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Concept

Cross

Summary

A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a saltire in heraldic terminology. The cross has been widely recognized as a symbol of Christianity from an early period. Before then, it was a pagan religious symbol throughout Europe and western Asia. The effigy of a man hanging on a cross was set up in the fields to protect the crops. It often appeared in conjunction with the female-genital circle or oval, to signify the sacred marriage, as in Egyptian amulet Nefer with male cross and female orb, considered as an amulet of blessedness, a charm of sexual harmony. The word cross is recorded in 11th-century Old English as cros, exclusively for the instrument of Christ's crucifixion, replacing the native Old English word rood. The word's history is complicated; it appears to have entered English from Old Irish, possibly via Old Norse, ultimately from the Latin crux (or its accusative crucem and its genitive crucis), "stake, cross". The English verb to cross arises from the noun 1200, first in the sense "to make the sign of the cross"; the generic meaning "to intersect" develops in the 15th century. The Latin word was influenced by popular etymology by a native Germanic word reconstructed as *krukjo (English crook, Old English crycce, Old Norse krokr, Old High German krucka). This word, by conflation with Latin crux, gave rise to Old French crocier (modern French crosse), the term for a shepherd's crook, adopted in English as crosier. Latin crux referred to the gibbet where criminals were executed, a stake or pole, with or without , on which the condemned were impaled or hanged, but more particularly a cross or the pole of a carriage. The derived verb crucio means "to put to death on the cross" or, more frequently, "to put to the rack, to torture, torment", especially in reference to mental troubles.

Official source

https://en.wikipedia.org/wiki/Cross

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Related publications (2)

Simple topological graphs

Andres Jacinto Ruiz Vargas

This thesis is devoted to the understanding of topological graphs. We consider the following four problems: 1. A \emph{simple topological graph} is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once, at endpoint or at a crossing. Let

G

be a complete simple topological graph on

n

vertices. The three edges induced by any triplet of vertices in

G

form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an \emph{empty triangle}. In 1998, Harborth proved that

G

has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least

2n/3

. We settle Harborth's conjecture in the affirmative. 2. A \emph{monotone cylindrical} graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called \emph{simple} if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are \emph{disjoint} if they do not intersect. We show that every simple complete monotone cylindrical graph on

n

vertices contains

\Omega(n^{1-\epsilon})

pairwise disjoint edges for any

\epsilon>0

. As a consequence, we show that every simple complete topological graph (drawn in the plane) with

n

vertices contains

\Omega(n^{\frac 12-\epsilon})

pairwise disjoint edges for any

\epsilon>0

. By extending some of the ideas here we are then able to get rid of the

\epsilon

term in the exponent, showing that in fact we can always guarantee a set with

\Omega(n^{\frac 12})

pairwise disjoint edges. This improves the previous lower bound of

\Omega(n^\frac 13)

by Suk and independently by Fulek. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges. 3. A {\em tangle} is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a {\em degenerate tangle}. We settle a problem of Pach, Radoi\v ci'c, and T'oth \cite{TTpaper}, by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles. 4. We show that for a constant

t\in \NN

, every simple topological graph on

n

vertices has

O(n)

edges if the graph has no two sets of

t

edges each such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is

K_{t,t}

-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i'c, and T'oth: Every

n

-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most

O(n)

edges.

