The Dome that Touches the Heavens - Institute of Mathematics and

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The Dome that Touches the Heavens
Snezana Lawrence FIMA
Haec Auguste, tamen, quae vertice sidera pulsat, par
domus est caelo sed minor est domino (Yet though
this Palace, Augustus, whose summit touches the
stars, rivals heaven, it is not so great as its lord)
Martial, Epigrams, VIII, vol. 36, no. 11–12.
Bohn’s Classical Library (1897)
T
his year is the 50th year of the IMA and my 50th year
too. Birthdays are times for celebration and reflection – to
identify what really inspires us. Appropriately then, very
soon, on my birthday, I will give a talk to a group of architects in
Paris about when architecture inspired new mathematics and vice
versa. One example is the invention of the catenary curve, which
is popularly attributed to Sir Christopher Wren (1632–1723) and
his design of St Paul’s Cathedral (built 1675–1720).
Wren’s cathedral was the highest building in London from
1710 to the not so distant 1962. It was built after the Great Fire
of London brought destruction to much of the city and changed
the landscape not only of London, but of the building trade in
England too. Until 1666, the masons’ guilds had an elaborate
system of trade control over the whole of the region for which
they were responsible. After the Great Fire, the need for rebuilding the city so far exceeded the skilled labour available that the
whole structure of the existing trade controls had to be relaxed in
order to attract and maintain a sufficient number of skilled workers. At the same time, the set of skills for master builders changed:
a new group emerged, that of architects. Some decades earlier,
the first coherent designs which could be replicated easily began
to circulate: Inigo Jones (1573–1652) studied and developed his
own style of Palladian villa and became probably the first famous
English architect. The role of architect as a designer and supervisor was from thereon forever divided from the actual execution of
the building. Wren came into this architectural world from Oxford when this practice was only just taking hold. Whilst Wren
remained very close to the builders with whom he collaborated,
he was very much a new type of architect too – one whose ideas
were based on mathematical principles.
Wren studied Latin and Natural Philosophy at Oxford, but
became known as someone who connected scientific geometry
with architecture. As such he was approached to redesign the
fortification at Tangier. He didn’t accept this task, but the offer
demonstrates that his interest and knowledge of geometry and
mathematics applied to architecture were well known. As a result it wasn’t long before Wren worked on the overdue repairs of
the Old St Paul’s in 1661. Over the next five years, Wren also
designed the chapel of Pembroke College in Cambridge and the
Sheldonian Theatre next to the Bodleian Library in Oxford.
As part of the repairs on the Old St Paul’s, Wren submitted
the proposals to redesign its dome. This design was accepted on
27 August 1666, but only a week later the Great Fire reduced
the Old St Paul’s to rubble. Wren’s design therefore had to be
re-examined. The redesign included several new mathematical
approaches for the new cathedral and dome.
To go and see St Paul’s dome is a multi-layered experience
– what you see outside is not what you see inside. The dome
of the cathedral is actually a three-dome nested structure. The
inner dome is lower than the outside one so that the dome is
striking when viewed from both inside and outside. The middle
dome, nested between the inside and outside domes gives structural support to the outer dome and provides enough strength for
the lantern which the external dome could not support alone.
Figure 1: Sketch showing how a cubic was used to model the
nested domes of St Paul’s Cathedral.1
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Contrary to the quite widely spread popular belief, Wren did
not use the model of a catenary curve for his designs of St Paul’s
dome. His sketch (Figure 1) from around 1690 shows that he used
the simplest form of cubic, y = x3 , for the middle dome.
Things are not so simple though: Wren and Robert Hooke
worked on the catenary and were certainly close collaborators
who shared some insight on this topic. Robert Hooke’s (1635–
1703) paper A Description of Helioscopes, and Some Other Instruments [1] gave an anagram for how to build a perfect arch in an
appendix. For the next 30 years, this puzzle remained unsolved,
and the secret remained safe. The anagram referred to a simple,
effective, and quite beautiful finding that the shape of a flexible
cord or a chain, which can carry a certain load, can be used inverted to model an arch that can carry the same load. The answer
was not published until Hooke’s death; in 1705 the executor of
his will provided the full text in Latin: Ut pendet continuum flexile, sic stabit contiguum rigidum inversum – As hangs a flexible
cable so inverted stand the touching pieces of an arch.
