Conway polyhedron notation
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, tC represents a truncated cube, taC, parsed as t, is a truncated cuboctahedron; the simplest operator dual swaps vertex and face elements. Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators abdegjkmost, while Hart added r and p. Conway's basic operations are sufficient to generate the Archimedean and Catalan solids from the Platonic solids; some basic operations can be made as composites of others. Implementations named further operators, sometimes referred to as "extended" operators. In general, it is difficult to predict the resulting appearance of the composite of two or more operations from a given seed polyhedron.
For instance, ambo applied twice is the expand operation: aa = e, while a truncation after ambo produces bevel: ta = b. Many basic questions about Conway operators remain open, for instance, how many operators of a given "size" exist. In Conway's notation, operations on polyhedra are applied from right to left. For example, a cuboctahedron is an ambo cube, i.e. a = a C, a truncated cuboctahedron is t = t = t a C. Repeated application of an operator can be denoted with an exponent: j2. In general, Conway operators are not commutative; the resulting polyhedron has a fixed topology, while exact geometry is not specified: it can be thought of as one of many embeddings of a polyhedral graph on the sphere. The polyhedron is put into canonical form. Individual operators can be visualized in terms of "chambers", as below; each white chamber is a rotated version of the others. For achiral operators, the red chambers are a reflection of the white chambers. Achiral and chiral operators are called local symmetry-preserving operations and local operations that preserve orientation-preserving symmetries although the exact definition is a little more restrictive.
The relationship between the number of vertices and faces of the seed and the polyhedron created by the operations listed in this article can be expressed as a matrix M x. When x is the operator, v, e, f are the vertices and faces of the seed, v ′, e ′, f ′ are the vertices and faces of the result M x =; the matrix for the composition of two operators is just the product of the matrixes for the two operators. Distinct operators may have the same matrix, for p and l; the edge count of the result is an integer multiple d of that of the seed: this is called the inflation rate, or the edge factor. The simplest operators, the identity operator S and the dual operator d, have simple matrix forms: M S = = I 3, M d = Two dual operators cancel out; when applied to other operators, the dual operator corresponds to horizontal and vertical reflections of the matrix. Operators can be grouped into groups of four by identifying the operators x, xd, dx, dxd. In this article, only the matrix for x is given. Hart introduced the reflection operator r.
This is not a LOPSP, since it does not preserve orientation. R has no effect on achiral seeds, rr returns the original seed. An overline can be used to indicate the other chiral form of an operator. R does not affect the matrix. An operation is irreducible if it cannot be expressed as a composition of operators aside from d and r; the majority of Conway's original operators are irreducible: the exceptions are e, b, o, m. Some open questions about Conway
Quilting is the process of sewing two or more layers of fabric together to make a thicker padded material to create a quilt or quilted garment. Quilting is done with three layers: the top fabric or quilt top, batting or insulating material and backing material, but many different styles are adopted; the process of quilting uses a needle and thread to join two or more layers of material to make a quilt. The quilter's hand or sewing machine passes the needle and thread through all layers and brings the needle back up; the process is repeated across the entire area. Rocking, straight or running stitches are used with these stitches being purely functional or decorative. Quilting is done to create bed spreads, art quilt wall hangings, a variety of textile products. Quilting can make a project thick, or with dense quilting, can raise one area so that another stands out; the whole process of creating a quilt or quilted garment involves other steps such as designing, appliqué, binding. A person who works at quilting is termed a quilter.
Quilting can be done by a specialized longarm quilting system. Quilt stores sell fabric, thread and other goods that are used for quilting, they have group sewing and quilting classes where one can learn how to sew or quilt. The origins of quilting remain unknown but sewing techniques of piecing, appliqué, quilting have been used for clothing and furnishings in diverse parts of the world for several millennia; the earliest known quilted garment is depicted on the carved ivory figure of a Pharaoh dating from the ancient Egyptian First Dynasty. In 1924 archaeologists discovered a quilted floor covering in Mongolia, estimated to date between 100 BC and 200 AD. In Europe, quilting has been part of the needlework tradition from about the fifth century, with early objects containing Egyptian cotton, which may indicate that Egyptian and Mediterranean trade provided a conduit for the technique. However, quilted objects were rare in Europe until the twelfth century, when quilted bedding and other items appeared after the return of the Crusaders from the Middle East.
The medieval quilted gambeson and arming doublet were garments worn under or instead of armor of maille or plate armor. These developed into the quilted doublet worn as part of fashionable European male clothing from the fourteenth to seventeenth century; the earliest known surviving European bed quilt is from late-fourteenth-century Sicily: the Tristan quilt made of linen and padded with wool. The blocks across the center are scenes from the legend of Tristan; the quilt is in the Victoria and Albert Museum in London. The word quilt comes from the Latin culcita meaning a stuffed sack, but it came into the English language from the French word cuilte. In American Colonial times, quilts were predominantly whole-cloth quilts–a single piece of fabric layered with batting and backing held together with fine needlework quilting. Broderie perse quilts were popular during this time and the majority of pierced or appliqued quilts made during the 1170-1800 period were medallion-style quilts ). Patchwork quilting in America dates to the 1770s, the decade the United States gained its independence from England.
