What is Euclidean Geometry?
Euclidean Geometry can be described in the following points;
- In Euclidean geometry, the essential vital ideas are points and lines.
- Points are considered the essential pieces of geometry, and lines are groups of points that go on forever in both directions.
- Euclidean geometry also contains triangles, circles, and other two-dimensional and three-dimensional patterns, corners, parallel lines, perpendicular lines, and lines that cross each other.
- One of the most critical parts of Euclidean geometry is its axiomatic approach.
- Euclid made a set of basic assumptions, or axioms, from which all of the other ideas and theorems in geometry can be obtained.
- These axioms are identity truths that are approved without proof.
- They provide a solid basis for the logical development of geometric ideas.
Definition of Euclidean Geometry
History of Euclid Geometry
Around 300 BC, Euclid was a Greek mathematician who lived in Alexandria. He devised a way to prove math that is still used today. He used it to show that the math of his time was correct, and he also wrote down several theorems in plane geometry and number theory. Euclid showed that there are infinite prime numbers and that almost all geometry can be done with a ruler and compass. He is now known as the “father of geometry.”
The excavations at Harappa and Mohenjo-Daro show that the towns of the Indus Valley Civilization were very well planned (about 3300-1300 BC). The Egyptians used geometry a lot to build the Pyramids so well. In India, the Sulba Sutras, which are geometry books, show that geometry was used during the Vedic period.
Geometry was changing slowly until Euclid, a math teacher in Alexandria, Egypt, put together most of these changes in his famous book, which he called “Elements.” Euclid is known as the “father of geometry” because he created how people do geometry now. In his book “Elements,” Euclid put together all of the math that was known at the time and a lot of his math, then proved it all using math. Because of how well-written his proofs were and how influential his work was, “Elements” is still in print today.
Euclidean Geometry in terms of Plane Geometry & Solid Geometry
Euclidean geometry is an axiomatic system, meaning that all the theorems come from a small number of simple statements. Euclidean geometry is called “plane geometry” about points, lines, angles, squares, triangles, and other shapes. It looks at how things work and how they connect.
|Plane Geometry||Solid Geometry|
|1. Congruence of triangles|
2. Similarity of triangles
4. Pythagorean theorem
6. Regular polygons
7. Conic sections
2. Regular solids
Euclidean plane Geometry
- For Points, Lines, and Planes
Euclidean geometry is about points, lines, and flat surfaces. A point is a place in space that has no size. On the other hand, a line is a straight path that can go in either direction forever. A plane is a flat surface that goes on forever in all directions.
- For Euclidean Axioms
Euclidean configuration is based on axioms, which are apparent truths on which all theorems and proofs are built. The most important of these axioms is the parallel postulate, which says that two simultaneous lines never meet, and the Pythagorean theorem, which shows how the sides of a right triangle are related.
- For Congruence
The idea of congruence is that two shapes are the same size and shape. Two shapes are congruent if they can be turned into each other by moving, rotating, and reflecting.
- For Similarity
The idea of similarity is that two shapes are the same, but their sizes are different. If the angles and the lengths of the sides are the same, then the shapes are similar.
- For Triangles
In Euclidean geometry, triangles belong to the most basic shapes. They are polygons with three sides, and in geometry, much time is spent studying their properties and how they fit together.
- For Circles
Another basic shape in Euclidean Plane geometry is the circle. They are a group of locations in a plane all the same distance from the center.
- For Geometric Structures
Geometric construction projects, in which a compass and straightedge are used to make geometric patterns and figures, are also a part of Euclidean geometry.
- For Coordinate Geometry
Coordinate geometry is the use of algebraic techniques for investigating geometry. It involves giving points in a plane’s exact location and using formulae to describe shapes and figures.
Euclidean Solid Geometry
- For Points, Lines, and Planes
This is what Euclidean geometry comprises at its most basic level. A point is a place in space, a line is a straight path that goes on forever in two different directions, and a plane is a flat surface that goes on forever in all directions.
- For Shapes and Figures
Euclidean geometry studies many different shapes and figures, such as polygons (like triangles, quadrilaterals, pentagons, etc.), circles, cylinders, spheres, cones, and pyramids.
- For Properties of Solids
Euclidean geometry allows us to talk about and study the properties of solid things. Some essential properties include volume, surface area, angles, symmetry, and congruence.
- For Transformations
There are different kinds of transformations in Euclidean geometry that can be used to change shapes. Some of these are translations, rotations, reflections, and dilations.
- For Proofs
Euclidean geometry is known for its strict proofs, which are based on axioms and theorems and use logical arguments. Proofs are vital to Euclidean geometry because they show that statements about geometry are factual.
Euclidean Geometry Examples
Angles and circles are two examples of Euclidean geometry that are often used. Angles are now two straight lines that meet. A circle is a flat shape with points all the same distance from the center. This distance is called the radius.
Elements of Euclidean Geometry
In Euclidean geometry, Euclid’s Elements is an arithmetical and fractal geometry work consisting of 13 texts published by ancient Greek scientist Euclid in Alexandria, Ptolemaic Egypt. Also, the “Elements” were broken up into thirteen textbooks that made geometry famous worldwide. These Aspects are a collection of concepts, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions.
Books 1 through Book 4 and Book 6 talk about plane geometry. He came up with five rules for plane geometry. These rules are called Euclid’s Postulates, and the geometry they describe is called Euclidean. Because of his work, we all have a common place to learn about geometry. It is the basis for geometry as we know it today.
Euclidean axioms are a collection of five postulates or assertions that the ancient Greek mathematician Euclid wrote about in his book “Elements.” They are the basis of Euclidean geometry. These are the axioms:
- Any two points can be joined together with a straight-line segment.
