Los SECRETOS MATEMÁTICOS de la ALHAMBRA

Mathematics in the Alhambra

Introduction to Mathematical Beauty

• The Alhambra is highlighted as a monument that embodies mathematical beauty, being the only structure built before group theory was discovered, showcasing examples of all planar crystallographic groups.

Exploring the Alhambra with Mathematicians

• Mathematicians Joaquín Valderramas and Francisco Fernández Morales have been visiting the Alhambra for over 20 years, aiming to spark interest in mathematics among students.

• Even after retirement, they continue to explore and promote mathematical concepts through visits to Granada and the Alhambra.

Geometric Proportions in Architecture

• The basic geometric shape discussed is the rectangle, with Islamic proportion emphasizing squares; specifically, rectangles with proportions of √2 are prevalent.

• The irrational nature of √2 complicates its mathematical handling but simplifies geometric applications.

Understanding Rectangles and A4 Paper

• The "Puerta del Vino" features a rectangle whose dimensions correspond to √2, constructed by drawing diagonals on a square.

• An A4 sheet's dimensions also reflect this ratio (1:√2), which can be verified using Pythagorean theorem principles.

Practical Applications of Proportions

• The design of the "Puerta del Vino" mirrors an A4 sheet's proportions, optimizing material use without waste.

• Folding an A4 sheet demonstrates how these proportions maintain consistency across various sizes (e.g., 2:5 when folded).

Architectural Harmony Through Geometry

• The orientation of arches at the "Puerta del Vino" creates an equilateral triangle that enhances aesthetic appeal through mathematical harmony.

Mosaics: Infinite Patterns and Symbolism

• Mosaics in the Alhambra exemplify geometric decoration; they symbolize unity within diversity in religious contexts.

• Francisco explains how mosaics are created using isometric paper patterns featuring equilateral triangles as foundational elements.

Creating Mosaic Patterns

• To form a basic tile for mosaics, one must find a central point on an equilateral triangle and rotate it to create symmetrical designs.

• Translations maintain shape and size during mosaic creation; moving tiles according to defined vectors generates infinite patterns.

Conclusion on Mosaics' Mathematical Basis

• All observed mosaics adhere to one of 17 types derived from four movements (rotation, symmetry, translation), marking the Alhambra as a unique site where these types converge.

Understanding Frisos and Their Mathematical Foundations

The Concept of Frisos

• A friso is a mathematical element that operates with algorithms, similar to mosaics, but differs in that it translates in only one direction to fill the plane.

• Joaquín demonstrates how to create a friso using a notebook by folding it like an accordion and making cuts, illustrating the formation based on specified proportions.

Symmetry and Reflection

• Each fold acts as an axis of reflection; unfolding reveals symmetrical figures on either side, indicating parallel axes of symmetry or translations.

• The discussion highlights how various transformations—translations, rotations, and symmetries—can generate up to seven distinct types of frisos based on mathematical algorithms.