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Rotational Symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids. Formal treatment Formally the rotational symmetry is symmetry with respect to some or all rotations in ''m''-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of ''E''+(''m'') (see Euclidean group). Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Armoured Triskelion On The Flag Of The Isle Of Man
''The'' () is a grammatical Article (grammar), article in English language, English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the Most common words in English, most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with pronouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropy
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Mathematics Within mathematics, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ; Isotropic quadratic form: A quadratic form ''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an isotropic vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Of David
The Star of David (). is a generally recognized symbol of both Jewish identity and Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles. A derivation of the ''seal of Solomon'', which was used for decorative and mystical purposes by Muslims and Kabbalah, Kabbalistic Jews, its adoption as a distinctive symbol for the Jews, Jewish people and their religion dates back to 17th-century Prague. In the 19th century, the symbol began to be widely used among the History of the Jews in Europe, Jewish communities of Eastern Europe, ultimately coming to be used to represent Jewish identity or religious beliefs."The Flag and the Emblem" (MFA). It became representative of Zionism after it was Flag of Israel#Origin of the flag, chosen as the central symbol for a Jewish national flag at the First Zionist Congress in 1897. By the end of World War I, it had become an internationally accepted symbol for the Jewish people, being used on the gravestones of fallen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The word ''swastika'' comes from sa, स्वस्तिक, svastika, meaning "conducive to well-being". In Hinduism, the right-facing symbol (clockwise) () is called ', symbolizing ("sun"), prosperity and good luck, while the left-facing symbol (counter-clockwise) () is called ''sauwastika'', symbolising night or tantric aspects of Kali. In Jain symbolism, it represents Suparshvanathathe seventh of 24 Tirthankaras ( spiritual teachers and sav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borromean Rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing. The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as a coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan. They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer, giving them the alternative n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triskelion
A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of the European Neolithic and Bronze Age with continuation into the Iron Age especially in the context of the La Tène culture and related Celtic traditions. The actual ''triskeles'' symbol of three human legs is found especially in Greek antiquity, beginning in archaic pottery and continued in coinage of the classical period. In the Hellenistic period, the symbol becomes associated with the island of Sicily, appearing on coins minted under Dionysius I of Syracuse beginning in BCE. It later appears in heraldry, and, other than in the flag of Sicily, came to be used in the flag of the Isle of Man (known as ''ny tree cassyn'' "the three legs"). Greek (''triskelḗs'') means "three-legged". While the Greek adjective "three-legge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union Flag
The Union Jack, or Union Flag, is the ''de facto'' national flag of the United Kingdom. Although no law has been passed making the Union Flag the official national flag of the United Kingdom, it has effectively become such through precedent. It is sometimes asserted that the term ''Union Jack'' properly refers only to naval usage, but this assertion was dismissed by the Flag Institute in 2013 following historical investigations. The flag has official status in Canada, by parliamentary resolution, where it is known as the Royal Union Flag. It is the national flag of all British overseas territories, being localities within the British state, or realm, although local flags have also been authorised for most, usually comprising the blue or red ensign with the Union Flag in the canton and defaced with the distinguishing arms of the territory. These may be flown in place of, or along with (but taking precedence after) the national flag. Governors of British Overseas Territories ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yin And Yang
Yin and yang ( and ) is a Chinese philosophical concept that describes opposite but interconnected forces. In Chinese cosmology, the universe creates itself out of a primary chaos of material energy, organized into the cycles of yin and yang and formed into objects and lives. Yin is the receptive and yang the active principle, seen in all forms of change and difference such as the annual cycle (winter and summer), the landscape (north-facing shade and south-facing brightness), sexual coupling (female and male), the formation of both men and women as characters and sociopolitical history (disorder and order). Taiji or Tai chi () is a Chinese cosmological term for the "Supreme Ultimate" state of undifferentiated absolute and infinite potential, the oneness before duality, from which yin and yang originate. It can be compared with the old '' wuji'' (, "without pole"). In the cosmology pertaining to yin and yang, the material energy, which this universe has created itself out o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical. Symmetric function In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). The s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells. Hints at a general definition Given an action of a group ''G'' on a topological space ''X'' by homeomorphisms, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Groups In Three Dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded (finite) 3D object have one or more common fixed points. We follow the usual convention by choosing the origin as one of them. The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its proper symmetry group, the intersection of its full symmetry group with E+(3), which consists of all ''direct isometries'', i.e., isometries preserving orientation. For a bounded object, the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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