Both Hooke and his friend Wren knew that this type of curve,
called a catenary (derived from catena, Latin for chain), is not
the same as a parabola or cubic parabola (see Figure 2), but they
were unable to give an exact mathematical equation to describe
it. Hooke presented a paper to the Royal Society in 1670 on the
application which could, in three dimensions, be used to build
perfect domes. In his paper a dome is formed by the cubicoparabolic conoid, which is created by the the rotation of half
the cubic parabola y = x3 about the y-axis. Both Wren and
Hooke were aware that the cubico-parabolic conoid was an approximation to the ideal surface but did not come up with the
curve’s mathematical formulation: that was to come from continental mathematicians James Bernoulli (1655–1705), Leibniz
(1646–1716) and Huygens (1629–1695).
basket in a shop in Oxford (see Figure 3). He apparently
noticed that if one of the ‘osiers’ led around the axis
of a cylinder and preserving that oblique position
with respect to the axis would describe that concavoconvex surface, and so cylindroids of that sort could
be made on a lathe by means of a straight steel tool
held in a position oblique to the axis of the cylinder, the section of which through the axis will be that
curved line [3].
These two examples of Wren’s mathematics are quite different
in scale but both deal with mathematics of surfaces. They show
us how mathematics can help us
both make sense of and build our
world. Wren used a simple object
– a wicker basket – as the inspiration to describe and classify a geometric surface. He also used mathematics to build a sophisticated, elaborate, grandiose cathedral that will
continue to adorn London’s skyline
for centuries to come.
Whilst the dome of St Paul’s
may ‘touch the skies’ above it,
it relies on the same mathematics
Figure 3: Ruled
that helps us understand the hyperhyperboloid.2
boloid. The dome suddenly does not
seem more important than the hyperboloid, and the ‘lord’ (as in
the starting quote of this article) of both is mathematics. A true
source of inspiration to celebrate my birthday, and the IMA’s.
Notes
1. © The Trustees of the British Museum.
2. Department of Mathematics at the University of Arizona,
Further reading
1 Bennett, J. (2002) The Mathematical Science of Christopher
Wren, Cambridge University Press.
2 Heyman, J. (1998) Hooke’s cubico-parabolical conoid, Notes
and Records of the Royal Society of London, vol. 52, no. 1,
pp. 39–50.
3 Lehmann, K. (1945) The dome of heaven, The Art Bulletin,
vol. 27, no. 1, pp. 1–27. Published by College Art Association.
Figure 2: The perfect dome line compared with the cubic
parabola and the hemisphere.
As well as having a lasting influence on the landscape of London’s architecture via his structure of St Paul’s dome, Wren made
some other mathematical discoveries. One of these is his understanding that a hyperboloid of revolution is a ruled surface. A surface is said to be ruled if through every point on it, a straight line
can be drawn that lies on the surface. Wren described this in the
paper he presented to the Royal Society in 1669 [2]. The anecdotal evidence suggests that Wren came up with the idea about
describing and defining ruled surfaces by seeing a round wicker
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References
1 Hooke, R. (1675) A Description of Helioscopes, and Some Other Instruments, London.
2 Wren, C. (1669) Generatio Corporis Cylindroidis Hyperbolici, Elaborandis Lenti bus Hyperbolicis Accommodati, Auth. Christophoro
Wren L L D. et Regiorum Aedisiciorum Praefecto, Nec non Soc.
Regiae Sodali, Phil. Trans., January 1, vol. 4, pp. 961–962;
doi:10.1098/rstl.1669.0018.
3 Hall, A.R. and Hall M.B., Editors (1965) Correspondence of Henry
Oldenburg, vol. 6, pp. 237–239, University of Wisconsin Press,
Madison.