[330] Architecture and Mathematics: From the Parthenon to the Sagrada Familia
Architecture is not an application of mathematics. For thousands of years, architecture and mathematics have developed together, stimulating one another.
Attempting to create beautiful buildings requires mathematics, and pursuing mathematical questions leads to architectural ideas. This cycle has produced some of humanity's most beautiful structures.
Ancient Greece: The Golden Ratio and the Parthenon
The Parthenon in Athens, construction of which began in 447 BC, is often cited as a classic example of design conscious of the golden ratio ([077] On the Golden Ratio: The Eternal Mystery of Beauty and Mathematics reference).
It is said that the golden ratio φ ≈ 1.618 appears in the ratio of the width to the height of the facade, the ratio of the spacing of the columns to their height, and so on. However, there is ongoing debate among researchers, with some arguing that there was an intentional use of the golden ratio, while others view it as merely a retrospective interpretation.
What is mathematically clear in Greek architecture is "entasis." The columns of the Parthenon are not perfect cylinders; they bulge slightly in the center. Straight columns appear visually pinched in the middle, so they were intentionally bulged to compensate for this.
The idea of using geometry to correct optical illusions is a prime example of the application of mathematical thinking to architecture.
Islamic Architecture: Group Theory and Geometric Patterns
The geometric patterns of tiles decorating the interiors of Islamic architecture possess profound mathematical depth.
It has been mathematically proven that there are only 17 types of "wallpaper groups"—patterns of symmetry that fill a plane with repeating tiling (a theorem of group theory).
When classifying the tile patterns of Islamic architecture (such as the Alhambra), it is found that they cover many of these 17 types. 14th-century Islamic craftsmen discovered these 17 wallpaper groups almost intuitively, without the language of modern group theory.
M. C. Escher, who visited the Alhambra in Spain, drew inspiration from its tile patterns to create his famous series of "transforming tiling" works.
Gothic Architecture: The Fusion of Mechanics and Mathematics
Medieval European Gothic cathedrals (such as Cologne Cathedral and Notre-Dame Cathedral) are products of mechanics and mathematics born from the demand to be "as high as possible and with windows as large as possible."
While stone walls are strong against gravity, they have the problem of being weak against lateral forces (wind pressure, horizontal thrust of arches). The "flying buttress" was devised to receive this lateral thrust from the outside.
The force applied to an arch is determined by its shape. A semicircular arch has high horizontal thrust, requiring thicker walls. In contrast, the "pointed arch" has lower horizontal thrust, allowing for thinner walls and larger windows.
The mathematics of force vector decomposition fundamentally changed architectural design.
The Renaissance: Domes and Catenary Curves
The dome of the Cathedral of Santa Maria del Fiore in Florence, completed in 1436, is a feat of engineering by Filippo Brunelleschi.
He constructed the 44-meter-diameter dome without the use of auxiliary wooden scaffolding. The secret lies in the brick arrangement pattern called "herringbone." By placing them not only horizontally but also diagonally in an alternating fashion, he ensured the dome would not collapse under its own weight before completion.
The mathematical basis for this method lies in the control of 'hoop stress,' where force is transmitted in a specific direction.
Gaudí: The Concept of Catenary Curves and Inverted Hanging Models
The Sagrada Família (Basílica i Temple Expiatori de la Sagrada Família) in Barcelona, Spain, is a building designed by Antoni Gaudí; construction began in 1882 and continues to this day, remaining unfinished.
Gaudí's genius idea was the 'inverted hanging model'.
When a chain is suspended from both ends, it forms a curve (a catenary curve) due to its own weight. If you invert this, it becomes an arch that functions entirely through 'compression' (with no tension or bending). Since stone and brick are strong in compression but weak in tension, this is the structurally optimal arch shape.
The catenary curve is expressed by the hyperbolic cosine function.
$${\displaystyle y = a \cosh\left(\frac{x}{a}\right) = \frac{a}{2}\left(e^{\frac{x}a} + e^{-\frac{x}a}\right)}$$
Gaudí created inverted hanging models by combining countless such curves and simulated the structure by attaching weights to strings. In an era without computers, he performed complex mechanical calculations using physical models.
The curved columns and ceilings of the Sagrada Família are born from the mathematics of the catenary curve.
Modern Architecture: Fractals and Computer-Aided Design
Complex curved architecture, such as the Guggenheim Museum Bilbao (1997) designed by Frank Gehry, would be impossible to design or construct without computers.
In modern architectural design, a method called 'parametric design' has become widespread. It involves defining shapes using parameters (numerical values) and having a computer calculate the optimal structure.
Furthermore, the concept of 'fractals,' where the same structure appears even when zoomed in (see [135] Fractals: The Beautiful World of Infinitely Continuing Self-Similarity), has also been incorporated into architecture. Buildings where large-scale structures and small-scale decorations have a self-similar relationship provide a visually rich impression.
Mathematics does not constrain architecture; it enables it. Mathematics is the language used to give form to the intuition that 'this is beautiful'.