Mary II was Queen of England and Ireland, co-reigning with her husband, King William III & II, from 1689 until her death. Popular histories refer to their joint reign as that of William and Mary. Although their father James, Duke of York, was Roman Catholic and her younger sister Anne were raised as Anglicans at the wishes of their uncle, King Charles II. Charles lacked legitimate children, she married her Protestant first cousin, William of Orange, in 1677. Charles died in 1685 and James took the throne, making Mary heir presumptive. James's attempts at rule by decree and the birth of his son, James Francis Edward Stuart, led to his deposition in the Glorious Revolution and the adoption of the English Bill of Rights. William and Mary became queen regnant. Though William relied on Mary, she wielded less power than William and, when he was in England, ceded most of her authority to him, she did, act alone when William was engaged in military campaigns abroad, proving herself to be a powerful and effective ruler.

Mary's death left William as sole ruler until his own death in 1702, when he was succeeded by Mary's sister, Anne. Mary, born at St James's Palace in London on 30 April 1662, was the eldest daughter of the Duke of York, his first wife, Anne Hyde. Mary's uncle was King Charles II, who ruled the three kingdoms of England and Ireland, she was baptised into the Anglican faith in the Chapel Royal at St James's, was named after her ancestor, Queen of Scots. Her godparents included Prince Rupert of the Rhine. Although her mother bore eight children, all except Mary and her younger sister Anne died young, King Charles II had no legitimate children. For most of her childhood, Mary was second in line to the throne after her father; the Duke of York converted to Roman Catholicism in 1668 or 1669 and the Duchess about eight years earlier, but Mary and Anne were brought up as Anglicans, pursuant to the command of Charles II. They were moved to their own establishment at Richmond Palace, where they were raised by their governess Lady Frances Villiers, with only occasional visits to see their parents at St James's or their grandfather Lord Clarendon at Twickenham.

Mary's education, from private tutors, was restricted to music, drawing and religious instruction. Her mother died in 1671, her father remarried in 1673, taking as his second wife Mary of Modena, a Catholic, only four years older than Mary. From about the age of nine until her marriage, Mary wrote passionate letters to an older girl, Frances Apsley, the daughter of courtier Sir Allen Apsley. Mary signed herself'Mary Clorine'. In time, Frances became uncomfortable with the correspondence, replied more formally. At the age of fifteen, Mary became betrothed to her cousin, the Protestant Stadtholder of Holland, William III of Orange. William was the son of the King's late sister, Princess Royal, thus fourth in the line of succession after James and Anne. At first, Charles II opposed the alliance with the Dutch ruler—he preferred that Mary wed the heir to the French throne, the Dauphin Louis, thus allying his realms with Catholic France and strengthening the odds of an eventual Catholic successor in Britain.

The Duke of York agreed to the marriage, after pressure from chief minister Lord Danby and the King, who incorrectly assumed that it would improve James's popularity among Protestants. When James told Mary that she was to marry her cousin, "she wept all that afternoon and all the following day". William and a tearful Mary were married in St James's Palace by Bishop Henry Compton on 4 November 1677. Mary accompanied her husband on a rough sea crossing back to the Netherlands that month, after a delay of two weeks caused by bad weather. Rotterdam was inaccessible because of ice, they were forced to land at the small village of Ter Heijde, walk through the frosty countryside until met by coaches to take them to Huis Honselaarsdijk. On 14 December, they made a formal entry to The Hague in a grand procession. Mary's animated and personable nature made her popular with the Dutch people, her marriage to a Protestant prince was popular in Britain, she was devoted to her husband, but he was away on campaigns, which led to Mary's family supposing him to be cold and neglectful.

Within months of the marriage Mary was pregnant. She suffered further bouts of illness that may have been miscarriages in mid-1678, early 1679, early 1680, her childlessness would be the greatest source of unhappiness in her life. From May 1684, King Charles's illegitimate son, James Scott, Duke of Monmouth, lived in the Netherlands, where he was fêted by William and Mary. Monmouth was viewed as a rival to the Duke of York, as a potential Protestant heir who could supplant the Duke in the line of succession. William, did not consider him a viable alternative and assumed that Monmouth had insufficient support. Upon the death of Charles II without legitimate issue in February 1685, the Duke of York became king as James II in England and Ireland and James VII in Scotland. Mary was playing cards when her husband informed her of her father's accession, that she was heir presumptive. When

Crystal Skulls is a Los Angeles, California-based indie pop band, named after a legend that there are 13 ancient crystal skulls hidden worldwide, which have mystical powers. Known for their intelligent, smooth poppy sound, the band was started in 2004 in Seattle by Christian Wargo, with a lineup comprising drummer Casey Foubert, bass guitarist Yuuki Matthews, guitarist Ryan J. Phillips, they toured the United States with Black Mountain and Helio Sequence, released albums in 2005 and 2006, during which time they shared a practice space with fellow Seattle band Fleet Foxes. In 2007 Crystal Skulls went on hiatus when Wargo joined Fleet Foxes, Foubert and Matthews joined Sufjan Stevens. Crystal Skulls was relaunched in 2014 with a new lineup comprising Wargo, drummer Aaron Sperske, Todd Dahlhoff, George Holdcroft, Thomas Hunter. Christian Wargo — vocals, guitar Aaron Sperske — drums Todd Dahlhoff — bass George Holdcroft — keyboards Thomas Hunter — guitar Blocked Numbers The Cosmic Door EP Outgoing Behavior MPR Radio Interview with Crystal Skulls Crystal Skulls on Myspace One of Scientific's albums on Allmusic Scientific's discography on Discogs.com

Euclidean space is the fundamental space of classical geometry. It was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane, it was introduced by the Ancient Greek mathematician Euclid of Alexandria, the qualifier Euclidean is used to distinguish it from other spaces that were discovered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical universe, their great innovation was to prove all properties of the space as theorems by starting from a few fundamental properties, called postulates, which either were considered as evident, or seemed impossible to prove. After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition.

It is this definition, more used in modern mathematics, detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is only one Euclidean space of each dimension. Therefore, in many cases, it is possible to work with a specific Euclidean space, the real n-space R n, equipped with the dot product. An isomorphism from a Euclidean space to R n associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point. Euclidean space was introduced by ancient Greeks as an abstraction of our physical space, their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few basic properties, which are abstracted from the physical world, cannot be mathematically proved because of the lack of more basic tools. These properties are called axioms in modern language.

This way of defining Euclidean space is still in use under the name of synthetic geometry. In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers; this reduction of geometry to algebra was a major change of point of view, as, until the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance. Euclidean geometry was not applied in spaces of more than three dimensions until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of n dimensions using both synthetic and algebraic methods, discovered all of the regular polytopes that exist in Euclidean spaces of any number of dimensions. Despite the wide use of Descartes' approach, called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century; the introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition.

This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition, now most used for introducing Euclidean spaces. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance; the other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures of the plane should be considered equivalent if one can be transformed into the other by some sequence of translations and reflections. In order to make all of this mathematically precise, the theory must define what is a Euclidean space, the related notions of distance, angle and rotation.

When used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, so on. A purely mathematical definition of Euclidean space ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres; the standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is to define a Euclidean space as a set of points on which acts a real vector space, the space of translations, equipped with an inner product. The action of translations makes the space an affine space, this allows defining lines, subspaces and parallelism; the inner product allows defining distance and angles. The set R n of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism b