Dome Truths
The English have always loved spires: crooked medieval spires; spires made from stacked triumphal arches; spires that are somehow also Greek columns; architectural style be damned: the spire will out. Just look at The Shard. But for architects of the Italian Renaissance and Baroque, and their imitators, there was nothing like a dome. Nothing in the world.
Was this down to the effortless simplicity of the dome’s geometric form? Not a bit of it. These maestri chose domes for the same reason John F. Kennedy chose to go to the moon (and do the other things): not because they were easy, but because they were hard. They knew that to build a successful dome required lavish displays of engineering and swagger, mathematics and theatre, in equal measure.
The great biographer of the Renaissance, Giorgio Vasari, tells us how the star architects of their day were invited to Florence in 1420 to thrash out the problem of how a dome might be installed on top of the otherwise complete Cathedral (or Duomo) of Santa Maria Del Fiore. A succession of increasingly elaborate and infeasible solutions were demonstrated and refuted. Finally, an architect called Filippo Brunelleschi, who had waited for the general hubbub to subside, announced that he knew exactly how the dome might be designed and built.
“Where are your models and plans?” scoffed the crowd, to which Brunelleschi pulled an egg from his pocket and held it aloft (doubtless to gasps and sniggers).
“You should give me the contract for the dome because I am the only person here who can make an egg stand on end on a flat piece of marble.” said Brunelleschi.
The assembled experts grumbled, but were forced to admit, finally, that none of them knew how to make an egg stand on end on a flat piece of marble. Brunelleschi then carefully broke the shell at the base of the egg, allowing him to perform the feat with ease.
“But if we had known that’s what you would do,” his rivals complained, “then any one of us could have done the same.”
“And if I showed you my plans and models for the dome of the Duomo, then you would all know that secret too,” countered Brunelleschi.
Brunelleschi got the job, of which more later. But just what is it about a dome that is so difficult?
Self-supporting stone arches
Forget what you might have learned in school: masonry arches (and similarly domes) are not as strong as you think. Shape is vital.
Take a semi-circle — on the face of it a natural and pleasing form; the basis of Romanesque architecture. Suppose I try to build a semi-circular-shaped arch that is free-standing (that is, not part of a wall), using a dry-stone construction (no mortar). To give myself the best chance of success I’ll use blocks with a slight wedge shape, so that they fit together into a perfect semi-circle, and I’ll screw the blocks at the two ends into the ground to secure them. I’ll also use a temporary support underneath the arch that can be dismantled and removed once the stonework is complete. (Renaissance builders called this kind of temporary support centering.)
Picture the completed construction. Do you imagine, when I remove the centering, that my semi-circular brick arch will stay standing? I think it’s easier for you to imagine it collapsing (which it will almost every time).
The problem is one of balancing compressive forces. Each block is pressing against the others, and the other blocks are pushing back. (Stone is good at this: if you stand on it, it pushes back — hard.) These compressive forces push along the direction of the arch at any given point (in mathematical terms, the forces are tangential to the curve of the arch).
Take any section of blocks in my arch: the blocks to the left of this section are pushing it to the right (roughly speaking); the blocks to the right of it are pushing to the left; one or both of these compressive forces is also pushing the whole section up; meanwhile, the weight of the whole section is pulling it down. For the arch to be stable, all of these forces need to be perfectly balanced, everywhere along the arch. But for most arch shapes the physics of this cannot be made to work.
The only other things that can keep the arch standing (if the compressive forces and weights are not perfectly balanced) are frictional forces between the blocks, and frictional forces between blocks are not something you would trust your grandmother’s life on. Of course, you could mortar between the blocks, which would offer additional security against the shearing effect of the imbalanced compressive forces, but mortar (like all masonry) is weak under tension, and the imbalanced forces will still slowly tear the mortared blocks apart.
Fortunately, there are arch shapes where the compressive forces and weights are perfectly balanced, holding every section of the arch securely in place. Such curves are called catenaries.
Catenaries
It turns out that catenaries are easy to visualise. You just need to turn the problem upside-down.
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Imagine a heavy chain (or rope), hung between two hooks across an empty span. Suppose the length of the chain is greater than the distance to be spanned, so that it hangs with some slack in it. Then the shape of the chain will be a catenary. Any given section of the chain is pulled by forces of tension from the left and the right (steel chains are good at this: if you hang from one it pulls back — hard), and pulled downwards by its weight, and when the chain comes to rest these forces must be in perfect balance.
