital letters like A, B, or C. We can use these points to name shapes. For example, triangle ABC has points at A, B, and C.

Angles that add up to a right angle (90 degrees) are called complementary angles. Angles that add up to a straight angle (180 degrees) are called supplementary angles. Today, we measure angles in degrees or radians. Geometry also talks about lines that go on forever, rays that stop at one end, and line segments that have two ends.

Some important or well known results

The pons asinorum or bridge of asses theorem states that in an isosceles triangle, the angles at the base are equal. This theorem often served as an early test of understanding in Euclid's Elements.

The triangle angle sum theorem tells us that the three angles of any triangle always add up to 180 degrees.

The Pythagorean theorem is a famous rule that says in a right triangle, the area of the square on the longest side (the hypotenuse) equals the combined areas of the squares on the other two sides.

Thales' theorem explains that if you have a diameter line on a circle, then the angle opposite that line is always a right angle.

Triangles can be proven to match exactly if all three sides are the same (SSS), or if two sides and the angle between them match (SAS), or if two angles and one side match (ASA).

We also know that the area of shapes grows with the square of their size, and the volume of solids grows with the cube of their size. Euclid explored these ideas in special cases like circles and certain solids.

System of measurement and arithmetic

Euclidean geometry uses two main types of measurements: angle and distance. Angles are measured using a right angle as the basic unit. For example, a 45-degree angle is half of a right angle. Distances are measured by choosing a specific line segment as the unit length, and all other distances are compared to this unit.

We can also measure area and volume using distances. For example, a rectangle that is 3 units wide and 4 units long has an area of 12 square units. Euclid described shapes as "equal" if they have the same size, whether in length, area, or volume. Shapes are congruent if one can be moved to exactly match the other in size and shape. Similar shapes have the same shape but different sizes, with their corresponding angles equal and sides in proportion.

In engineering

Euclidean geometry is very important in engineering. It helps engineers design and analyze many things. For example, it is used in designing gears, lenses, and heat exchangers. It also helps in creating 3D models in computer-aided design (CAD) systems, which are used to make cars, airplanes, and many other products.

Euclidean geometry is also used in analyzing vibrations in machines, designing airplane wings, and calculating satellite orbits. It even helps in designing circuits and antennas. In short, Euclidean geometry provides the basic shapes and measurements that engineers need to build and improve all sorts of technology.

Other general applications

Euclidean geometry has many practical uses. For example, surveyors use it to measure and map land accurately. It also helps in packing objects efficiently, like stacking oranges in a grocery store.

Geometry is important in architecture, helping designers build structures. It is also used in art, to create beautiful patterns, and in designing origami, where paper folding follows geometric rules. Even everyday objects like water towers use geometry in their shape and volume calculations.

Later history

See also: History of geometry and Non-Euclidean geometry § History

Archimedes was one of the greatest ancient mathematicians, remembered for his original work on volumes and areas of various shapes. Apollonius of Perga is known for studying conic sections.

In the 1600s, René Descartes created analytic geometry, a new way to describe shapes using algebra. This method uses coordinates to represent points and equations to represent lines and curves. It changed how mathematicians understand geometry.

During the 1700s, mathematicians tried to prove one of Euclid’s postulates using only the others but couldn’t succeed. They also discovered that some classic geometry problems, like trisecting an angle, couldn’t be solved with just a compass and straightedge.

In the 1800s, mathematicians explored geometry in more than three dimensions. They also developed non-Euclidean geometry, where the rules about parallel lines are different from Euclid’s. This new geometry later helped explain space and time in Einstein’s theories of relativity.

Main article: Non-Euclidean geometry

As a description of the structure of space

Euclid believed that his axioms were obvious truths about how space works. He thought that moving shapes around, like sliding, flipping, or turning them, wouldn’t change certain important properties like side lengths and angles. These movements are called Euclidean motions.

Euclid’s ideas describe space as smooth and continuous, without gaps or edges. His rules suggest that space looks the same in every direction and can be flat, like a plane. Later, Albert Einstein’s theory of relativity changed this understanding.

Treatment of infinity

Euclid often talked about finite lines and sometimes about infinite lines, but he only made these differences when they were important for his work. Later thinkers, like Proclus, looked closely at questions about infinity and tried to prove ideas such as the endless divisibility of a line.

In the 20th century, mathematicians like Otto Stolz, Paul du Bois-Reymond, Giuseppe Veronese, and Abraham Robinson explored models of Euclidean geometry where distances could be infinitely large or infinitely small. They built strong logical reasons to support these ideas.

Logical basis

Euclidean geometry is a way of understanding shapes and spaces that was first described by the ancient Greek mathematician Euclid in his book Elements. Euclid used a special method called proof by contradiction to show that certain ideas were true. This method works because, in classical logic, every statement is either true or false — there are no in-between possibilities. If assuming something is false leads to a silly or impossible result, then that something must be true.

Over time, mathematicians have tried to make Euclidean geometry more clear and complete by using a set of basic ideas, called axioms, from which many other facts can be carefully proven. Different mathematicians, like Hilbert and Tarski, created their own sets of axioms to make Euclidean geometry stricter and easier to understand. These efforts help us see exactly how the rules of Euclidean geometry fit together and what they mean.