A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam
The Elements is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates and mathematical proofs of the propositions; the books cover plane and solid Euclidean geometry, elementary number theory, incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics, it has proven instrumental in the development of logic and modern science, its logical rigor was not surpassed until the 19th century. Euclid's Elements has been referred to as the most successful and influential textbook written, it was one of the earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482, with the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students.
Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read. Scholars believe that the Elements is a compilation of propositions based on books by earlier Greek mathematicians. Proclus, a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Pythagoras was the source for most of books I and II, Hippocrates of Chios for book III, Eudoxus of Cnidus for book V, while books IV, VI, XI, XII came from other Pythagorean or Athenian mathematicians; the Elements may have been based on an earlier textbook by Hippocrates of Chios, who may have originated the use of letters to refer to figures. In the fourth century AD, Theon of Alexandria produced an edition of Euclid, so used that it became the only surviving source until François Peyrard's 1808 discovery at the Vatican of a manuscript not derived from Theon's.
This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 is a tiny fragment of an older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century; the Arabs received the Elements from the Byzantines around 760. 800. The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation; the first printed edition appeared in 1482, since it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered in 1533. In 1570, John Dee provided a respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley.
Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford. The manuscripts available are of variable quality, invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text. Ancient texts which refer to the Elements itself, to other mathematical theories that were current at the time it was written, are important in this process; such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text. Of importance are the scholia, or annotations to the text; these additions, which distinguished themselves from the main text accumulated over time as opinions varied upon what was worthy of explanation or further study. The Elements is still considered a masterpiece in the application of logic to mathematics. In historical context, it has proven enormously influential in many areas of science. Scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, Sir Isaac Newton were all influenced by the Elements, applied their knowledge of it to their work.
Mathematicians and philosophers, such as Thomas Hobbes, Baruch Spinoza, Alfred North Whitehead, Bertrand Russell, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, studied it late at night by lamplight. Edna St. Vincent Millay wrote in her sonnet "Euclid alone has looked on Beauty bare", "O
A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about 8.4 lb of honey to secrete 1 lb of wax, so it makes economic sense to return the wax to the hive after harvesting the honey. The structure of the comb may be left intact when honey is extracted from it by uncapping and spinning in a centrifugal machine—the honey extractor. If the honeycomb is too worn out, the wax can be reused in a number of ways, including making sheets of comb foundation with hexagonal pattern; such foundation sheets allow the bees to build the comb with less effort, the hexagonal pattern of worker-sized cell bases discourages the bees from building the larger drone cells. Fresh, new comb is sometimes sold and used intact as comb honey if the honey is being spread on bread rather than used in cooking or as a sweetener. Broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet, called travel stain by beekeepers when seen on frames of comb honey.
Honeycomb in the "supers" that are not used for brood stays light-colored. Numerous wasps Polistinae and Vespinae, construct hexagonal prism-packed combs made of paper instead of wax. However, the term "honeycomb" is not used for such structures; the axes of honeycomb cells are always quasihorizontal, the nonangled rows of honeycomb cells are always horizontally aligned. Thus, each cell has two vertically oriented walls, with the upper and lower parts of the cells composed of two angled walls; the open end of a cell is referred to as the top of the cell, while the opposite end is called the bottom. The cells slope upwards, between 9 and 14°, towards the open ends. Two possible explanations exist as to why honeycomb is composed of hexagons, rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, while minimizing the total perimeter of the cells. Known in geometry as the honeycomb conjecture, this was given by Jan Brożek and proved much by Thomas Hales.
Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. A second reason, given by D'Arcy Wentworth Thompson, is that the shape results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency; the closed ends of the honeycomb cells are an example of geometric efficiency, albeit three-dimensional. The ends are trihedral sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume; the shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells. Individual cells do not show this geometric perfection: in a regular comb, deviations of a few percent from the "perfect" hexagonal shape occur.
