The Loire Valley

The Loire, France’s longest river (1000 km), arises in the Massif Central in the south of the country. On its way to the sea, it flows first north then west through a region known as the ‘Garden of France’, renowned for its mild climate, fruitful agriculture and exquisite châteaux, more than 120 of them.

In the département of Loir et Cher, between Orléans and Tours, the Loire Valley is particularly appealing. It is home to some delightful wines: Vouvray, Montlouis and Cherveny. To the north of the river lies the Beauce, the granary of France; to the south, the Sologne, a region of horticultural land, forests and marshes, ideal for hunting. The landscape is varied and attractive. Little wonder that so many châteaux de plaisance were built here, where kings, princes and other nobility chose to live and dally.

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Blois and Chambord

Two of the most famous châteaux in the region are the Château de Blois (shown here), and the Château de Chambord (see title page), which is the Loire’s largest residence. Both are rated by the Michelin Guide as *** (= worth a journey). Whereas Chambord was built by King François I in the 16th century in the new Italian decorative style of Renaissance architecture, the château at Blois has a mixture of styles reflecting the various periods of its building from the 13th to the 17th century. Despite their differing styles, they have one thing in common, the result of the caprice of François I: both have magnificent spiral staircases.





















The staircases

The staircase at Chambord is unusual in that it consists of two superimposed spirals which do not meet, so that it is possible to ascend on one spiral and descend on the other. The staircase at Blois (shown here), while having only one spiral, is more finely decorated.

It has been suggested that both were designed by Leonardo da Vinci as ‘premier peintre et ingénieur et architecte du roi’, for Leonardo spent his last years in the Loire Valley under the patronage of François I.




















The circular helix

The shape of the rail of a spiral staircase is a circular helix. It is the shape taken by a straight line (not vertical or horizontal) on a plane when the plane is wrapped around a cylinder. A helix can be easily constructed by drawing a line, roughly diagonal, with an OHP pen on an acetate sheet, then wrapping this around a cylinder. A transparent plastic cylinder (such as used for presenting flowers) allows the helix to be seen to best advantage. Try this for yourself.

The helix is generated as the product of a rotation about a point of the plane and a translation perpendicular to the plane. One can think of the path of the point as a screw displacement or continuous twist about a straight line, so that it moves along the surface of a cylinder radius a, a distance z proportional to the angle of twist.In terms of cylindrical coordinates (r, , z), the equations of the helix are thus r = a , z = c where x = r cos , y = r sin , z are the equivalent Cartesian coordinates

Investigate Find the Cartesian equations of the helix.    





















Phyllotaxis

The arrangement of buds, stems or leaves around a branch is know as phyllotaxis. As a branch grows, it moves outwards in a spiral fashion, giving rise to buds, stems and leaves at regular intervals along a helical path.

To find the phyllotactic ratio of branch, locate a bud (stem or leaf) directly above another along the branch. Count the number b of spaces between the buds up to the one above, and the number a of complete turns the branch went through in generating those buds. The ratio a/b is the phyllotactic ratio for that particular plant. This ratio tends to be the same for any given species, although it may change along a single branch. Oddly enough, a and b are typically members of the Fibonacci sequence, either consecutive or alternate, as can be seen in the following table.

TYPICAL PHYLLOTACTIC RATIOS
2
1/3
2/3
2/5
3/8
5/13

elm
beech, blackberry, grasses, hazel
elm, grasses
apple, apricot, cherry, holly, plum
pear, poplar, weeping willow
almond, bottle brush, pussy willow

  Project  Investigate the phyllotactic ratio of common trees and shrubs. List your findings, and include sketches or photographs of the various trees and shrubs. How many different ratios of Fibonacci numbers can you find?



















The DNA molecule

Probably the best known example of the double helix is the molecule of DNA, deoxyribose nucleic acid, the ‘blueprint of life’, which occurs in the cells of every living thing. Its structure was first identified by American biologist James Watson and British physicist Francis Crick in 1953 based on X-ray crystallographic data. An account of their discovery is given in Watson’s book The Double Helix. It’s a good read.

Further reading

Garland, T.H. (1987), Fascinating Fibonacci: Mystery and Magic in Numbers,Dale Seymour Publications.

Watson, J.D., (1968), The Double Helix: a personal account of the discovery of the structure of DNA, Weidenfeld & Nicolson.


http://www.castles.org/castles/Europe/Western_Europe/France/france110.htm