Euclidean geometry

Euclidean geometry is a way to study shapes and spaces. It was created by Euclid, an ancient Greek mathematician. Euclid wrote a book called Elements. In this book, he started with a few simple rules called axioms. From these rules, he figured out many other ideas about shapes. One key rule is the parallel postulate. This rule explains how parallel lines act on a flat surface called a Euclidean plane. Euclid was the first to put these ideas into a logical system where every answer is proved using the rules and what was already proven.

The Elements book starts with plane geometry. This is the first type of geometry that many students learn in secondary school. It is also the first example of an axiomatic system and helps students learn about mathematical proofs. The book also talks about solid geometry in three dimensions. It includes ideas from algebra and number theory, but these are shown using shapes.

For a long time, people thought Euclidean geometry was the only kind of geometry that existed. But now we know there are other types called non-Euclidean geometries. These are hyperbolic and elliptic geometry. Because of Albert Einstein's theory of general relativity, we learned that space is not always Euclidean. It only seems that way over short distances or where gravity is weak.

Euclidean geometry is a type of synthetic geometry. This means it starts with simple ideas about points and lines and builds more complicated ideas from there. This is different from analytic geometry. Analytic geometry was started by René Descartes. It uses coordinates and algebraic formulas to explain geometry.

The Elements

Main article: Euclid's Elements

The Elements is a book about geometry written by Euclid. It brings together many ideas 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 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 named with 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.

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, or if two sides and the angle between them match, or if two angles and one side match.

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.

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 for engineers. It helps them design and understand many things. For example, it is used to make gears, lenses, and heat exchangers. It also helps create 3D models in computer-aided design (CAD) systems. These models are used to build cars, airplanes, and many other products.

Euclidean geometry is used to study vibrations in machines, design airplane wings, and calculate satellite orbits. It also helps in making circuits and antennas. In short, Euclidean geometry gives engineers the basic shapes and measurements they need to build and improve technology.

Other general applications

Euclidean geometry has many useful applications. Surveyors use it to measure and map land accurately. It also helps in packing objects well, 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. 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. He did important work on shapes and their sizes. Apollonius of Perga studied special curves called conic sections.

In the 1600s, René Descartes created analytic geometry. This is a new way to describe shapes using math. It uses numbers to show points and equations to show lines and curves. This changed how people study geometry.

During the 1700s, mathematicians tried to prove one of Euclid’s rules using the others but could not. They also found that some old geometry problems, like trisecting an angle, could not be solved with just a compass and straightedge.

In the 1800s, mathematicians looked at geometry in more than three dimensions. They also made non-Euclidean geometry, where the rules about parallel lines are different. 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 thought his axioms were simple truths about space. He believed that moving shapes, like sliding or turning them, would not change important parts like side lengths and angles. These movements are called Euclidean motions.

Euclid’s ideas show space as smooth and without gaps. 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 how we understand space.

Treatment of infinity

Euclid talked about lines that end and lines that go on forever. He only made this difference when it mattered for his work. Later thinkers, like Proclus, studied questions about infinity. They tried to prove ideas such as how a line can be divided again and again without end.

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

Logical basis

Euclidean geometry is a way to understand shapes and spaces. It 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.

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 worked to make Euclidean geometry clearer. They use basic ideas called axioms. From these axioms, many other facts can be proven. Mathematicians like Hilbert and Tarski created their own sets of axioms. These efforts help us understand the rules of Euclidean geometry better.