In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are distorted. Cells are angled up about 13° from horizontal to prevent honey from dripping out. In 1965, László Fejes Tóth discovered the trihedral pyramidal shape used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would be.035% more efficient. This difference is too minute to measure on an actual honeycomb, irrelevant to the hive economy in terms of efficient use of wax, considering wild comb varies from any mathematical notion of "ideal" geometry. Bees used their antennae and legs to manipulate the wax during comb construction, while warming the wax. During the construction of hexagonal cells, wax temperature is between 33.6 and 37.6°C, well below the 40°C temperature at which wax is assumed to be liquid for initiating new comb construction. The body temperature of bees is a factor for regulating an ideal wax temperature for building the comb.
Honeycomb structure Wax foundation Hive frame Jan Dzierzon
The Ancient Greek language includes the forms of Greek used in Ancient Greece and the ancient world from around the 9th century BCE to the 6th century CE. It is roughly divided into the Archaic period, Classical period, Hellenistic period, it is succeeded by medieval Greek. Koine is regarded as a separate historical stage of its own, although in its earliest form it resembled Attic Greek and in its latest form it approaches Medieval Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects. Ancient Greek was the language of Homer and of fifth-century Athenian historians and philosophers, it has contributed many words to English vocabulary and has been a standard subject of study in educational institutions of the Western world since the Renaissance. This article contains information about the Epic and Classical periods of the language. Ancient Greek was a pluricentric language, divided into many dialects; the main dialect groups are Attic and Ionic, Aeolic and Doric, many of them with several subdivisions.
Some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are several historical forms. Homeric Greek is a literary form of Archaic Greek used in the epic poems, the "Iliad" and "Odyssey", in poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic and other Classical-era dialects; the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period, they differ in some of the detail. The only attested dialect from this period is Mycenaean Greek, but its relationship to the historical dialects and the historical circumstances of the times imply that the overall groups existed in some form. Scholars assume that major Ancient Greek period dialect groups developed not than 1120 BCE, at the time of the Dorian invasion—and that their first appearances as precise alphabetic writing began in the 8th century BCE.
The invasion would not be "Dorian" unless the invaders had some cultural relationship to the historical Dorians. The invasion is known to have displaced population to the Attic-Ionic regions, who regarded themselves as descendants of the population displaced by or contending with the Dorians; the Greeks of this period believed there were three major divisions of all Greek people—Dorians and Ionians, each with their own defining and distinctive dialects. Allowing for their oversight of Arcadian, an obscure mountain dialect, Cypriot, far from the center of Greek scholarship, this division of people and language is quite similar to the results of modern archaeological-linguistic investigation. One standard formulation for the dialects is: West vs. non-west Greek is the strongest marked and earliest division, with non-west in subsets of Ionic-Attic and Aeolic vs. Arcadocypriot, or Aeolic and Arcado-Cypriot vs. Ionic-Attic. Non-west is called East Greek. Arcadocypriot descended more from the Mycenaean Greek of the Bronze Age.
Boeotian had come under a strong Northwest Greek influence, can in some respects be considered a transitional dialect. Thessalian had come under Northwest Greek influence, though to a lesser degree. Pamphylian Greek, spoken in a small area on the southwestern coast of Anatolia and little preserved in inscriptions, may be either a fifth major dialect group, or it is Mycenaean Greek overlaid by Doric, with a non-Greek native influence. Most of the dialect sub-groups listed above had further subdivisions equivalent to a city-state and its surrounding territory, or to an island. Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, Northern Peloponnesus Doric; the Lesbian dialect was Aeolic Greek. All the groups were represented by colonies beyond Greece proper as well, these colonies developed local characteristics under the influence of settlers or neighbors speaking different Greek dialects; the dialects outside the Ionic group are known from inscriptions, notable exceptions being: fragments of the works of the poet Sappho from the island of Lesbos, in Aeolian, the poems of the Boeotian poet Pindar and other lyric poets in Doric.
