THE GEOMETRY OF LOVE

Do you want to know where I found my model?  An upright tree; it bears its branches and these, in turn, their twigs, and these, in turn, their leaves.  And every individual part has been growing harmoniously, magnificently, ever since God the artist created it.

Antoni Gaudí

Antoni Gaudí (1852-1926) was a Catalan architect grounded in mathematics and inspired by the geometry of nature to express the Catholic liturgy in sandstone.  Geometry, nature, and religion are all related to mathematics and all are visible in Gaudí's ecclesiastical architecture.  That religion, based on belief, is related to mathematics, based on proofs, is no paradox, since both seek to describe the concept of infinity. 

Consider Gaudí's finest achievement, the Basílica i Temple Expiatori de la Sagrada Família (the Basilica and Temple of the Sacrifice of the Holy Family) in Barcelona.  Its design is all threes and fives.  Architecturally the plan is a Latin cross and seems like a conventional basilica, until you take into account the symbolism of the mathematical module on which it is based.  Starting with the ground floor plan, there are five aisles in the nave.  They represent to me the Holy Family:  Mary (1), Joseph (+1), and Jesus, who embodies the Trinity (+3).  There are three aisles in the transept.  They represent in my view the Father, the Son, and the Holy Spirit. 

Continuing with the structural system as laid out on the floor, the centers of the columns are on a 7.5 meter grid, which is 1/12 (one for each apostle?) of the 90-meter length of the interior.  Both dimensions are divisible by three and five. Turning to the elevations, the side nave vaults are 30 meters high, while the central nave vault is 45 meters high.  The central vault of the crossing of the nave and the transept is 60 meters high, and the apse is capped by a vault reaching 75 meters.  Like the column grid and the length of the basilica, all vault heights are divisible by three and five.  The vaults reach a deliberate, gradual increase in height, a crescendo building from the main entrance (the Glory Façade) to the horseshoe-shaped apse, which is crowned with a triangular constellation of hyperboloid windows through which a natural golden light streams.    

In addition to this deceptively simple arithmetic, there is the Gaudian geometry:  Euclidian geometry in three-dimensional space, as seen in nature.  Gaudí favored geometrical forms called ruled surfaces, defined as the surface created when a straight line sweeps through space.  A simple example of a ruled surface is the cylinder you get if you connect each point on a circle with its corresponding point on another circle.   

https://en.wikipedia.org/wiki/Hyperboloid#/media/File:Cylinder_-_hyperboloid_-_cone.gif

Ruled surfaces are everywhere in nature, unlike the triangle, the pyramid, the square, the cube, (and to a lesser extent) the circle and the sphere, which have few natural examples.  The images below depict several types of ruled surfaces and how Gaudí employed them in the Sagrada Familia.

Gaudí used the helicoid, 

the hyperboloid, 

and the hyperbolic paraboloid

Roof of the School for Workers’ Children at the Sagrada Familia

—all ruled surfaces—in the Sagrada Familia.  The crossing of the nave and the transept (penultimate photo) supports a great hyperboloid surrounded by two rings of 12 hyperboloids (a total of 24, divisible by three).  The hyperboloid is an ideal form to capture, concentrate, and diffuse light, a metaphor for God.  

Gaudí also pioneered the catenary arch which, like the ruled surface, is based on the straight line.  The catenary arch is the ideal shape for an arch because its line of thrust (the line of compressive stress—that is, the weight of the structure the arch is supporting) is concentrated in the center of the arch, making it unusually strong, yet light.  

By using the catenary arch, rather than the Gothic arch, Gaudí was able to achieve soaring heights in the interior, while eliminating buttresses on the exterior, of the basilica.  In effect, the interior columns form catenary arches to create a kind of forest of trees leaning inward and branching upward toward organic cut-out daisy-like forms that suggest a canopy of leaves through which sunlight filters below, much as trees in nature do.  Wiki describes the intricate geometry of the columns:

The columns of the interior are a unique Gaudí design.  Besides branching to support their load, their ever-changing surfaces are the result of the intersection of various geometric forms. The simplest example is that of a square base evolving into an octagon as the column rises, then a sixteen-sided form, and eventually to a circle. This effect is the result of a three-dimensional intersection of helicoidal columns (for example a square cross-section column twisting clockwise and a similar one twisting counter-clockwise). Essentially none of the interior surfaces [apart from the floor] are flat.... Even detail-level work such as the iron railings for balconies and stairways are full of curvaceous elaboration.

Finally, there is the Catholic liturgy.  The three entrances symbolize the three virtues: Faith, Hope, and Love and each is dedicated to a part of Christ's life.   

The Nativity Façade is dedicated to Christ's birth.  It culminates in a cypress tree on which doves perch, symbolizing the tree of life and the resurrection.   

