Euclidean geometry
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What is Euclidean geometry?
Euclidean geometry is the study of plane and solid figures based on axioms and theorems as employed by the Greek mathematician Euclid (c. 300 BCE). Until the 19th century, geometry was synonymous with Euclidean geometry, representing general mathematical thinking through insight, theorem application, generalization, and proof.
Euclid's Elements used only a ruler and compass for geometrical constructions, a restriction still followed today. The Elements remained a model for scientific exposition until David Hilbert's Foundations of Geometry (1899). Modern Euclidean geometry involves Euclidean coordinate spaces of multiple dimensions, where distances are measured using a generalization of the Pythagorean theorem.
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Who was Euclid and why is this geometry named after him?
Euclid (flourished c. 300 BCE) was a prominent mathematician in Greco-Roman antiquity, known for his geometry treatise, the Elements. Euclidean geometry is named after him because it is based on the axioms and theorems found in the Elements.
Euclid gathered and systematized the mathematical knowledge of his predecessors. The Elements covers plane and solid figures, employing only a ruler and compass for geometrical constructions. His work presents theorems and proofs of plane geometric figures, properties of parallel lines, area relations, basic properties of circles, constructions of polygons, and more. Euclid's Elements became the standard for mathematical rigor and organization.
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What are the basic shapes and objects studied in Euclidean geometry?
Euclidean geometry, based on the work of the Greek mathematician Euclid, explores shapes and figures in both plane and solid forms.
Euclid's Elements begins with undefined terms like "a point is that which has no part" and "a line is a length without breadth." From these basics, more complex concepts are defined, including angles, circles, triangles, and various other polygons. Polygons are closed curves formed by line segments; examples include triangles, quadrilaterals, and pentagons.
Solid geometry introduces three-dimensional shapes like spheres, tetrahedrons, and cubes. Regular polyhedra, known as Platonic solids, include the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
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What are Euclid's five main postulates (axioms)?
Euclidean geometry is based on axioms and theorems developed by the Greek mathematician Euclid around 300 BCE. Euclid's Elements begins with undefined terms and builds to five common notions and five unprovable postulates, or axioms, that serve as the basis for logical deductions. These postulates are:
- Two points can be joined by one straight line.
- A straight line segment can be extended indefinitely.
- A circle can be constructed with a given center and radius.
- All right angles are equal.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines will meet on that side.
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How do lines and angles behave in Euclidean geometry?
In Euclidean geometry, a line is defined as an interval between two points that can be extended indefinitely in either direction. An angle is the inclination between two straight lines.
Euclid's parallel postulate states that through a point not on a line, there is exactly one line parallel to the given line in the same plane. This postulate is the basis for the uniqueness of parallel lines.
Triangles are formed by three line segments (edges) intersecting at vertices. The sum of a triangle's interior angles is 180 degrees.
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How does Euclidean geometry differ from non-Euclidean geometry?
Euclidean geometry, based on the work of Euclid around 300 BCE, studies plane and solid figures using axioms and theorems. It relies on specific postulates, including the parallel postulate, which states that through a point not on a given line, there is only one line parallel to the given line.
Non-Euclidean geometry encompasses geometries that differ from Euclidean geometry by modifying or rejecting Euclid's fifth postulate. Riemannian geometry is a type of non-Euclidean geometry that rejects the parallel postulate, stating that there are no lines parallel to a given line, and modifies the second postulate, so that all straight lines are of the same length. Hyperbolic geometry is another type of non-Euclidean geometry.
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Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300 bce).
In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th century, when non-Euclidean geometries attracted the attention of mathematicians, geometry meant Euclidean geometry. It is the most typical expression of general mathematical thinking.
Rather than the memorization of simple algorithms to solve equations by rote, Euclidean geometry demands true insight into the subject, clever ideas for applying theorems in special situations, an ability to generalize from known facts, and an insistence on the importance of proof. In Euclid’s great work, the Elements, the only tools employed for geometrical constructions were the ruler and the compass—a restriction retained in elementary Euclidean geometry to this day.
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In its rigorous deductive organization, the Elements remained the very model of scientific exposition until the end of the 19th century, when the German mathematician David Hilbert wrote his famous Foundations of Geometry (1899). The modern version of Euclidean geometry is the theory of Euclidean (coordinate) spaces of multiple dimensions, where distance is measured by a suitable generalization of the Pythagorean theorem. See analytic geometry and algebraic geometry.
Fundamentals
Euclid realized that a rigorous development of geometry must start with the foundations. Hence, he began the Elements with some undefined terms, such as “a point is that which has no part” and “a line is a length without breadth.” Proceeding from these terms, he defined further ideas such as angles, circles, triangles, and various other polygons and figures. For example, an angle was defined as the inclination of two straight lines, and a circle was a plane figure consisting of all points that have a fixed distance (radius) from a given center.
As a basis for further logical deductions, Euclid proposed five common notions, such as “things equal to the same thing are equal,” and five unprovable but intuitive principles known variously as postulates or axioms. Stated in modern terms, the axioms are as follows:
- 1. Given two points, there is a straight line that joins them.
- 2. A straight line segment can be prolonged indefinitely.
- 3. A circle can be constructed when a point for its center and a distance for its radius are given.
- 4. All right angles are equal.
- 5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.
Hilbert refined axioms (1) and (5) as follows:
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- 1. For any two different points, (a) there exists a line containing these two points, and (b) this line is unique.
- 5. For any line L and point p not on L, (a) there exists a line through p not meeting L, and (b) this line is unique.
The fifth axiom became known as the “parallel postulate,” since it provided a basis for the uniqueness of parallel lines. (It also attracted great interest because it seemed less intuitive or self-evident than the others. In the 19th century, Carl Friedrich Gauss, János Bolyai, and Nikolay Lobachevsky all began to experiment with this postulate, eventually arriving at new, non-Euclidean, geometries.) All five axioms provided the basis for numerous provable statements, or theorems, on which Euclid built his geometry.
The rest of this article briefly explains the most important theorems of Euclidean plane and solid geometry.