After the conquests of Alexander the Great in the late 4th century BCE, a new international dialect known as Koine or Common Greek developed based on Attic Greek, but with influence from other dialects. This dialect replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, spoken in the region of modern Sparta. Doric has passed down its aorist terminations into most verbs of Demotic Greek. By about the 6th century CE, the Koine had metamorphosized into Medieval Greek. Ancient Macedonian was an Indo-European language at least related to Greek, but its exact relationship is unclear because of insufficient data: a dialect of Greek; the Macedonian dialect (or l
In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged; the term isotoxal is derived from the Greek τοξον meaning arc. An isotoxal polygon is an equilateral polygon; the duals of isotoxal polygons are isogonal polygons. In general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is an isotoxal polygon with D2 symmetry. All regular polygons are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges. Regular polyhedra are isohedral and isotoxal. Quasiregular polyhedra are not isohedral. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal.
For instance, the truncated icosahedron has two types of edges: hexagon-hexagon and hexagon-pentagon, it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge. An isotoxal polyhedron has the same dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, six more as quasiregular star polyhedra and their duals. There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal, infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings, non-right groups. Table of polyhedron dihedral angles Vertex-transitive Face-transitive Cell-transitive Peter R. Cromwell, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Coxeter, Harold Scott MacDonald.
"Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.
Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment
In geometry a simple polygon is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pair-wise to form a closed path. If the sides intersect the polygon is not simple; the qualifier "simple" is omitted, with the above definition being understood to define a polygon in general. The definition given above ensures the following properties: A polygon encloses a region which always has a measurable area; the line segments that make-up a polygon meet only at their endpoints, called vertices or less formally "corners". Two edges meet at each vertex; the number of edges always equals the number of vertices. Two edges meeting at a corner are required to form an angle, not straight. Mathematicians use "polygon" to refer only to the shape made up by the line segments, not the enclosed region, however some may use "polygon" to refer to a plane figure, bounded by a closed path, composed of a finite sequence of straight line segments. According to the definition in use, this boundary may not form part of the polygon itself.
Simple polygons are called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it. A polygon in the plane is only if it is topologically equivalent to a circle, its interior is topologically equivalent to a disk. If a collection of non-crossing line segments forms the boundary of a region of the plane, topologically equivalent to a disk this boundary is called a weakly simple polygon. In the image on the left, ABCDEFGHJKLM is a weakly simple polygon according to this definition, with the color blue marking the region for which it is the boundary; this type of weakly simple polygon can arise in computer graphics and CAD as a computer representation of polygonal regions with holes: for each hole a "cut" is created to connect it to an external boundary. Referring to the image above, ABCM is an external boundary of a planar region with a hole FGHJ; the cut ED connects the hole with the exterior and is traversed twice in the resulting weakly simple polygonal representation.
In an alternative and more general definition of weakly simple polygons, they are the limits of sequences of simple polygons of the same combinatorial type, with the convergence under the Fréchet distance. This formalizes the notion. However, this type of weakly simple polygon does not need to form the boundary of a region, as its "interior" can be empty. For example, referring to the image above, the polygonal chain ABCBA is a weakly simple polygon according to this definition: it may be viewed as the limit of "squeezing" of the polygon ABCFGHA. In computational geometry, several important computational tasks involve inputs in the form of a simple polygon. Point in polygon testing involves determining, for a simple polygon P and a query point q, whether q lies interior to P. Simple formulae are known for computing polygon area. Polygon partition is a set of primitive units, which do not overlap and whose union equals the polygon. A polygon partition problem is a problem of finding a partition, minimal in some sense, for example: a partition with a smallest number of units or with units of smallest total side-length.
A special case of polygon partition is Polygon triangulation: dividing a simple polygon into triangles. Although convex polygons are easy to triangulate, triangulating a general simple polygon is more difficult because we have to avoid adding edges that cross outside the polygon. Bernard Chazelle showed in 1991 that any simple polygon with n vertices can be triangulated in Θ time, optimal; the same algorithm may be used for determining whether a closed polygonal chain forms a simple polygon. Boolean operations on polygons: Various Boolean operations on the sets of points defined by polygonal regions; the convex hull of a simple polygon may be computed more efficiently than the convex hull of other types of inputs, such as the convex hull of a point set. Voronoi diagram of a simple polygon Medial axis/topological skeleton/straight skeleton of a simple polygon Offset curve of a simple polygon Minkowski sum for simple polygons Star domain Weisstein, Eric W. "Simple polygon". MathWorld