Following the tradition of the time, it would follow that Ingund was named after her father's mother. Siblings included a sister, Chlodosind (born about 569) and a brother Childebert (born 570). Sigebert became ruler of the Frankish kingdom of Austrasia in 561 on the death of his father Chlothar I.

In 575, Sigebert was embroiled in a civil war with his half brother, Chilperic I, king of Neustria. On the verge of victory, Sigebert was assassinated. With the death of Sigebert, Brunhilda and the children were in great fear for their safety. Childebert, only five years old, faced almost certain death from Chilperic. Duke Gundovald immediately came to Paris, where Brunhilda and the children were living, took possession of Childebert and secured his safety among the Austrasian nobility. When Chilperic came to Paris, he seized Brunhilda and ordered Ingund and Chlodosind to be held in custody in the monastery of Meaux. Ingund would have been only seven or eight during this traumatic time.

Contents 1 Marriage of Hermenegild and Ingund 2 Revolt of Hermenegild 3 Leovigild's Response 4 Aftermath 5 Speculation 6 Sources 7 References

Marriage of Hermenegild and Ingund Map showing Baetica and Lusitania

In 569 Leovigild was elevated to co-rule the Visigoths in Hispania and Septimania with his brother Liuva. Soon afterwards, in order to legitimize his kingship, he married Goiswintha, widow of the previous Visigothic King Athanagild. Leovigild had two sons, Hermenegild and Reccared, from a previous marriage. About 578 Leovigild negotiated the marriage of his eldest son Hermenegild to Ingund, daughter of Brunhilda now regent for her son Childebert.

Ingund travelled from France to Toledo through Septimania, the part of Gaul still held by the Visigoths. Septimania stretches from the eastern end of the Pyrenees, along the Mediterranean, to the Rhone. As Ingund passed through the Visigothic town of Agde she met the local Catholic bishop, Phronimius, who warned her not to accept the 'poison' of Arianism.

In 579 Prince Hermenegild married Ingund, he being an Arian and she a Catholic. At first Ingund was warmly received by Queen Goiswintha. However, the queen was determined that Ingund should be re-baptized in the Arian faith. Ingund, still only twelve, firmly refused. According to Gregory of Tours: "the Queen lost her temper completely" and "seized the girl by her hair and threw her to the ground: then she kicked her until she was covered with blood, had her stripped naked and ordered her to be thrown into the baptismal pool". Whether because of this fracas, or, more likely, because of Leovigild's desire to assure the succession of his sons (consistent with his previous actions to associate his sons with himself as rulers of the kingdom), he sent Hermenegild and Ingund to Seville to rule a portion of his kingdom - presumably the province of Baetica and southern Lusitania. Revolt of Hermenegild

It was at Seville that Ingund came into contact with Leander, a Catholic monk. Leander belonged to an elite and influential family of Hispano-Roman stock. His two brothers later became bishops and his sister an Abbess. The vast majority of the population of southern Spain was Hispano-Roman and Catholic. Also a significant segment of the Visigoth nobility were Catholic, not to mention that portion of the nobility whose roots were Hispano-Roman. Leander either was already bishop of Seville when Hermenegild and Ingund arrived there, or became bishop soon afterwards. There can be no doubt of the influence the bishop held, nor can there be any doubt that he saw in this Catholic princess an opportunity to advance the Catholic cause, for the history of this period contains numerous examples (real or mythical) of queens influencing their husband's religious conversion. Map showing Byzantine Spain c. 580.

Hermenegild's Baetica bordered Spania, the Byzantine controlled cities of southeastern Spain. These cities were predominantly Latin Christian.

The sixth century experienced a flight of Catholic clergy to southern Spain, many from Africa, but other areas as well. Persecution and the Three-Chapter Controversy would account for much of the flight. Examples of the new arrivals are the African Nanctus, Donatus and the Greek named Paul. So when Hermenegild and Ingund arrived in Seville, they would have been met by a strong and possibly active Catholic party.

In the winter of 579-80 Hermenegild proclaimed himself king at Seville, and yet, he continued to also refer to his father as 'King'. Whether or not Hermenegild held the Orthodox Christian belief in the Trinity at this time cannot be known, for it is not till 582 that he "officially" accepted the Catholic faith. However, from the beginning, he seems to have been supported by those who support the Catholic cause. For already in 580 Leander travelled to Constantinople to plead the rebel's cause and seek aid from the Byzantine Empire.

Sometime between 580 and 582 Hermenegild and Ingund had a son named Athanagild after Ingund's maternal grandfather. Leovigild's Response

Leander travelled to Constantinople to gain support from Emperor Tiberius in 580, returning in 582. Hermenegild converted to Catholicism in 582 - as Leander was absent in the years prior, it would follow that Ingund was a major influence for his conversion.

Leovigild more or less ignored his son's transgression until 582 when he marched on Merida and captured the city. It is difficult to determine whether this was because of Hermenegild's new found Catholicism or a coincidence. Nevertheless, Leovigild saw in Arianism Visigothic identity and any threat to this identity as a threat to Visigoth legitimacy to rule. He viewed Catholicism as the 'Roman' religion and Arianism as the Visigoth religion. Leovigild's response may have been primarily a reaction to Hermenegild and other Visigoth nobles who had, at one time or another, converted to Catholicism.

