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33-416: The reservation, in the shape of a square , is almost evenly divided in land area between the two states, with the northern 50.2 percent in southern Owyhee County, Idaho and the southern 49.8 percent in northwestern Elko County, Nevada . The total land area is 450.391 square miles (1,166.5 km). A resident population of 1,265 persons was reported in the 2000 census , more than 80 percent of whom lived on

66-401: A rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle , central angle , and external angle are all equal (90°). A square with vertices ABCD would be denoted ◻ {\displaystyle \square } ABCD . A quadrilateral is a square if and only if it is any one of the following: A square is a special case of

99-438: A rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely: A square has Schläfli symbol {4}. A truncated square, t{4},

132-444: A rhombus . These two forms are duals of each other, and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid , and p2 is the symmetry of a kite . g2 defines the geometry of a parallelogram . Only the g4 subgroup has no degrees of freedom, but can be seen as a square with directed edges . Every acute triangle has three inscribed squares (squares in its interior such that all four of

165-424: A consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. A vertex

198-447: A horizontal or vertical radius of r . The square is therefore the shape of a topological ball according to the L 1 distance metric . The following animations show how to construct a square using a compass and straightedge . This is possible as 4 = 2 , a power of two . The square has Dih 4 symmetry, order 8. There are 2 dihedral subgroups: Dih 2 , Dih 1 , and 3 cyclic subgroups: Z 4 , Z 2 , and Z 1 . A square

231-409: A polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve, there will be a point of extreme curvature near each polygon vertex. A vertex of a plane tiling or tessellation is a point where three or more tiles meet; generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally,

264-404: A property is that of a three by six rectangle. In classical times , the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power. The area can also be calculated using the diagonal d according to In terms of the circumradius R , the area of a square is since the area of the circle

297-478: A simple polygon P is called an ear if the diagonal [ x (i − 1) , x (i + 1) ] that bridges x i lies entirely in P . (see also convex polygon ) According to the two ears theorem , every simple polygon has at least two ears. A principal vertex x i of a simple polygon P is called a mouth if the diagonal [ x (i − 1) , x (i + 1) ] lies outside the boundary of P . Any convex polyhedron 's surface has Euler characteristic where V

330-400: A square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih 2 , order 4. It has the same vertex arrangement as the square, and is vertex-transitive . It appears as two 45-45-90 triangles with a common vertex, but the geometric intersection is not considered a vertex. A crossed square is sometimes likened to a bow tie or butterfly . the crossed rectangle

363-417: A square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of

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396-402: A square. The coordinates for the vertices of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points ( x i , y i ) with −1 < x i < 1 and −1 < y i < 1 . The equation specifies the boundary of this square. This equation means " x or y , whichever

429-534: A tessellation can be viewed as a kind of topological cell complex , as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces. A polygon vertex x i of a simple polygon P is a principal polygon vertex if the diagonal [ x (i − 1) , x (i + 1) ] intersects the boundary of P only at x (i − 1) and x (i + 1) . There are two types of principal vertices: ears and mouths . A principal vertex x i of

462-473: Is π R 2 , {\displaystyle \pi R^{2},} the square fills 2 / π ≈ 0.6366 {\displaystyle 2/\pi \approx 0.6366} of its circumscribed circle . In terms of the inradius r , the area of the square is hence the area of the inscribed circle is π / 4 ≈ 0.7854 {\displaystyle \pi /4\approx 0.7854} of that of

495-425: Is a transcendental number rather than an algebraic irrational number ; that is, it is not the root of any polynomial with rational coefficients. In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles. In spherical geometry , a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry,

528-404: Is a corner point of a polygon , polyhedron , or other higher-dimensional polytope , formed by the intersection of edges , faces or facets of the object. In a polygon, a vertex is called " convex " if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angles ); otherwise, it

561-401: Is a special case of many lower symmetry quadrilaterals: These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order. Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilaterals . r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a rectangle , and p4 is the symmetry of

594-438: Is an octagon , {8}. An alternated square, h{4}, is a digon , {2}. The square is the n = 2 case of the families of n - hypercubes and n - orthoplexes . The perimeter of a square whose four sides have length ℓ {\displaystyle \ell } is and the area A is Since four squared equals sixteen, a four by four square has an area equal to its perimeter. The only other quadrilateral with such

