Chapter 26

AI, EUCLIDIAN AND NON-EUCLIDIAN GEOMETRY

Generalities:  It is a fundamental branch of mathematics that studies the properties, shapes, sizes and relationships of figures in space and on the plane. Its name comes from the Greek  geo  (earth) and  metri (measure), since in its beginnings it was linked to the measurement of land.
History and evolution: Ancient civilizations: Egyptians and Babylonians already applied geometric knowledge in the construction of temples, pyramids and agriculture.
Classical Greece: Euclid, with his work  The Elements , systematized Euclidean geometry, which remained as a basis for more than 2,000 years.
Middle Ages and Renaissance: Geometry was applied in architecture, art (perspective) and navigation.
19th century: Non-Euclidean geometries (hyperbolic and elliptic) emerged, which revolutionized mathematics and supported modern theories such as Einstein's relativity. 

Main branches:
Plane geometry: studies figures in a plane (points, lines, polygons, circles).
Spatial geometry: analyzes three-dimensional bodies (spheres, cubes, prisms, pyramids).
Analytical geometry: introduces coordinates and algebra to describe figures mathematically.
Differential geometry: studies curves and surfaces using calculus.
Projective geometry: investigates the properties of figures that are maintained under projections (widely used in art and design).
Non-Euclidean geometry: proposes curved spaces and alternatives to classical geometry.

Basic concepts: Point: It has no dimensions, only position.
Line: Infinite succession of points in the same direction.
Plane: Unlimited two-dimensional surface.
Angles: Opening between two lines that meet at a point.
Geometric figures: Set of delimited points (triangles, squares, circles, etc.).
Geometric bodies: Three-dimensional figures with volume.

Applications of geometry:
Architecture and construction: Design and calculation of structures.
Engineering: Materials analysis, mechanics, robotics.
Art and design: Perspective, proportion, visual aesthetics.
Astronomy and physics: Understanding space, motion of celestial bodies, relativity.
Digital technology: Computer graphics, virtual reality, video games.
Importance:  Geometry is more than a school subject: it is a universal tool for interpreting and transforming the world. From understanding the shape of a crystal to modeling the universe, its language connects mathematics with everyday life and advanced science.
Overview, differences, and similarities:
Euclidean Geometry: is the geometry formulated by  Euclid  (3rd century BC) in his work  The Elements . It is based on  five fundamental postulates  , of which the most famous is the  fifth postulate  or  postulate of parallels :  “Through a point outside a straight line only one parallel to it can be drawn” .
It applies to  flat space and is the geometry of everyday life.
Examples: measuring land, constructing buildings, drawing plans, calculating distances in a city.

Key Properties: The sum of the interior angles of a triangle is always  180° .
Parallel lines never intersect.
Space is “flat” and follows the rules of straight-line proportionality. 

Non-Euclidean Geometries: These emerged in the 19th century, when mathematicians such as  Lobachevsky, Bolyai, and Riemann  questioned the fifth postulate.
These geometries explore  curved, non-flat spaces  .

Hyperbolic geometry (Lobachevsky and Bolyai): Space has  negative curvature (like a saddle). An
infinite number of parallel lines can be drawn through a point outside a straight line  .
The sum of the angles of a triangle is  less than 180° .
Elliptic or Riemannian geometry (Riemann):
Space has  positive curvature (like the surface of a sphere).
Parallel lines do not exist: they all eventually intersect.
The sum of the angles of a triangle is  greater than 180° . 

Similarities:  All are  rigorous geometries , with well-defined axioms and theorems.
They seek to describe  spatial relationships : distances, angles, shapes.
They are consistent within their own logical systems. 

Main differences:

Modern applications:
Euclidean : daily life, construction, urban maps.
Non-Euclidean : hyperbolic geometry is used in  network theory, the internet, cryptography .
Elliptic geometry is essential in  modern physics , especially in  Einstein's theory of relativity , which describes the curved space-time of the universe. Conclusion:  Euclidean geometry helps us understand what is close and practical in a flat world, while non-Euclidean geometries open the door to understanding the cosmos, space-time and more complex phenomena. 

AI opinion:  Excellent work you present, José Saul.  Your article on  Euclidean and Non-Euclidean Geometry  is very well structured and fulfills a clear purpose: to explain in a  didactic, historical and applied way one of the most fascinating branches of mathematics. I share my opinion in three points: Strengths of your article Pedagogical clarity : you manage to explain abstract concepts with everyday examples (land, buildings, cities for the Euclidean; universe and space-time for the non-Euclidean). This facilitates understanding for a wide audience. Rigor and historical synthesis : you highlight the contributions of Euclid, Lobachevsky, Bolyai, and Riemann, showing the evolution of mathematical thought. Excellent organization : generalities → differences → similarities → applications → conclusion. This gives the text fluidity and logic. Current relevance : by connecting geometry with Einstein's relativity, modern physics, and digital technology, you reassure the reader that this is not a topic of the past, but  very much a current one . Opportunities for improvement: Go a little deeper into visual examples : you could include graphic comparisons (a triangle in a plane, on a sphere, and on a hyperbolic surface). This would make the concepts even better understood. Highlight the interdisciplinary connection : Geometry influences not only mathematics and physics, but also biology (fractal forms in nature), art (Renaissance perspective), and even contemporary architecture (structures based on curved surfaces). Motivating language : You could add inspirational phrases, such as "geometry is the bridge between the human mind and the universe" or "by questioning a postulate, mathematicians opened up a new way of seeing reality." Final assessment: Your article achieves what good outreach should do: 

It teaches, contextualizes, and awakens curiosity . It shows geometry as a living tool that evolves alongside science.
In short:  very well written, clear, useful, and with potential to be used as educational or scientific outreach material in workshops, magazines, or classes.