- Any part of a straight line can be made longer and longer in a straight line.
- We can draw a circle with any straight line segment as the radius and one endpoint as the center.
- Every right angle is the same.
- If a straight line that crosses two straight lines makes the angles on the same side of it add up to less than two right angles, then the two lines will meet on the side where the angles add up to less than two right angles if they are stretched out forever.
We need to know a few terms before we can talk about Euclid’s postulate. From solids to points, Euclid talks about a three-step process: solids, surface, lines, and points. At each step, the shape loses one dimension. So, a solid is a shape with three dimensions, a surface has two, a line has one, and points have dimensions. Surface refers to something that has only length and width.
On the other hand, a point has no parts, is long, etc. These words will help you get a better grasp of the postulate. Let’s look at five of Euclid’s postulates:
Euclid’s Postulate 1
This postulate says that at least one straight line goes through two different points, but he didn’t say there can’t be more than one such line. Even though, in all of his work, he has assumed that only one line connects two points.
Euclid’s Postulate 2
Euclid said a “Line segment” is just a “Line that ends.” So, this postulate tells us that we can make a line by extending a line that ends or a line segment in either direction. In the picture below, the line segment AB can be made into a line by extending it as shown.
Euclid’s Postulate 3
We can draw any circle from the end or beginning of a circle, and the circle’s diameter will be the length of the line segment.
Euclid’s Postulate 4
Right angles, with a measure of 90°, are always congruent, or the same, no matter how long the sides are or which way they are facing.
Euclid’s Postulate 5
Angles 1 and 2 don’t add up to more than 180° in the diagram, so lines n and m will meet on the side of angles 1 and 2.
Now, let’s look at John Playfair’s version of Euclid’s fifth postulate, which is the same. His words:
“In a plane, if we have a line and a point that is not on the line, we can only draw one line through the point that is parallel to the line.”
In easy-to-understand language, the above sentence can be written as:
“There is a unique line m that goes through every point P that is not on line l and is parallel to line l.”
In the picture above, think about line l and a point P that isn’t on l. Now, we know that an infinite number of lines can reach any point. This means that any number of lines can go through point P. But are all lines parallel to line l, some lines parallel to line l, or none parallel to line l? Playfair says that only one line goes through P and is parallel to line l.
Draw a line through point P. Let’s call it line s. If you draw a transversal that cuts lines s and l and goes through point P, you can see that on one side, the sum of the co-interior angles will be less than 180° so that both lines will meet in that direction. There is only one option, line m, in which the sum of the angles on both sides is precisely 180°. In this case, this means that the two lines will never meet.
Properties of Euclidean Geometry
- Euclidean geometry is founded on five axioms, which are essential truths.
- A line segment can be stretched out to make a long line.
- Any center and radius can be used to draw a circle.
- All right angles are parallel.
- If you draw two lines that cross a third line in a way that the total of the inner angles on one side is less than two right angles, then if you extend the two lines very far, they will cross each other on that side.
- Euclidean geometry is based on the idea that space is flat and never-ending.
- The sum of the three angles in a triangle is always 180 degrees.
- Two parallel lines never meet.
- The shapes of similar figures are the same, but their sizes may differ.
- Congruent shapes are the same size and shape.
- Euclidean geometry supports the Pythagorean theorem, which says that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
- The Pythagorean distance formula tells us how far away two points in Euclidean space are from each other.
- The length and width of a rectangle are added together to get its area.
- Euclidean geometry is used extensively in engineering, physics, and architecture.
Which terms are considered undefined terms in euclidean geometry?
In geometry, some basic terms are not defined. The point, the line, the plane, and the set are these four things. These terms are significant for building theorems and other mathematical ideas. Undefined terms in geometry are ideas that can’t be explained formally and are usually explained through examples and pictures. Some examples are point, line, plane, and set.
Is Euclidean geometry rigorous?
Geometry based on Euclid is rigorous. Euclid’s axioms and theorems had holes and were often based on assumptions that had not been proven. Hilbert and Birkhoff were able to fix this problem in the end by giving a strict set of axioms from which everything can be derived. Euclidean geometry studies two-dimensional and three-dimensional shapes based on Euclid’s axioms and theorems (c. 300 BCE). Euclidean geometry is the kind of plane, and solid geometry usually taught in high school.
What are the basic tools of euclidean geometry?
The compass and the straightedge. The name comes from the first three Euclidean postulates describing the most basic geometric constructions. These are connecting points with segments, extending segments, and drawing circles with a given center and a given radius. There are two kinds of Euclidean geometry: plane geometry and solid geometry. Plane geometry is Euclidean geometry in two dimensions, and solid geometry is Euclidean geometry in three dimensions. A point, a line, and a plane are the most basic terms in geometry. A point has no length or width, but it has a place.
What are the 3 types of geometry?
The most famous examples of geometry are plane, solid, and spherical. Plane geometry deals with points, lines, circles, triangles, and polygons. Solid geometry deals with things like lines, spheres, and polyhedrons (dealing with objects like the spherical triangle and spherical polygon).
What shapes are in Euclidean geometry?
The tetrahedron, or pyramid, has four triangular faces; the cube has six square faces; the octahedron has eight equilateral triangular faces; the dodecahedron has twelve pentagonal faces; and the icosahedron has twenty equilateral triangular faces.
Why is it called Euclidean geometry?
Euclidean geometry is a well-known maths system credited to Euclid of Alexandria, a Greek mathematician. Euclid’s book Elements was the first book to talk about geometry systematically. It has been one of the most important books in history, both for how it was written and for what it said about maths.