By 584 the revolt had decidedly turned against Hermenegild and its outcome became all too clear. Ingund with their young son fled to the neighboring Byzantine cities of Spain, who later refused to turn them over to Leovigild.

On her way to Constantinople with her son Athanagild, Ingund died (584) in Carthage, Africa and was buried there. The cause of her early death is not recorded, but one of the world's greatest plagues ravaged the Mediterranean at this time. Athanagild survived the journey to the Byzantine capital of Constantinople where he was brought up by Emperor Maurice.

The Byzantines used their custody of Ingund and her son to induce Ingund's brother, King Childebert II, to attack the Lombards of northern Italy. Childebert while only fourteen years of age at this time, would have also been much influenced by his strong-willed mother Brunhilda, who was also committed to securing Ingund and her grandson.

Leovigild besieged Seville for a year before he was able to capture the city in 584. The tenacity of the resistance is evidence of the support for this Catholic usurper. Convinced that resistance was now futile, Hermenegild surrendered to his father. Hermenegild was imprisoned at Tarragona and repeatedly urged to abjure Catholicism. He refused and was executed by Duke Sigisbert on 13 April 585. Aftermath

When in 579 Ingund arrived in Toledo, who would have guessed how resolute a spirit resided in this young girl and the effect she would have on the course of Spanish history. Her example, as attested by Gregory of Tours, profoundly influenced her husband's acceptance of Catholicism and eventual conversion. A conversion not solely based on political expediency, for when Hermenegild faced denying his Catholic faith or execution, he remained steadfast to his new faith. Regardless, the revolt of Hermenegild made apparent the weakening influence of the Arian doctrine in Spain.

Soon after the death of Hermenegild and Ingund, King Leovigild died and was succeeded by Reccared, Hermenegild's younger brother. By the second year of his reign, Reccared embraces Catholicism and begins the task of unifying the Spanish people under a single religion. It is plain that the spirit of Ingund and the example of Hermenegild had an influence on Spanish society, and particularly on the new king Reccared. Reccared's not supporting his father's actions against Hermenegild and the retribution he took on his brother's executioner indicates a bond between the two brothers. Pope Gregory's words further confirms Hermenegild's influence: "Reccared, following not his faithless father but his martyr brother, was converted from the perverseness of the Arian heresy." Speculation

In 680 Erwig became king of the Visigoths as a result of a palace coup. The Chronicle of Alfonso III asserts that Erwig was the son of Ardabast, who had travelled to Spain from Greece in the mid-7th century; and that Ardabast was the son of Athanagild. Sources Collins, Roger Early Medieval Spain: Unity in Diversity, 400-1000 Second Edition. New York: St. Martin's Press Thompson, E.A. The Goths in Spain. Oxford: Clarendon Press, 1969 Treadgold, Warren T. A History of the Byzantine State and Society. Stanford: Stanford University Press, 1997 Gibbons, The History of the Decline and Fall of the Roman Empire Volume IV. London: The Folio Society Gregory of Tours, The History of the Franks, England: Penguin Books Ltd. 1974

In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess.

Classically the defect arises in two ways: the defect of a vertex of a polyhedron; the defect of a hyperbolic triangle;

and the excess arises in one way: the excess of a spherical triangle.

In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).

In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem.

Contents 1 Defect of a vertex 2 Examples 3 Descartes' theorem 4 A potential error 5 References 6 External links

Defect of a vertex

For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If the sum of the angles exceeds a full turn, as occurs in some vertices of most (not all) non-convex polyhedra, then the defect is negative. If a polyhedron is convex, then the defects of all of its vertices are positive.

The concept of defect extends to higher dimensions as the amount by which the sum of the dihedral angles of the cells at a peak falls short of a full circle. Examples

The defect of any of the vertices of a regular dodecahedron (in which three regular pentagons meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles is 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.

The same procedure can be followed for the other Platonic solids: Descartes' theorem

Descartes' theorem on the "total defect" of a polyhedron states that if the polyhedron is homeomorphic to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4π radians). The polyhedron need not be convex.

A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron. This is a special case of the Gauss–Bonnet theorem which relates the integral of the Gaussian curvature to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the Gaussian curvature is zero and the integral of Gaussian curvature at a vertex is equal to the defect there.

This can be used to calculate the number V of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect. This total will have one complete circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron.

A converse to this theorem is given by Alexandrov's uniqueness theorem, according to which a metric space that is locally Euclidean except for a finite number of points of positive angular defect, adding to 4π, can be realized in a unique way as the surface of a convex polyhedron. A potential error

It is tempting to think that every non-convex polyhedron has some vertices whose defect is negative. Here is a counterexample. Consider a cube where one face is replaced by a square pyramid: this elongated square pyramid is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same and so are all positive.

Negative defect indicates that the vertex resembles a saddle point, whereas positive defect indicates that the vertex resembles a local maximum or minimum.

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