EPFL2016

Villes à la frontière et transformation de l'espace

Lena Poschet El Moudden

Few borders exist which separate two countries as dramatically disparate as the Dominican Republic and Haiti. Both share the territory of the Hispaniola island, but a profound process of differentiation rooted in the colonial division of the island between France and Spain and ideologically maintained by both modern nations, has lead to what Théodat (1998, 2003) calls the "twofold insularity" – each country behaves as if it were located on a separate island. In spite of all prejudices and ethnical, historical, linguistic, social, economic or religious differences, people living in the border regions established cross-border relations which, even if largely based on economic interests, are much more frequent and important to both nations than officially admitted (De Jesus Cerdano and Dilla, 2005; Sillé & Segura & Cabral, 2002). The local dynamic created by this exchange becomes palpable when one observes the busy binational market in the border towns of Ouanaminthe (Haiti) and Dajabón (Dominican Republic). Twice a week, people from nearly the whole northern part of both countries travel to these towns located on either side of the Masacre River, in order to buy and sell goods or to offer services in the market of Dajabón. The other days, tons of cargo transit to Haiti, to the extent that the once small borderland settlements have become an important junction in the economic subsystem of the northern part of the island. Moreover, the construction of a free zone in Ouanaminthe in 2002 –to produce textiles for the US-market– has connected the borderland towns to the international economy. This recent development is set within a context of progressive opening of borders (Martinez, 1994). It is the result of international policies that have been decided in the capitals, without consulting the concerned regions. In Ouanaminthe and Dajabón, local institutions are weak and receive no or little financial and administrative support from central administration. They do not have the means to cope with growing pressure on land and infrastructure, as the emergence of legal and illegal livelihood opportunities in the borderland has generated a threefold increase of the overall population in the past twenty years (Dajabón grew from 8'500 inhabitants in 1980 to 16'000 in 2002 and Ouanaminthe from 7'000 in 1981 to 39'000 in 2003). How did these recent economic, political and societal transformations shape the urban space of the two borderland towns? I addressed this issue by analyzing the motivation of stakeholders involved in the process of spatial transformation, and the impact of their actions. In my work I refer to the conceptual framework of urban intermediation (Bolay and Rabinovich, 2004), i.e. the spatial issues that each town has to address according to its internal characteristics, its function in national and international networks and its relation to the direct rural environment. In order to use this framework for the borderland issues, the conceptual part of this study proposes to develop the concept of "borderland intermediation". To the existing urban intermediation concept, it adds the influence of the characteristics of the border (based on policy and political aspects, economic issues, local identities, social networks, national defense issues and geographical insertion). By doing so, it relates the bi-national system of borderland towns to the intermediation concept. Data has been collected over three and a half years during several periods of fieldwork: observation and mapping of occupation and use of urban space, cartography, qualitative interviews with local stakeholders and residents, completed by other primary and secondary sources. This study shows that the recent transformations have accentuated the existing unbalanced relationship between the two towns. The powerlessness of the municipality of Ouanaminthe to meet population growth and to impose regulations or taxes related to the growing trade has enhanced its dependency to the Dominican town. For a growing number of Haitians, it is of existential importance to cross over to Dajabón in order to seek income, healthcare and education and this, in spite of latent racism and mistreatments. Founded on a strong sentiment of exploitation, Haitians often criticize the capitalist or even colonialist relationship imposed by their neighbors: not only the fact that the bi-national market takes place in Dajabón, but also that Dominican enterprises develop subsidiaries in Ouanaminthe, taking advantage of cheap labor and the absence of local production. On the contrary, the majority of Dajabón residents interviewed do not need to cross the river for their living. Most of them have never been to Ouanaminthe, but consider it insalubrious, out of control and dangerous. Both towns lack application of regulations in urban planning matters. Thus, spatial growth has entirely been ruled by landowner structure and land occupation practices. Around Dajabón, the adjacent land belongs mainly to two landowners, whose strategy –to enhance soil value through development of equipped middleclass neighborhoods– has restricted the urban sprawl. In Ouanaminthe land belongs to a multitude of small peasants. The selling and illegal occupation of rather small plots leads to sprawling urban growth, guided by rural pathways and leaving entire parts of neighborhoods inaccessible to motor vehicles. Population growth has also been absorbed by high densification of existing neighborhoods. No formal collaborations exist between the two municipalities. Informal ones do, through the implication of the two mayors, but overall, cross-border collaboration is left to NGOs and professional associations who militate for human rights, environmental issues and fair treatment inside the bi-national market. International development agencies present in the borderland initiate projects with bi-national concern, but these are disconnected from the local level and strongly influenced by the fact that these agencies relate directly to the central governments. We also observe a process of internal segregation in both towns, as commercial establishments progressively occupy parts of the town centers and the emergent middle class gathers in specific neighborhoods. Ouanaminthe and Dajabón are functionally interlinked and their transformation is undeniably governed by the progressive permeability of the border. Nevertheless, this border separates two completely different urban landscapes that are deeply marked by the internal characteristics of each town. Furthermore, the unequal and complex relationship of both populations creates the paradox that their spatial proximity accentuates the segregation in both towns, as it allows the citizens of both countries to stay in their environment and to commute –if necessary– even daily. This work shows that the application of the intermediation concept is relevant for the study of complex relationships that govern borderland space. The analysis based on this concept has enabled to study, on a local level, processes that are taking place on a global level. In so doing, it has allowed us to provide some answers to the central question: "How do local populations, national and international firms and governments react when a territorial economy, based on agriculture, is progressively linked to international networks? "

EPFL2006

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Cross

Swastika

The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by the Nazi Party and by neo-Nazis. It continues to be used as a symbol of divinity and spirituality in Indian religions, including Hinduism, Buddhism, and Jainism. It generally takes the form of a cross, the arms of which are of equal length and perpendicular to the adjacent arms, each bent midway at a right angle.

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The Goths (𐌲𐌿𐍄𐌸𐌹𐌿𐌳𐌰; Gothi, Gótthoi) were Germanic people who played a major role in the fall of the Western Roman Empire and the emergence of medieval Europe. In his book Getica (c. 551), the historian Jordanes writes that the Goths originated in southern Scandinavia, but the accuracy of this account is unclear. A people called the Gutones - possibly early Goths - are documented living near the lower Vistula River in the 1st century, where they are associated with the archaeological Wielbark culture.

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