627-402: Is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex, if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and is concave otherwise. Polytope vertices are related to vertices of graphs , in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of

660-401: Is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to 2 . {\displaystyle {\sqrt {2}}.} Then the circumcircle has the equation Alternatively the equation can also be used to describe the boundary of a square with center coordinates ( a , b ), and

693-429: Is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3- simplex ( tetrahedron ). Vertex (geometry) In geometry , a vertex ( pl. : vertices or vertexes ) is a point where two or more curves , lines , or edges meet or intersect . As

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726-448: Is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals . The interior of a crossed square can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A square and a crossed square have the following properties in common: It exists in the vertex figure of a uniform star polyhedra , the tetrahemihexahedron . The K 4 complete graph

759-448: Is the number of vertices, E is the number of edges , and F is the number of faces . This equation is known as Euler's polyhedron formula . Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a cube has 12 edges and 6 faces, the formula implies that it has eight vertices. In computer graphics , objects are often represented as triangulated polyhedra in which

792-667: The Coeur d'Alene , Kootenai Tribe of Idaho , Nez Perce , and Shoshone-Bannock . It is one of several federally recognized tribes in Nevada, some of which include other Shoshone and Paiute bands. 42°00′N 116°08′W  /  42°N 116.14°W  / 42; -116.14 Square (geometry) In Euclidean geometry , a square is a regular quadrilateral , which means that it has four straight sides of equal length and four equal angles (90- degree angles, π/2 radian angles, or right angles ). It can also be defined as

825-657: The Duck Valley Western Shoshone Reservation by Executive Order ; it was also used for Northern Paiute people . Despite the Native Americans having a designated reservation, local settlers and some politicians tried to force the tribal members off the valuable Duck Valley lands in 1884, suggesting they should join their Western Shoshone kinsmen at the reservation at Fort Hall, Idaho . The bands' chiefs successfully resisted these efforts to be displaced from their lands. Meanwhile,

858-490: The Nevada side. In October 2016 the Nevada Native Nations Land Act was passed to put Bureau of Land Management (BLM) and Forest Service lands into trust for six federally recognized tribes in the state. The Shoshone-Paiute Tribe will have 82 acres (33 ha) of Forest Service land added to their reservation. Some other tribes are receiving thousands of acres of trust lands. Gaming is prohibited on

891-810: The Northern Paiute band joined with another branch of Shoshone in the Bannock War of 1878. Survivors were sent to a prisoner-of-war camp at the Yakama Indian Reservation in Yakima County, Washington . Upon their release, the Northern Paiute returned to the Duck Valley. President Grover Cleveland expanded the reservation by Executive Order on May 4, 1886 to accommodate the Paiute. President William Howard Taft expanded

924-432: The angles of such a square are larger than a right angle. Larger spherical squares have larger angles. In hyperbolic geometry , squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles. Examples: A crossed square is a faceting of the square, a self-intersecting polygon created by removing two opposite edges of

957-489: The new lands. The only significant community on the reservation is Owyhee, Nevada , at an elevation of 5,400 feet (1,650 m) above sea level . Owyhee is nearly equidistant from the two nearest major cities: 98 miles (158 km) north of Elko, Nevada , the county seat of the county by that name; and 97 miles (156 km) south of Mountain Home, Idaho . On April 16, 1877, President Rutherford B. Hayes established

990-414: The polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. However, in graph theory , vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve , its points of extreme curvature: in some sense the vertices of

1023-416: The reservation to its current size by Executive Order on July 1, 1910. It was unusual to have two federal government actions to enlarge the reservation after it was established; most federal actions have been taken to reduce the size of Indian reservations. The Shoshone-Paiute Tribe of Duck Valley is one of five federally recognized tribes in the state of Idaho, each of which have reservations. The others are

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1056-400: The square. Because it is a regular polygon , a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds: with equality if and only if the quadrilateral is

1089-525: The triangle's longest side. The fraction of the triangle's area that is filled by the square is no more than 1/2. Squaring the circle , proposed by ancient geometers , is the problem of constructing a square with the same area as a given circle , by using only a finite number of steps with compass and straightedge . In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem , which proves that pi ( π )

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