The Glory Façade is dedicated to Christ’s life on Earth and is to be decorated with the words from the Apostles’ Creed.  The inside face of its bronze door reproduces the Lord’s Prayer in Catalan, surrounded by multiple variations of "Give us this day our daily bread" in all the major languages of the world.    

The Passion Façade (above) is dedicated to Christ’s suffering.  Its doors reproduce excerpts of the Passion from the New Testament in various languages.  The apse spire bears the Latin text of the Hail Mary prayer.  Other spires are decorated with words such as "Hosanna," "Excelsis," and "Sanctus."  The roof tops are decorated with ceramic grapes and wine chalices, and with sheaves of wheat and the Host, together symbolizing the blood and the body of Christ celebrated in the rite of Communion.  The entire basilica is rich in religious symbolism, using liturgical texts and sculptural referents molded into plant and animal forms.

In the same way that an opera is a tour de force combination of music, theater, and lyrics, the Sagrada Familia is a dense, complex, and complete work.  And yet it is not actually complete.  It is unfinished.  Construction of the Sagrada Familia began in 1882 under the architect Francisco de Paula de Villar.  When he resigned in 1883, Gaudí took over the project as its chief architect, redesigning the basilica with his own unique, modern structural system.  Through his use of the catenary, Gaudí was able to remove an interior floor planned by Villar, open up the vertical and horizontal spaces to admit more light and improve acoustics, and add exterior cloisters in place of Villar's Gothic buttresses.  Gaudí devoted the last 12 years of his life to the project, living and working in the basilica’s basement.  When he died on June 10, 1926 after being hit by a tram, he was buried in the crypt.  Only about a quarter of the Sagrada Familia had been completed.

After Gaudí’s death, construction was at first hampered by a lack of public funds and then by the Spanish Civil War.  In 1936, revolutionaries set fire to the crypt and broke into the workshop, partially destroying Gaudí's original plans, drawings, and plaster models.  (It is believed there were very few plans or drawings, as Gaudí preferred to design in three-dimensional space, using plaster models.)  It took 16 years to piece together the jigsaw puzzle of fragments that were his master model so that construction faithful to Gaudí’s intentions could resume.  Today, advanced technologies such as computer aided design and 3-D printing have facilitated construction, which passed the 50% point in 2010.  

However, challenges remain, including sourcing the sandstone to match that used by Gaudí for ten of the unbuilt 18 spires.  Back to arithmetic:  The tallest spire, currently under construction and situated over the crossing of the nave and transept, is dedicated to Jesus Christ; the next tallest, over the horseshoe-shaped—womb-like?--apse, is dedicated to the Virgin Mary.  The four spires surrounding the tallest spire are dedicated to the Evangelists, and 12 spires (four on each of three sides of the temple) are dedicated to the apostles.  Construction is slated for completion by June 12, 2026, the centenary of Gaudí's death. 

Sometimes called “God’s Architect,” Gaudí left no theoretical discourses on his architecture, and very few writings remain to tell us how he conceived of his work, but here are a few clues in his own words:

I capture the purest and most pleasurable images from the nature, the nature that is always my teacher. Architecture creates the organism and that is why it must have a law in accordance with the law of nature.  In producing surfaces, geometry does not complicate the construction but rather simplifies it.  I am a geometrician; that is to say, I synthesize. 

And this:

The hyperboloid is light; the helicoid is movement; the hyperbolic paraboloid, the father of geometry.

Nature and geometry, and the geometry of nature, expressed in a UNESCO World Heritage architectural gem devoted to religion.  I felt the power of math as I walked through the Sagrada Familia.  What begins as a simple, orthogonal floor plan of straight lines is developed into a seemingly riotous volume of ruled surfaces masquerading as organic forms.  But, like a Mandelbrot set of fractals, it’s all in cosmic order.  And more importantly, it’s all about straight lines, the simplest mathematical form after the point.  So it’s no wonder mathematicians are drawn to Gaudí.  The monograph, “An Exploratory Study of the Geometrical Elements in Gaudí’s Architecture,” published in the International journal of Arts & Sciences, illustrates the fascination:

As God’s Architect, it is without doubt that religious metaphor is necessarily hidden in his design.  Since all generatrices of hyperbolic paraboloid rest on two straight lines, the two sets of straight lines are like the Father and Son, forming the Holy Spirit….  In this manner, Gaudí did promote geometry to the level of theology, as what it had been in ancient Greece.

Crossing of Nave and Transept Vaults

My own theory about Gaudian geometry and its expression in the Sagrada Familia is this.  All ruled surfaces are generated by a straight line, which is the shortest distance between two points.  I think (actually, I feel) that Gaudí conceived of one point as God and the other as man, and he sought to connect them by the shortest distance possible—the straight line of love--that generates the ruled surfaces of his basilica.

I hope to return in 2026 to see this masterwork in its full realization and walk again in the presence of Gaudí’s genius.

Keep it real!

Marilyn