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+----- Thread: An introduction to the mathematics in the Alhambra of Granada (/showthread.php?tid=1980)
An introduction to the mathematics in the Alhambra of Granada - mklabgr - 09-15-2026
The article “An introduction to the mathematics in the Alhambra of Granada” by Miguel Ortega explores the mathematical ideas underlying the extraordinary geometric decoration of the Nasrid Palaces. The author explains how medieval artisans used Euclidean geometry and the four basic isometries of the plane—translations, rotations, reflections, and glide reflections—to construct elaborate patterns from relatively simple geometric lattices of squares, triangles, and rectangles. Examples include the famous pajarita (bow-tie), the “bone” pattern, and the black-and-white “airplane” tiling.
The article then introduces rosettes, patterns with finite rotational or reflectional symmetry, and explains that mathematically they can have either cyclic or dihedral symmetry. Particularly remarkable are the many order-9 rosettes found in the Alhambra, since a regular nonagon cannot be constructed exactly with the classical ruler-and-compass methods.
The second major theme is plane crystallographic groups, which provide a mathematical framework for understanding repeating decorative patterns. Ortega introduces the crystallographic restriction theorem, showing that rotational symmetries in a plane crystallographic group can only have orders 2, 3, 4, or 6, and uses Conway-style notation to describe the different symmetry combinations. He then identifies examples corresponding to all 17 plane crystallographic groups among the Alhambra's decorations, including patterns based on translations, reflections, glide reflections, and rotations.
The central message is that the Nasrid artists, despite working centuries before modern mathematical terminology and technology, achieved an astonishing variety and precision of geometric constructions. Their artistic achievements provide a remarkable visual demonstration of concepts from group theory, geometry, symmetry, tilings, and crystallography.
ARTICLE [PDF]