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37-509: A six-sided polygon is a hexagon , one of the three regular polygons capable of tiling the plane . A hexagon also has 6 edges as well as 6 internal and external angles . 6 is the second smallest composite number . It is also the first number that is the sum of its proper divisors, making it the smallest perfect number . 6 is the first unitary perfect number , since it is the sum of its positive proper unitary divisors , without including itself. Only five such numbers are known to exist. 6

74-417: A 6 that looks like a "b" is not practical. Just as in most modern typefaces , in typefaces with text figures the character for the digit 6 usually has an ascender , as, for example, in [REDACTED] . This digit resembles an inverted 9 . To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels. Indeed, We created

111-404: A circle. The sometimes rather striking graphic similarity they have with the hieratic and demotic Egyptian numerals, while suggestive, is not prima facie evidence of an historical connection, as many cultures have independently recorded numbers as collections of strokes. With a similar writing instrument, the cursive forms of such groups of strokes could easily be broadly similar as well, and this

148-494: A given perimeter, the one with the largest area is regular (and therefore cyclic). Many specialized formulas apply to the areas of regular polygons . The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by This radius is also termed its apothem and is often represented as a . The area of a regular n -gon in terms of the radius R of its circumscribed circle can be expressed trigonometrically as: The area of

185-401: A regular n -gon inscribed in a unit-radius circle, with side s and interior angle α , {\displaystyle \alpha ,} can also be expressed trigonometrically as: The area of a self-intersecting polygon can be defined in two different ways, giving different answers: Using the same convention for vertex coordinates as in the previous section, the coordinates of

222-399: A simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1. In every polygon with perimeter p and area A , the isoperimetric inequality p 2 > 4 π A {\displaystyle p^{2}>4\pi A} holds. For any two simple polygons of equal area,

259-501: A simple polygon given by a sequence of line segments. This is called the point in polygon test. Brahmi numerals Brahmi numerals are a numeral system attested in the Indian subcontinent from the 3rd century BCE. It is the direct graphic ancestor of the modern Hindu–Arabic numeral system . However, the Brahmi numeral system was conceptually distinct from these later systems, as it

296-470: A solid polygon is its body , also known as a polygonal region or polygonal area . In contexts where one is concerned only with simple and solid polygons, a polygon may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygons and other self-intersecting polygons . Some sources also consider closed polygonal chains in Euclidean space to be

333-465: A total of six convex regular polytopes . In the classification of finite simple groups , twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant , the largest sporadic group: five first generation Mathieu groups , seven second generation subquotients of the Leech lattice , and eight third generation subgroups of

370-500: A type of polygon (a skew polygon ), even when the chain does not lie in a single plane. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes. The word polygon derives from the Greek adjective πολύς ( polús ) 'much', 'many' and γωνία ( gōnía ) 'corner' or 'angle'. It has been suggested that γόνυ ( gónu ) 'knee' may be

407-465: Is a "perfect ruler". The six exponentials theorem guarantees that under certain conditions one of a set of six exponentials is transcendental . The smallest non- abelian group is the symmetric group S 3 {\displaystyle \mathrm {S_{3}} } which has 3! = 6 elements. 6 the answer to the two-dimensional kissing number problem . A cube has 6 faces . A tetrahedron has 6 edges . In four dimensions , there are

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444-404: Is a polygon with n sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a solid polygon . The interior of

481-575: Is assumed throughout. Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: In this section, the vertices of the polygon under consideration are taken to be ( x 0 , y 0 ) , ( x 1 , y 1 ) , … , ( x n − 1 , y n − 1 ) {\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})} in order. For convenience in some formulas,

518-439: Is commonly called the shoelace formula or surveyor's formula . The area A of a simple polygon can also be computed if the lengths of the sides, a 1 , a 2 , ..., a n and the exterior angles , θ 1 , θ 2 , ..., θ n are known, from: The formula was described by Lopshits in 1963. If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives

555-451: Is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines). The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of

592-662: Is one of the primary hypotheses for the origin of Brahmi numerals. Another possibility is that the numerals were acrophonic , like the Attic numerals , and based on the Kharoṣṭhī alphabet. For instance, 4 "chatur" early on took a shape much like the Kharosthi letter 𐨖 "ch", while 5 "pancha" looks remarkably like Kharosthi 𐨤 "p"; and so on through 6 "ssat" and 𐨮, then 7 "sapta" and 𐨯, and finally 9 nava and 𐨣. However, there are problems of timing and lack of records. The full set of numerals

629-447: Is the largest of the four all-Harshad numbers . 6 is the 2nd superior highly composite number , the 2nd colossally abundant number , the 3rd triangular number , the 4th highly composite number , a pronic number , a congruent number , a harmonic divisor number , and a semiprime . 6 is also the first Granville number , or S {\displaystyle {\mathcal {S}}} -perfect number. A Golomb ruler of length 6

666-461: The Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides do determine the area. Of all n -gons with given side lengths, the one with the largest area is cyclic. Of all n -gons with

