Euclidean Geometry

A Euclidean Geometry is a geometry that deals with the study of plane and solid figures in 2D and 3D.

• Context:
  • It is based on 5 main axioms (Euclid's postulates):
    • 1. A straight line segment can be drawn between any two points.
    • 2. Any straight line segment can be extended indefinitely in a straight line.
    • 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
    • 4. All right angles are congruent (i.e., equal to each other).
    • 5. If a straight line intersects two other straight lines, and the interior angles on one side of the intersection sum to less than 180 degrees, then the two lines will intersect on that side if extended far enough.
• Example(s):
  • Euclidean measure distances between two points in a straight line.
  • …
• Counter-Example(s):
  • Fractal Geometry,
  • Hyperbolic Geometry,
  • Spherical Geometry.
• See: Euclidean Vector Space, Euclid's Elements, Axiom, Proposition, Theorem, Non-Euclidean Space.

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References

2023a

• (ChatGPT-OpenAi, 2023) ⇒ https://chat.openai.com Retrieved: 2023-4-16.
  • Q: What is Euclidean Geometry?
  • A: Euclidean geometry is a branch of mathematics that deals with the study of plane and solid figures in two and three dimensions, respectively. It is named after the Greek mathematician Euclid, who wrote the book "Elements" around 300 BCE, which is considered one of the most influential works in the history of mathematics.

    Euclidean geometry is based on a set of axioms, or postulates, which are self-evident truths that are assumed to be true without proof. These axioms form the foundation for all the theorems and propositions in Euclidean geometry.

    The five main axioms of Euclidean geometry are:

    • 1. A straight line segment can be drawn between any two points.
    • 2. Any straight line segment can be extended indefinitely in a straight line.
    • 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
    • 4. All right angles are congruent (i.e., equal to each other).
    • 5. If a straight line intersects two other straight lines, and the interior angles on one side of the intersection sum to less than 180 degrees, then the two lines will intersect on that side if extended far enough.

Using these axioms, Euclidean geometry proves a vast number of theorems and propositions, including the Pythagorean theorem, the law of sines and cosines, and many other important geometric concepts. Euclidean geometry has many practical applications, including in architecture, engineering, and physics.

2023b

• (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Euclidean_geometry Retrieved:2023-4-16.
  • Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, 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. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

    The Elements begins with plane geometry , still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

    For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).

    Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast to analytic geometry, introduced almost 2,000 years later by René Descartes, which uses coordinates to express geometric properties as algebraic formulas.

300BC

• (Euclid, 300BC) ⇒ Euclid. (300BC). “Elements (Stoicheia).”

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