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Geometry

Geometry is a branch of mathematics that explores shapes, sizes, and the properties of space.

The term "geometry" comes from the Greek words meaning "earth measurement."

This discipline has ancient roots, with civilizations like the Egyptians, Chinese, Babylonians, Romans, and Greeks employing it for practical purposes such as land surveying, navigation, and astronomical observations.

The ancient Egyptians used geometry to construct pyramids, redefine agricultural boundaries after the Nile’s floods, and in their study of the stars. The Babylonians recorded various geometric problems on cuneiform tablets, applying them mainly in astronomy. The Chinese relied on geometry for dividing farmland and in architectural design. The Greeks incorporated it into philosophy and architecture, while the Romans primarily used it in engineering projects (like aqueducts and roads), city planning, and art.

The Greeks, in particular, were instrumental in shaping geometry as we know it. They aimed to systematize and rationalize geometric concepts through logical reasoning and proofs.

Renowned Greek philosophers and mathematicians such as Thales, Pythagoras, Plato, and Aristotle made significant contributions to this field.

However, it was Euclid who authored the Elements, a seminal text that organized and consolidated the geometric knowledge of his time. This work became so influential that it served as the foundational reference for geometry for over two millennia.

As a result, what we now refer to as "plane geometry" is often called "Euclidean geometry" in honor of Euclid.

The fundamental concepts of geometry

Today, when we discuss geometry, we typically mean either "plane geometry" (concerned with flat surfaces) or "spatial geometry" (focused on three-dimensional space). Here, space refers to the world around us, which has three dimensions: length, width, and height.

The basic building blocks of geometry are the point, the line, and the plane.

  • Point
    A point is like a tiny dot. It has no dimensions—no length, width, or height. It simply marks a position in space.
    an example of a point
  • Line
    Picture a straight path that extends infinitely in both directions. That’s a line. It has only one dimension: length.
    the line
  • Plane
    A plane is like a limitless sheet of paper. It has two dimensions: length and width.
    the plane

By combining these elements, we can create other shapes, such as line segments (parts of a line), angles (formed where two lines intersect), geometric figures, polygons (like triangles and squares), and polyhedra (such as cubes and pyramids).

Not all shapes are flat, though. Some geometric figures, like circles and ellipses, are curved.

Geometry doesn’t stop there. With advanced mathematics, it’s possible to explore spaces with more than three dimensions, even if we can’t easily visualize or imagine them. These are known as vector spaces.

Finally, not all geometries adhere to the same rules. Most of the principles we know come from Euclidean geometry.

However, there are also geometries where these rules don’t apply, called non-Euclidean geometries.

An example of non-Euclidean geometry is the surface of our planet. A triangle drawn on Earth’s curved surface may have properties different from one drawn on a flat sheet of paper.

The Evolution of Geometry: From Inductive Reasoning to Deductive Logic

Imagine observing the world around you: the shapes of mountains, the contours of buildings, or the circular ripples in water after a stone is tossed. This is the essence of early geometry, rooted in intuition and direct experience.

In ancient times, the study of geometry relied on an inductive approach: starting from observations of concrete, repeatable phenomena to identify general patterns, known as conjectures.

For example, after measuring the angles of countless triangles, one might conjecture that the sum of the interior angles in any triangle is always 180°.
practical example

Conjectures were propositions that seemed plausible based on repeated observations, but they had not yet been formally proven.

However, the inductive approach had its limitations: it could only suggest that a property was valid in the observed cases, without guaranteeing it would hold universally.

As knowledge advanced, geometry demanded greater rigor. It was no longer enough for an idea to "seem" true; it needed to be proven.

For instance, if you draw a triangle on a curved surface, like a sphere, the sum of its angles can exceed 180°. Conversely, on a concave surface, the sum of the angles will be less than 180°. Thus, the idea that the sum of a triangle's angles equals 180° is not universally true—it depends on the type of surface (flat, concave, or convex) on which the triangle is drawn.
triangle in non-Euclidean geometries

This marked the rise of the deductive method, which transformed geometry into a logical science based on axioms and proofs.

The deductive approach begins with a few foundational principles:

  • Primitive concepts: These are undefined terms accepted as intuitive, such as "point" or "line." This approach avoids a circular process where defining these basic entities would require other terms that would also need definitions.
  • Axioms or postulates: These are statements assumed to be true without proof, such as "through any two distinct points, there exists exactly one line." They are taken as self-evident truths.
  • Definitions: These introduce new terms to describe more complex mathematical objects, like "triangle" or "parallelogram." Definitions build on primitive concepts, axioms, and previously established definitions.
  • Theorems: These are statements proven through logical reasoning, known as "proofs," derived from axioms and earlier propositions.

A quintessential example is the system devised by Euclid, a Greek mathematician who, in the 3rd century BCE, structured geometry using the axiomatic-deductive method.

In his 13-volume work, "Elements," Euclid outlined five fundamental axioms and built the framework of Euclidean geometry, proving each theorem step by step.

For example, Euclid postulated that a straight line can be drawn between any two points and that a line intersecting two parallel lines creates equal corresponding angles.

This new method not only made geometry more rigorous but also broadened its scope. It showed that proving a general property didn’t require observing every specific instance—it could be logically deduced.

Moreover, it paved the way for the development of alternative geometries, or "non-Euclidean geometries", such as spherical and hyperbolic geometry, where Euclid’s axioms were modified.

In conclusion, the inductive approach remains a natural starting point for those exploring geometry: observing, hypothesizing, and conjecturing.

However, it is the axiomatic-deductive method that gives geometry its scientific rigor, ensuring that every truth is backed by a formal proof.

This shift from conjecture to proof elevates geometry into a universal, logical science, capable of explaining the world—and even transcending the limits of what is observable.




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