46-461: A loaf ( pl. : loaves ) is a (usually) rounded or oblong quantity of food, typically and originally of bread . It is common to bake bread in a rectangular bread pan or loaf pan because some kinds of bread dough tend to collapse and spread out during the cooking process if not constrained; the shape of less viscous doughs can be maintained with a bread pan whose sides are higher than the uncooked dough. More viscous doughs can be hand-molded into
92-485: A parallelogram containing a right angle. A rectangle with four sides of equal length is a square . The term " oblong " is used to refer to a non- square rectangle. A rectangle with vertices ABCD would be denoted as [REDACTED] ABCD . The word rectangle comes from the Latin rectangulus , which is a combination of rectus (as an adjective, right, proper) and angulus ( angle ). A crossed rectangle
138-431: A pair of lines in congruent corresponding angles then all transversals must do so. Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines. Because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines,
184-493: A pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements . Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate . Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius' definition as well as its modification by
230-431: A pentagon. The unique ratio of side lengths is a b = 0.815023701... {\displaystyle \displaystyle {\frac {a}{b}}=0.815023701...} . A crossed quadrilateral (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of
276-413: A plane q in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect. Equivalently, they are parallel if and only if the distance from a point P on line m to the nearest point in plane q is independent of the location of P on line m . Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in
322-406: A rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex. A crossed quadrilateral is sometimes likened to a bow tie or butterfly , sometimes called an "angular eight". A three-dimensional rectangular wire frame that is twisted can take
368-434: A rectangle is a rhombus , as shown in the table below. A rectangle is a rectilinear polygon : its sides meet at right angles. A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position (comprising two of translation and one of rotation ), one for shape ( aspect ratio ), and one for overall size (area). Two rectangles, neither of which will fit inside
414-435: A rectangle. A parallelogram with equal diagonals is a rectangle. The Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted A , B , C , and D , for any point P on the same plane of a rectangle: For every convex body C in
460-414: A unique rectangle with sides a {\displaystyle a} and b {\displaystyle b} , where a {\displaystyle a} is less than b {\displaystyle b} , with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape – a triangle and
506-605: Is ∥ {\displaystyle \parallel } . For example, A B ∥ C D {\displaystyle AB\parallel CD} indicates that line AB is parallel to line CD . In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 (∥) and U+2226 (∦), respectively. In addition, U+22D5 (⋕) represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space ,
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#1732779570454552-432: Is a convex quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which
598-549: Is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram , and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical , elliptic , and hyperbolic , have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling
644-486: Is evidently a symmetric relation . According to Euclid's tenets, parallelism is not a reflexive relation and thus fails to be an equivalence relation . Nevertheless, in affine geometry a pencil of parallel lines is taken as an equivalence class in the set of lines where parallelism is an equivalence relation. To this end, Emil Artin (1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common. Then
690-496: Is replaced by the more general concept of a geodesic , a curve which is locally straight with respect to the metric (definition of distance) on a Riemannian manifold , a surface (or higher-dimensional space) which may itself be curved. In general relativity , particles not under the influence of external forces follow geodesics in spacetime , a four-dimensional manifold with 3 spatial dimensions and 1 time dimension. In non-Euclidean geometry ( elliptic or hyperbolic geometry )
736-477: Is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with
782-441: Is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a. Lewis Carroll ), wrote a play, Euclid and His Modern Rivals , in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction . According to Wilhelm Killing
828-584: Is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles . The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes , allowing all rotations and reflections. There are also tilings by congruent polyaboloes . The following Unicode code points depict rectangles: Parallel (geometry) In geometry , parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in
874-406: Is used in many periodic tessellation patterns, in brickwork , for example, these tilings: A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size,
920-446: The same direction , but are not parts of the same straight line, are called parallel lines ." Wilson (1868 , p. 12) Augustus De Morgan reviewed this text and declared it a failure, primarily on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson also devotes a large section of his play (Act II, Scene VI § 1) to denouncing Wilson's treatment of parallels. Wilson edited this concept out of
966-437: The bottom of the loaf. Loaves of rectangular shape can be made more or less identical, and can be packed and shipped efficiently. The modern English word loaf is derived from Old English hlaf , 'bread', which in turn is from Proto-Germanic *khlaibuz . Old Norse hleifr , Swedish lev , Old Frisian hlef , Gothic hlaifs , Old High German hleib and modern German Laib derive from this Proto-Germanic word, which
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#17327795704541012-502: The desired loaf shape and cooked on a flat oven tray. The same principle applies to non-bread products such as meatloaf and cakes that are cooked so as to retain their shape during the cooking process. In determining the size of the loaf, the cook or baker must take into consideration the need for heat to penetrate the loaf evenly during the cooking process, so that no parts are overcooked or undercooked. Many kinds of mass-produced bread are distinctly squared, with well-defined corners on
1058-408: The distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines and calculating the distance between them. Since the lines have slope m , a common perpendicular would have slope −1/ m and we can take the line with equation y = − x / m as a common perpendicular. Solve the linear systems and to get the coordinates of
