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Serpentine refers to the curved shape of an object or design which resembles the letter 's'. The serpentine shape is a geometric form that has historically been used in many fields such as art, architecture, anatomy and topography. The origin of the word was derived from the words snake and serpentine. A serpentine is a precious stone used throughout history for a variety of purposes such as seals, magic amulets, personal adornment and funerary equipment.
Serpentine, in geometry, is a cubic curve described by Sir Isaac Newton, and given by the cartesian equation y(a2 + x2) = abx. The origin is a point of inflection, the axis of x is an asymptote, and the curve lies between the parallel lines 2y = ±b. 
Italian witches during the Roman period believed that small pieces of serpentine stone could protect people from venomous creatures such as snakes or spiders. The reason is that the dark green color streaked with white color resembles the appearance of the snakeskin. Therefore, serpentine is believed to be a tool to draw out the toxins whenever a person got bitten by a venomous creature.
A poem attributed to the mythical Greek poet Orpheus, said to have been written in the fourth century AD, shows how far back the association between the mineral and snakes was made:
"No more the trailing serpent's tooth to fear. Let him who by the dragon's fang hath bled, On the dire wound Serpentine powdered spread, And in the stone his sure reliance place, For wounds inflicted by the reptile race."
(figure number refers to Hogarth's The Analysis of Beauty's figure plate)
Human's eye has a specific kind of enjoyment in winding walks, and serpentine rivers, and all sorts of objects, which obtain an elegant form of waving and serpentine lines.
As shown in [fig. 56], the cone represents a straight horn; it is relatively difficult to find any form of beauty on the account of its content.
Then if one observe in a manner and a degree where the beauty of this horn is increased, as shown in [Fig. 57] the shape is supposed to be bent in two different manners and degrees.
Finally, let the person attend increase the beauty even to grace and elegance, yet using the same horn,[Fig. 58], it then gets twisted round, and bent in two ways again at the same time(as in the last figure). In the first figure of these, the dashed line expresses the straight lines without any help from the curve lines, light or shade. It would hardly show contents.
Same thing in the second that by the bending the horn, the straight dashed line converts into a beautiful waving-line.
At last by twisting and bending the horn, it changes from waving into a serpentine-sape; it dips out of sight behind the horn in the middle, and returns again at the smaller end, not only provides spaces for imagination, and delightment for the eyes; but also informs the quantity and variety of the contents.
This horn example is an easy way to show the peculiar qualities of these serpentine-lines, and the advantage of bringing them into compositions, where contents to express contain grace and elegance. The same things of these serpentine-lines could be comprehended, that among the vast variety of waving-lines that may be conceived, there is only one that really deserves the name of the line of beauty, a precise serpentine-line - the line of grace.
— William Hogarth, The Analysis of Beauty, (London: 1753), p59–60
- Serpentine walls at the University of Virginia
Serpentine walls extend down the length of the main lawn at University of Virginia and flank both sides of the Rotunda. Each is an example of a crinkle crankle wall, and they are one of the many structures Thomas Jefferson created that combine aesthetics with utility. The sinusoidal path of the wall provides strength against toppling over, allowing the wall to be only a single brick thick.
- San Carlo alle Quattro Fontane, Rome, Italy (The Church of Saint Charles at the Four Fountains)
The monastic buildings and the cloister were completed first after which construction of the church took place during the period 1638-1641 and in 1646 it was dedicated to Saint Charles Borromeo. Although the idea for the serpentine facade must have been conceived fairly early on, probably in the mid-1630s, it was only constructed towards the end of Borromini's life and the upper part was not completed until after the architect's death. The concave-convex facade of San Carlo undulates in a non-classic way. Tall corinthian columns stand on plinths and bear the main entablatures; these define the main framework of two storys and the tripartite bay division. Between the columns, smaller columns with their entablatures weave behind the main columns and in turn they frame niches, windows, a variety of sculptures as well as the main door, the central oval aedicule of the upper order and the oval framed medallion borne aloft by angels. Above the main entrance, cherubim herms frame the central figure of Saint Charles Borromeo by Antonio Raggi and to either side are statues of St. John of Matha and St. Felix of Valois, the founders of the Trinitarian Order.
- London's famed Hyde Park and Kensington Gardens knows 'The Serpentine', a lake that spans both parks. It received the name from its snakelike, curving shape. A central bridge divides the lake into two parts, and defines the boundaries between Hyde Park and Kensington Gardens.
- In furniture industry, serpentine shape has been used in Louis XV commodes, and also in 18th-century English sideboards, cupboards, desks, and chests.
- The serpentine path at Castle Howard
Castle Howard has a couple of different gardens. There is a large formal one right behind the building, where it's situated on a ridge and therefore an English landscape park is created. It opens out from the formal garden and merges back with the park. When buildings and site elements are set into the landscape, a serpentine path connecting every spots is placed in between elements. The path merges into the landscape as well because of its natural shape, which makes it fit into the gardens very well.
- The serpentine rill, Rousham
Rousham House (or Rousham Park) is a country house at Rousham in Oxfordshire, England. It was built circa 1635 and remodeled by William Kent in the 18th century in a free Gothic style. The Paths lead through woods where the abundant water from the Cherwell is fully utilised, by using small rills lead to larger ponds and formal pools. It is in such way that the rill is a serpentine shape for easier flowing around the slopes.
Examples of Serpentine Shapes
- There are several Serpentine Rivers in the world, including
- Serpentine River, New Zealand
- Serpentine River (Tasmania), Australia
- Serpentine River (Western Australia)
- Serpentine River (British Columbia), Canada
- Serpentine River (New Brunswick), Canada
- The River Westbourne and Serpentine (lake), London, England (formerly known as the Serpentine River).
- This shape is often used while riding horses as a riding figure, to help train the animal and help improve the rider.
- Boustrophedon text is ordered in a serpentine fashion.
- The zigzag entropy coding used in JPEG images is evocative of the serpentine.
- Thomas Jefferson designed serpentine brick walls for the University of Virginia.  
- Sigmoid function
- Line of Beauty
References and footnotes
- 1911 Encyclopædia Britannica
- *O'Connor, John J.; Robertson, Edmund F., "Serpentine", MacTutor History of Mathematics archive, University of St Andrews.
- M. Owen Lee, Virgil as Orpheus: A Study of the Georgics. (Albany:State University of New York Press,1996), 9
- Elle Schroeder, Serpentine:Green stone magic, San Jose: 2003
- "Hyde Park History & Architecture". The Royal Parks. 2007. Retrieved 2012-03-29.
- Holly, "Things that inspire", August 12, 2007, http://www.thingsthatinspire.net/2007/08/serpentine-shape.html
|Wikisource has the text of the 1911 Encyclopædia Britannica article Serpentine (geometry).|