|Wikisource has the text of the 1911 Encyclopædia Britannica article Serpentine (geometry).|
- 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
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. 
Serpentine shape is a beautiful geometry that has historically been used in many fields such as art, architecture, anatomy and topography etc. However, the earliest origin of the word derived from the shape of a snake and a similar stone. Same as other commonly used stones, serpentine stone has been used throughout history for various kinds of objects: seals, magic amulets, personal adornment and funerary equipment etc.
Italian witches in Roman period well into the Middle Ages believed that small pieces of serpentine stone can 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 and 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."
Applications and observations
(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, who obtains 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
Recognized for flanking both sides of the Rotunda and extending down the length of the Lawn, with ten Pavilions interspersed with student rooms. Each of the room has its own classical architectural style, as well as its own walled garden separated by Jeffersonian Serpentine walls. These walls are called "serpentine" because they run a sinusoidal course, a course that lends strength to the wall and allows for the wall to be only one brick thick. It is one of many innovations that Jefferson attempted to combine aesthetics with utility.
- 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.
“Serpentine” is used both as the name of a rock and the name of a mineral. Mineralogists use “serpentine” as a group name for serpentine minerals. Petrologists refer to rocks composed mostly of serpentine minerals and minor amounts of talc, chlorite, magnetite, and brucite as serpentinites. Soils developed from serpentine (ultramafic) substrates are noted for their meager and strange biomass. The chemical infertility is the main controlling factor in the development ofplants in serpentine soils (Proctor and Woodell 1975, Kruckeberg 1984, Brooks 1987). See more on Serpentine soil
Almost all the muscles, and bones, of which the human form is composed, have more, or less of these kind of twists in them; and give in a less degree, the same kind of appearance to the parts which cover them, and are the immediate object of the eye: and for this reason it is that I have been so particular in describing these forms of the bent, and twisted, and ornamental horn. There is scarce a straight bone in the whole body. Almost all of them are not only bent different ways, but have a kind of twist, which in some of them is very graceful; and the muscles annexed to them, though they are of various shapes, appropriated to their particular uses, generally have their component fibres running in these serpentine-lines, surrounding and conforming themselves to the varied shape of the bones they belong to: more especially in the limbs. Anatomists are so satisfied of this, that they take a pleasure in distinguishing their several beauties. I shall only instance in the thigh-bone, and those about the hips. The thigh-bone [Fig. 62], has the waving and twisted turn of the horn, 58: but the beautiful bones adjoining, called the ossa innominata [Fig. 60], have, with greater variety, the same turns and twists of that horn when it is cut; and its inner and outward surfaces are exposed to the eye. How ornamental these bones appear, when the prejudice we conceive against them, as being part of a skeleton, is taken off, by adding a little foliage to them, may be seen in fig. || [Fig. 61] ― such shell-like winding forms, mixt with foliage, twisting about them, are made use of in all ornaments; a kind of composition calculated merely to please the eye. Divest these of their serpentine twinings and they immediately lose all grace, and return to the poor gothic taste they were in an hundred years ago [Fig. 63].
[Fig. 64] is meant to represent the manner, in which most of the muscles, (those of the limbs in particular) are twisted round the bones, and conform themselves to their length and shape; but with no anatomical exactness. As to the running of their fibres, some anatomists have compared them to skains of thread, loose in the middle, and tight at each end, which, when they are thus considered as twisted contrary ways round the bone, gives the strongest idea possible of a comparison of serpentine-lines. Of these fine winding forms then is the human body composed, and which, by their varied situations with each other, become more intricately pleasing, and form a continued waving of winding forms from one into the other, as may be best seen by examining a good anatomical figure, part of which you have here represented, in the muscular leg and thigh, [Fig. 65]: which shews the serpentine forms and varied situations of the muscles, as they appear when the skin is taken off. It was drawn from a plaster of paris figure cast off nature, the original of which was prepared for the mould by Cowper, the famous anatomist
Thus, in all other parts of the body, as well as these, wherever, for the sake of the necessary motion of the parts, with proper strength and agility, the insertions of the muscles are too hard and sudden, their swellings too bold, or the hollows between them too deep, for their out-lines to be beautiful; nature most judiciously softens these hardnesses, and plumps up these vacancies with a proper supply of fat, and covers the whole with the soft, smooth, springy, and, in delicate life, almost transparent skin, which, conforming itself to the external shape of all the parts beneath, expresses to the eye the idea of its contents with the utmost delicacy of beauty and grace.
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
- William Hogarth, The Analysis of Beauty (London: 1753), 61-64