The fascinating world of boring tunnels

That's Maths: Laser-guided machines are today used to drill tunnels from both ends, but how did the ancient Greeks do it?

A giant drill at the construction site for the Gotthard Base Tunnel near Sedrun, Switzerland. Photograph: Johannes Simon/Getty Images

A giant drill at the construction site for the Gotthard Base Tunnel near Sedrun, Switzerland. Photograph: Johannes Simon/Getty Images

 

The Gotthard Base Tunnel opened in June and will be fully operational by December, cutting the journey time from Zurich to Milan by more than an hour. The accuracy of modern tunnel construction is impressive: when the British and French tunnelling teams met in December 1990 under the English Channel, the two shafts were misaligned by less than a metre. Thanks to their skill, a rail trip from London to Brussels now takes less than two hours.

The Channel Tunnel was a magnificent engineering accomplishment. Boring tunnels from both ends makes sense: two teams boring in opposite directions simultaneously can halve the construction time. The one obvious – and essential – requirement is that the two shafts meet accurately in the centre. Modern laser-guided tunnel boring machines make this possible.

Geometric principles

Boring tunnels from both ends is nothing new. More than 2,500 years ago, a tunnel of more than a kilometre in length was constructed on the island of Samos in Greece. It was designed to serve as an aqueduct and was the first tunnel to make use of geometric principles. The engineer was Eupalinos, and the tunnel was bored under Mount Kastro to serve the ancient capital of Samos, today called Pythagoreio. The aqueduct was concealed underground to prevent enemies of the tyrant Polycrates from finding and destroying it. It served for more than 1,000 years until it fell into disuse, to be rediscovered around 1883.

Since parallel lines never meet, two shafts may fail to intersect if there is an error in position. To avoid such a catastrophe, Eupalinos arranged for the two straight shafts to zig-zag slightly near the centre point, to ensure a successful juncture even if the cuts were initially misaligned. As it turned out, the construction was so accurate that if the diggers had continued along straight lines they would have made an almost perfect connection.

Eupalinos also needed to ensure that the cuts were vertically aligned. He constructed a continuous duct, like a gutter, around the mountain between the two openings. When filled with water, this enabled him to locate the two openings at the same altitude. Similar arrangements ensured that the shafts remained horizontal as they were bored.

No one knows exactly how Eupalinos determined the directions of boring. Several methods of triangulation have been suggested. He may have set up a line of marker poles over the mountain-top between the openings. Alternatively, by laying off a series of short segments in the cardinal directions, he could determine the nett north-south and east-west displacements. The theorem of Pythagoras would then give the length of the tunnel. It would also enable the direction of the bores to be determined. Pythagoras was born in Samos around 572 BC. It is tempting to speculate that he had a role in the design of the tunnel, but we have no evidence of this.

The ancient Greeks had no laser-guided boring machines. Neither had they magnetic compasses, topographic maps or survey instruments. The tunnel was built in the sixth century BC, 200 years before the principles of geometry were formalised by Euclid.

It was one of the greatest engineering achievements of ancient times and is a testament to the power of mathematics and the ingenuity of the ancient Greeks.

Peter Lynch is emeritus professor at the school of mathematics and statistics, University College Dublin. He blogs at thatsmaths.com

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