Curve

1 Jan

A catenary describes the curve adopted by a chain suspended from two points –  gravity acting uniformly along its length. I have been trying to suss out the right curvature for the top rail of a new four poster bed and playing with chains has helped.

Catenary

As you can see, the chain is slightly more curved in the middle than at the ends, like the steam bent lath of oak on top of it.

Catenary curves are important in architecture – particularly in bridge building – because of the way that they resist bending moments. Gaudi loved them so much, all the spires of his great cathedral, the Sagrada Familia in Barcelona, are based upon the catenary curve – here is his fantastic inverted string model complete with tiny sand bags… a spider’s web of catenary curves.

Gaudi

View of Nativity Façade of Basilica and Expiatory Church of the Holy Family (Basílica i Temple Expiatori de la Sagrada Família) ( UNESCO World Heritage Site). Barcelona, Catalonia, Spain.

If you are curious there are many mathematical treatments of catenary curves and their analogies in nature (skeletons). You will find them everywhere if you care to look. The lovely ‘Winking Bridge’ across the Tyne in Gateshead, dogs on leads, electricity cables hanging from pylons…

Gateshead      Catenary curves

I would be the first to admit that I am no mathematician, but I do love symmetry in natural forms. The completed Ruskin Sculpture – Mir Jansen and I will be exhibiting at the Millennium Gallery, consists of a framework of steam bent, thin oak laths on a sturdy base attached to a circular annulus to make a light, airy framework. Within the framework hang a series of paintings by Mir in gouache on panels of oak all cut from the same tree. The paintings appear to float within the interior of the sculpture, each suspended on 3 or 4 powerful magnets.

The laths are identical to the one in the top picture.  They were bent over a hemispherical frame – the slight recoil on removing the dried piece 24 hours later yields a catenary curve  (rather like the opening curvature of the helix generated by the golden mean below). This gives the sculpture great stability and natural spring, and like the Earth, it is, as a result, an oblate spheroid.

Mir and Henk  IMG_4750

The globular gallery is designed with 37 steam bent ribs – a convenient opening at the front for people to step in to structure. I have always thought of it as John Ruskin’s Mind – ideas within leaking out, ideas without leaking in.

The design also allows disabled access as I have taken a bite out of the floor so that you can feel that you are right inside – even in a wheel chair, and sit comfortably too.

But why 37 ribs?

37 is a prime number in the Padovan sequence.

Padovan sequence

The equation for the Padovan Sequence is
 defined by the equation:
P(n) = P(n-2) + P(n-3)            also known as a recurrence relation where every subsequent number depends upon the numbers before it.
with the initial conditions P=(0) = P (1) = P (2) = 1
The first few Padovan numbers are :  1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265  (the Prime numbers are in Bold)
Another recurrence relation with which you will be familiar is the Fibonacci Sequence:
Fn = Fn-1 + Fn-2
with the initial conditions – F0=0, F1=1
giving the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … (The next number is found by adding up the two numbers before it). Without going into it in detail the formula which allows you to calculate the nth Fibonacci number relies on a special number called phi (1.618), or better know as the Golden Mean. Rectangles with sides 1:1.618 can be used to derive spirals, snail shells and so on.
fibonacci-plant  divine ratio (a sequence of golden rectangles – Yin and Yang)
The Golden Ratio…1.618 (approximately) lies at the heart of proportions of beauty in Greek Architecture.
Greek Architecture
John Ruskin certainly appreciated structure at a deep level, in fact he insisted upon the importance of underlying Natural Laws and Principles in architecture (The Seven Lamps of Architecture)  and it is no accident that the sculpture resonates with the maths. Mir’s paintings reflect other aspects of Ruskin’s thinking … come and see them at the Millennium Gallery from January 23rd 2016 when our piece will be on display as part of an exhibition on contemporary Art and Craft.
This is a chain of thought, I hope you enjoy the links. Happy New Year!
Acknowledgements:
The entire structure was made from a single oak tree – a very kind donation by the Guild of St. George from Ruskinland, through John Isles who supported our work and encouraged us. We were commissioned by Museums Sheffield and generously supported by Arts Council England.

One Response to “Curve”

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  1. Bridge | Woodenhenk - November 25, 2016

    […] Leonardo would have laughed. More recently I came across his design for a simple bridge made of poles that interlock in my search for a new engaging sculpture project, following the success of the Ruskin Sculpture. […]

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