In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the dual of polyhedron P is defined in terms of polar reciprocation about a sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius; when the sphere has radius r and is centered at the origin, i.e. defined by equation x 2 + y 2 + z 2 = r 2, P is a convex polyhedron its polar dual is defined as P ∘ = where y·x denotes the standard dot product of y and x. When no sphere is specified in the construction of the dual the unit sphere is used, meaning r = 1 in the above definition. For each face of P described by the linear equation x 0 x + y 0 y + z 0 z = r 2, the dual polyhedron P° will have a vertex; each vertex of P corresponds to a face of P°, each edge of P corresponds to an edge of P°. The correspondence between the vertices and faces of P and P° reverses inclusion.

For example, if an edge of P contains a vertex, the corresponding edge of P° will be contained in the corresponding face. For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'.

Some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models; the concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical.

It is the canonical dual, the two together form a canonical dual pair. When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way; such pairs of polyhedra are abstractly dual. The vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron; the same graph can be

In Chinese dialectology, Beijing Mandarin refers to a major branch of Mandarin Chinese recognized by the Language Atlas of China, encompassing a number of dialects spoken in areas of Beijing, Inner Mongolia and Tianjin, the most important of, the Beijing dialect, which provides the phonological basis for Standard Chinese. Beijing Mandarin and Northeastern Mandarin were proposed by Chinese linguist Li Rong as two separate branches of Mandarin in the 1980s. In Li’s 1985 paper, he suggested using tonal reflexes of Middle Chinese checked tone characters as the criterion for classifying Mandarin dialects. In this paper, he used the term “Beijing Mandarin” to refer the dialect group in which checked tone characters with a voiceless initial have dark level, light level and departing tone reflexes, he chose the name Beijing Mandarin. He subsequently proposed a split of Beijing Mandarin and Northeastern Mandarin in 1987, listing the following as reasons: Checked-tone characters with voiceless initials in Middle Chinese are far more distributed into the rising tone category in Northeastern Mandarin than in Beijing Mandarin.

The 2012 edition of Language Atlas of China added one more method for distinguishing Beijing Mandarin from Northeastern Mandarin: The modern pronunciations of the 精, 知, 莊 and 章 initials of Middle Chinese are two sets of sibilants—dental and retroflex—and these two sets are not merged or confused in Beijing Mandarin. Meanwhile, there are some scholars who regard Beijing Mandarin and Northeastern Mandarin as a single division of Mandarin. Lin noticed the phonological similarity between Northeastern Mandarin. Zhang suggested that the criteria for the division of Beijing Mandarin and Northeastern Mandarin as top-level Mandarin groups are inconsistent with the criterion for the division of other top-level Mandarin groups. Beijing Mandarin is classified into the following subdivisions in the 2012 edition of Language Atlas of China: Jīng–Chéng Jīngshī, including the urban area and some inner suburbs of Beijing. Huái–Chéng, including some suburbs of Beijing, parts of Langfang, most parts of Chengde and Duolun.

Cháo–Fēng, an area between the Huái–Chéng cluster and the Northeastern Mandarin, covering the cities of Chaoyang and Chifeng. This subgroup has characteristics intermediate of those of Beijing Mandarin and Northeastern Mandarin. Per the 2012 edition of Atlas, these subgroups are distinguished by the following features: Jīng–Chéng subgroup has a high dark level tone, the Cháo–Fēng subgroup a low one. Compared with the first edition, the second edition of the Atlas demoted Jīngshī and Huái–Chéng subgroups to clusters of a new Jīng–Chéng subgroup. Shí–Kè or Běijiāng subgroup, listed as a subgroup of Beijing Mandarin in the 1987 edition, is re-allocated to a Běijiāng subgroup of Lanyin Mandarin and a Nánjiāng subgroup of Central Plains Mandarin; the Cháo–Fēng subgroup covers a greater area in the 2012 edition. With regard to initials, the reflexes of kaikou hu syllables with any of the 影, 疑, 云 and 以 initials in Middle Chinese differ amongst the subgroups: a null initial is found in the Jīngshī cluster, while /n/ or /ŋ/ initials are present in the Huái–Chéng cluster and the Cháo–Fēng subgroup.

Dental and retroflex sibilants are distinct phonemes in Beijing Mandarin. This is contrary to Northeastern Mandarin, in which the two categories are either in free variation or merged into a single type of sibilants. In both Beijing Mandarin and Northeastern Mandarin, the checked tone of Middle Chinese has dissolved and is distributed irregularly among the remaining tones. However, Beijing Mandarin has fewer rising-tone characters with a checked-tone origin, compared with Northeastern Mandarin; the Cháo–Fēng subgroup has a lower tonal value for the dark level tone. The Cháo–Fēng subgroup has more words in common with that of Northeastern Mandarin; the intensifier 老 is used in the Cháo–Fēng subgroup. Chinese Academy of Social Sciences, Zhōngguó Yǔyán Dìtú Jí 中国语言地图集, Hànyǔ Fāngyán Juàn 汉语方言卷, Beijing: Commercial Press, ISBN 9787100070546Hou, Jingyi, Xiàndài Hànyǔ Fāngyán Gàilùn 现代汉语方言概论, Shanghai Educational Publishing House, ISBN 7-5320-8084-6Li, Rong, "Guānhuà Fāngyán de Fēnqū" 官话方言的分区, Fāngyán 方言: 2–5, ISSN 0257-0203Li, Rong, "Hànyǔ Fāngyán de Fēnqū" 汉语方言的分区, Fāngyán 方言: 241–259, ISSN 0257-0203Lin, Tao, "Běijīng Guānhuà Qū de Huàfēn" 北京官话区的划分, Fāngyán 方言: 166–172, ISSN 0257-0203Zhang, Shifang, Běijīng Guānhuà Yǔyīn Yánjiū 北京官话语音研究, Beijing Language and Culture University Press, ISBN 978-7-5619-2775-5

Medžitlija is a village in the municipality of Bitola, North Macedonia, along the border with Greece. The village is located 14 km south of Bitola at the Medžitlija-Níki border crossing; the village exists since the end of the 19th century, when it belonged to the Kazas of Monastir in the Ottoman Empire. Medžitlija was populated by Turks until the 1950s - 1960s when all migrated to Turkey and sold their properties to Albanians from the nearby village of Kišava. According to the census of 2002, the village has 155 inhabitants. According to the 2002 census, the village had a total of 155 inhabitants. Ethnic groups in the village include: Albanians 154 Macedonians 1 Location