1104-563: The following properties are equivalent: Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are then consequences of Euclid's Parallel Postulate . The definition of parallel lines as
1150-413: The idea may be traced back to Leibniz . Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, and the difference of their directions is the angle between them." Wilson (1868 , p. 2) In definition 15 he introduces parallel lines in this way; "Straight lines which have
1196-418: The other, are said to be incomparable . If a rectangle has length ℓ {\displaystyle \ell } and width w {\displaystyle w} , then: The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter , the square has the largest area . The midpoints of the sides of any quadrilateral with perpendicular diagonals form
1242-412: The outside and exceed 180°. A rectangle and a crossed rectangle are quadrilaterals with the following properties in common: [REDACTED] In spherical geometry , a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in
1288-531: The philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry , so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid,
1334-517: The plane by rectangles or tiling a rectangle by polygons . A convex quadrilateral is a rectangle if and only if it is any one of the following: A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular . A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length . A trapezium
1380-439: The plane, we can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C and the positive homothety ratio is at most 2 and 0.5 × Area ( R ) ≤ Area ( C ) ≤ 2 × Area ( r ) {\displaystyle 0.5{\text{ × Area}}(R)\leq {\text{Area}}(C)\leq 2{\text{ × Area}}(r)} . There exists
1426-410: The points. The solutions to the linear systems are the points and These formulas still give the correct point coordinates even if the parallel lines are horizontal (i.e., m = 0). The distance between the points is which reduces to When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included): their distance can be expressed as Two lines in
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1472-404: The problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry, simplified and explained requires a proof of the fact that if one transversal meets
1518-434: The same direction or opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid 's parallel postulate . Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry , lines can have analogous properties that are referred to as parallelism. The parallel symbol
1564-446: The same three-dimensional space that do not intersect need not be parallel. Only if they are in a common plane are they called parallel; otherwise they are called skew lines . Two distinct lines l and m in three-dimensional space are parallel if and only if the distance from a point P on line m to the nearest point on line l is independent of the location of P on line m . This never holds for skew lines. A line m and
1610-400: The same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines . Line segments and Euclidean vectors are parallel if they have
1656-417: The same vertex arrangement as isosceles trapezia). A rectangle is cyclic : all corners lie on a single circle . It is equiangular : all its corner angles are equal (each of 90 degrees ). It is isogonal or vertex-transitive : all corners lie within the same symmetry orbit . It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°). The dual polygon of
1702-653: The same plane can either be: In the literature ultra parallel geodesics are often called non-intersecting . Geodesics intersecting at infinity are called limiting parallel . As in the illustration through a point a not on line l there are two limiting parallel lines, one for each direction ideal point of line l. They separate the lines intersecting line l and those that are ultra parallel to line l . Ultra parallel lines have single common perpendicular ( ultraparallel theorem ), and diverge on both sides of this common perpendicular. In spherical geometry , all geodesics are great circles . Great circles divide
1748-401: The same three-dimensional space and contain no point in common. Two distinct planes q and r are parallel if and only if the distance from a point P in plane q to the nearest point in plane r is independent of the location of P in plane q . This will never hold if the two planes are not in the same three-dimensional space. In non-Euclidean geometry , the concept of a straight line
1794-511: The sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry. In elliptic geometry , an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. In hyperbolic geometry , a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length. The rectangle
1840-403: The shape of a bow tie. The interior of a crossed rectangle can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A crossed rectangle may be considered equiangular if right and left turns are allowed. As with any crossed quadrilateral , the sum of its interior angles is 720°, allowing for internal angles to appear on
1886-410: The sphere in two equal hemispheres and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of
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1932-506: The sphere. If l, m, n are three distinct lines, then l ∥ m ∧ m ∥ n ⟹ l ∥ n . {\displaystyle l\parallel m\ \land \ m\parallel n\ \implies \ l\parallel n.} In this case, parallelism is a transitive relation . However, in case l = n , the superimposed lines are not considered parallel in Euclidean geometry. The binary relation between parallel lines
1978-443: The third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better. The main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from
2024-582: The three Euclidean properties mentioned above are not equivalent and only the second one (Line m is in the same plane as line l but does not intersect l) is useful in non-Euclidean geometries, since it involves no measurements. In general geometry the three properties above give three different types of curves, equidistant curves , parallel geodesics and geodesics sharing a common perpendicular , respectively. While in Euclidean geometry two geodesics can either intersect or be parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to
2070-464: The tiling is imperfect . In a perfect (or imperfect) triangled rectangle the triangles must be right triangles . A database of all known perfect rectangles, perfect squares and related shapes can be found at squaring.net . The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a perfect tilling a square is 21, found in 1978 by computer search. A rectangle has commensurable sides if and only if it
2116-420: Was also borrowed into Slavic ( Polish chleb , Russian khleb ) and Finnic ( Finnish leipä , Estonian leib ) languages as well. Rectangle In Euclidean plane geometry , a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles . It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or
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