703-613: The Giant's Causeway in Northern Ireland , or at the Devil's Postpile in California . In biology , the surface of the wax honeycomb made by bees is an array of hexagons , and the sides and base of each cell are also polygons. In computer graphics , a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of

740-456: The geometrical vertices , as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials . Any surface is modelled as a tessellation called polygon mesh . If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2 n squared triangles since there are two triangles in a square. There are ( n + 1) / 2( n ) vertices per triangle. Where n

777-488: The regular star pentagon is also known as the pentagram . To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by Kepler , and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra , though not all sources use it. Polygons have been known since ancient times. The regular polygons were known to

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814-526: The ancient Greeks, with the pentagram , a non-convex regular polygon ( star polygon ), appearing as early as the 7th century B.C. on a krater by Aristophanes , found at Caere and now in the Capitoline Museum . The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century. In 1952, Geoffrey Colin Shephard generalized the idea of polygons to

851-453: The centroid of a solid simple polygon are In these formulas, the signed value of area A {\displaystyle A} must be used. For triangles ( n = 3 ), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3 . The centroid of the vertex set of a polygon with n vertices has the coordinates The idea of a polygon has been generalized in various ways. Some of

888-426: The complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons . Polygons appear in rock formations, most commonly as the flat facets of crystals , where the angles between the sides depend on the type of mineral from which the crystal is made. Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt , which may be seen at

925-535: The first three numerals seems clear: they are collections of 1, 2, and 3 strokes, in Ashoka 's era vertical I, II, III like Roman numerals , but soon becoming horizontal like the Han Chinese numerals . In the oldest inscriptions, 4 looks like a +, reminiscent of the X of neighboring Kharoṣṭhī , and perhaps a representation of 4 lines or 4 directions. However, the other unit numerals appear to be arbitrary symbols in even

962-469: The friendly giant. The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs ( Ly , O'N , Ru , J 4 , J 3 , and J 1 ). Hexa is classical Greek for "six". Thus: Sex- is a Latin prefix meaning "six". Thus: The SI prefix for 1000 is exa- (E), and for its reciprocal atto- (a). The evolution of our modern digit 6 appears rather simple when compared with

999-628: The heavens and the earth and everything in between in six Days, and We were not ˹even˺ touched with fatigue. Note 1: The word day is not always used in the Quran to mean a 24-hour period. According to Surah Al-Hajj (The Pilgrimage):47, a heavenly Day is 1000 years of our time. The Day of Judgment will be 50,000 years of our time - Surah Al-Maarij (The Ascending Stairways):4. Hence, the six Days of creation refer to six eons of time, known only by Allah. Note 2: Some Islamic scholars believe this verse comes in response to Exodus 31:17, which says, "The Lord made

1036-440: The heavens and the earth in six days, but on the seventh day He rested and was refreshed." Polygon In geometry , a polygon ( / ˈ p ɒ l ɪ ɡ ɒ n / ) is a plane figure made up of line segments connected to form a closed polygonal chain . The segments of a closed polygonal chain are called its edges or sides . The points where two edges meet are the polygon's vertices or corners . An n -gon

1073-833: The more important include: The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον ( polygōnon/polugōnon ), noun use of neuter of πολύγωνος ( polygōnos/polugōnos , the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek -derived numerical prefix with the suffix -gon , e.g. pentagon , dodecagon . The triangle , quadrilateral and nonagon are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon. Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example

1110-552: The notation ( x n , y n ) = ( x 0 , y 0 ) will also be used. If the polygon is non-self-intersecting (that is, simple ), the signed area is or, using determinants where Q i , j {\displaystyle Q_{i,j}} is the squared distance between ( x i , y i ) {\displaystyle (x_{i},y_{i})} and ( x j , y j ) . {\displaystyle (x_{j},y_{j}).} The signed area depends on

1147-422: The oldest inscriptions. It is sometimes supposed that they may also have come from collections of strokes, run together in cursive writing in a way similar to that attested in the development of Egyptian hieratic and demotic numerals, but this is not supported by any direct evidence. Likewise, the units for the tens are not obviously related to each other or to the units, although 10, 20, 80, 90 might be based on

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1184-423: The ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive x -axis to the positive y -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value . This

1221-427: The origin of gon . Polygons are primarily classified by the number of sides. Polygons may be characterized by their convexity or type of non-convexity: The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon . Euclidean geometry

1258-532: The other digits. The modern 6 can be traced back to the Brahmi numerals of India , which are first known from the Edicts of Ashoka c.  250 BCE . It was written in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped

1295-546: The part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G. On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal,

1332-426: The processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation. In computer graphics and computational geometry , it is often necessary to determine whether a given point P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} lies inside

1369-462: Was a non- positional decimal system, and did not include zero . Later additions to the system included separate symbols for each multiple of 10 (e.g. 20, 30, and 40). There were also symbols for 100 and 1000, which were combined in ligatures with the units to signify 200, 300, 2000, 3000, etc. In computers, these ligatures are written with the Brahmi Number Joiner at U+1107F. The source of

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