So, to visualise a catenary-shaped arch, imagine a hanging chain and turn the picture upside-down. The more slack there is in the chain, the deeper the shape it forms (and the taller the arch). A catenary-shaped dome has the same kind of outline.
A striking example of a catenary-shaped building can be found in Star Wars Episode VIII: The Last Jedi. The beehive-shaped huts on the ocean planet Ahch-To, to which Luke Skywalker has retreated to live in isolation, are perfect catenary domes. (The film location was a group of real-life, dry-stone huts, built by sixth century monks on the island of Skellig Michael in the Atlantic Ocean.)
Supporting walls
There is still one important condition to ensuring a catenary-shaped dome stays standing, stress-free: the curve of the catenary (and the physics behind it) must continue, uninterrupted, all the way down to the ground, just like those beehive-shaped huts in Star Wars. If you want to stick a dome on top of a building with vertical walls — on top of a drum-shaped wall, for example — then you have a problem: the dome will tend to push out the rim of the drum where they meet (known as hoop stress), causing the drum to collapse.
Are we back where we started? How can we construct buildings that have vertical walls and domed ceilings? One solution is to have very thick walls. This was the approach used for the most celebrated domed building of the ancient world — the Pantheon in Rome. You can think of the walls simply as being thick enough to hold back the weight of the roof on top, or you can picture the catenary curve of the dome continuing uninterrupted, within the solid wall, all the way down to the ground.
The disadvantage of this scheme is that the building is more pie-shaped than dome-shaped, and the thick walls feel crude, and make the interior dark, though the dome will be impressive from the inside. (The Romans punched a hole in the centre of the dome of the Pantheon to let the light in, to spectacular effect.)
The dome of St Paul’s
If you have read this far, I hope you will pause the next time you walk past an impressive dome to ask, “What on Earth is holding that up?”
The most prominent example in my own backyard (of London) is St Paul’s Cathedral, designed by Christopher Wren. Wren’s solution to the problem was extraordinary.
The dome you see from outside is of very light construction: not heavy enough to disturb the drum underneath, but also too light to support the stone lantern that sits on top of it. The lantern is supported, instead,by a second dome underneath the external dome. This second dome is in the form of a catenary, but a catenary that is almost conical. (It’s the weight of the lantern that stretches the appropriate catenary into a cone — rather like a heavy pendant stretching a necklace chain into a V-shape.) You might say that Wren hid a spire under his dome. Underneath the second dome is a third dome, which is the dome you see from inside the cathedral, thinking you are staring up at the dome you saw from outside.
This is not the only way in which the cathedral plays with it’s dimensions. In fact, St Paul’s Cathedral is like Dr Who’s TARDIS in reverse: much smaller on the inside. The walls of the cathedral are supported by buttresses holding back the weight of the roof over the nave, and these buttresses are, in turn, hidden by outer screens that pretend to be the cathedral walls.
On the mathematics of catenaries
Mathematical treatises on catenaries began to appear in the late seventeenth century, around the time St Paul’s was being built. (Wren’s friend and collaborator, Robert Hooke, was one of those who published on the subject.)
If you’re not interested in the maths, skip to the next section. In brief, you can show that to balance the different forces, the tangent of the angle of the curve to the horizontal at any point needs to be proportional to the total length of the curve (i.e. the weight of the chain) up to that point, measured from the middle. This can be solved to show that the curve is a hyperbolic cosine function.
Brunelleschi’s dome
And what did Brunelleschi do in Florence? The truth is we don’t entirely know. The dome of the Duomo is octagonal, and double-skinned, complicating understanding of the transmission of forces.
We know that to eliminate the need for centering that would have filled the whole cathedral, Brunelleschi suspended working platforms from the roof, and fitted the bricks of the dome in a herring-bone pattern that helped support them in place during construction. We can see that the overall shape of the dome is closer to a catenary than a hemisphere. Oh, and we know that Brunelleschi held the whole thing together with iron chains — a cheat employed by many of the dome-builders who followed him — you can’t make perfection without breaking an egg. Iron and steel are strong under tension (but you knew that), so are perfect for resisting hoop stress.
The dome of the Duomo was finally topped off in 1471 with a gilded ball made at the workshop of the Florentine jeweller and sculptor, Andrea del Verrocchio. Apprenticed to Verrocchio at this time was a young Leonardo da Vinci. In one of his manuscripts Leonardo was later to muse, “Remember how we soldered the ball of Santa Maria del Fiore.” Perhaps Leonardo himself climbed the dome to plant the flag: a true Renaissance man on the moon.