These late-eighteenth- and nineteenth-century patchwork quilts mixed wool, silk and cotton in the same piece, as well as mixing large-scale and small-scale patterns. Some antique quilts made in North America have worn-out blankets or older quilts as the internal batting layer, quilted between new layers of fabric and thereby extending the usefulness of old material. During American pioneer days, foundation piecing became popular. Paper was used as a pattern. Paper was a scarce commodity in the early American west so women would save letters from home, newspaper clippings, catalogs to use as patterns; the paper not only served as a pattern but as an insulator. The paper found between the old quilts has become a primary source of information about pioneer life. Quilts made without any insulation or batting were referred to as summer quilts, they were not made for warmth. There is a long tradition of African-American quilting beginning with quilts made by slaves, both for themselves and for their owners.
The style of these quilts was determined by time period and region, rather than race, the documented slave-made quilts resemble those made by white women in their region. After 1865 and the end of slavery in the United States, African-Americans began to develop their own distinctive style of quilting. Harriet Powers, a slave-born African American woman, made two famous story quilts, she was just one of the many African-American quilters. The first nationwide recognition of African-American quilt-making came when the Gee's Bend quilting community was celebrated in an exhibition that opened in 2002 and traveled to many museums, including the Smithsonian. Gee's Bend is a small, isolated community of African-Americans in southern Alabama with a quilt-making tradition that goes back several generations and is characterized by pattern improvisation, multiple patterning and contrasting colors, visual motion, a lack of rules; the contributions made by Harriet Powers and others quilters of Gee's Bend, Alabama have been recognized by the US Postal Service with a series of stamps.
The communal nature of the quilting process (and how it c
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, the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them; the symmetry group of an object is sometimes called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral; the point groups in three dimensions are used in chemistry to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, in this context they are called molecular point groups.
Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group is represented by a Coxeter -- Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E +, which consists of i.e. isometries preserving orientation. O is the direct product of SO and the group generated by inversion: O = SO × Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. There is a 1-to-1 correspondence between all groups of direct isometries H in O and all groups K of isometries in O that contain inversion: K = H × H = K ∩ SOFor instance, if H is C2 K is C2h, or if H is C3 K is S6. If a group of direct isometries H has a subgroup L of index 2 apart from the corresponding group containing inversion there is a corresponding group that contains indirect isometries but no inversion: M = L ∪ where isometry is identified with A.
An example would be C4 for H and S4 for M. Thus M is obtained from H by inverting the isometries in H ∖ L; this group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries; this is clarifying when see below. In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations is normal both in the group obtained by adding reflections in planes through the axis and in the group obtained by adding a reflection plane perpendicular to the axis; the isometries of R3 that leave the origin fixed, forming the group O, can be categorized as follows: SO: identity rotation about an axis through the origin by an angle not equal to 180° rotation about an axis through the origin by an angle of 180° the same with inversion, i.e. respectively: inversion rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis reflection in a plane through the originThe 4th and 5th in particular, in a wider sense the 6th are called improper rotations.
See the similar overview including translations. When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O. For example, two 3D objects have the same symmetry type: if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis. In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second; the conjugacy definition would allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. There are many infinite isometry groups.
We may create non-cyclical abelian groups by adding more rotations around the same axis. There are non-abelian groups generated by rotations around different axes; these are free groups. They will be infinite. All the infinite groups mentioned so far are not closed as topological subgroups of O. We now discuss
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
United States Postal Service
The United States Postal Service is an independent agency of the executive branch of the United States federal government responsible for providing postal service in the United States, including its insular areas and associated states. It is one of the few government agencies explicitly authorized by the United States Constitution; the U. S. Mail traces its roots to 1775 during the Second Continental Congress, when Benjamin Franklin was appointed the first postmaster general; the Post Office Department was created in 1792 from Franklin's operation. It was elevated to a cabinet-level department in 1872, was transformed by the Postal Reorganization Act in 1970 into the USPS as an independent agency; the USPS as of 2017 has 644,124 active employees and operated 211,264 vehicles in 2014. The USPS is the operator of the largest civilian vehicle fleet in the world; the USPS is obligated to serve all Americans, regardless of geography, at uniform price and quality. The USPS has exclusive access to letter boxes marked "U.
S. Mail" and personal letterboxes in the United States, but now has to compete against private package delivery services, such as United Parcel Service and FedEx. Since the early 1980s, many of the direct tax subsidies to the Post Office, with the exception of subsidies for costs associated with the disabled and overseas voters, have been reduced or eliminated in favor of indirect subsidies, in addition to the advantages associated with a government-enforced monopoly on the delivery of first-class mail. Since the 2006 all-time peak mail volume, after which Congress passed the Postal Accountability and Enhancement Act which mandated that $5.5 billion per year be paid to prefund employee retirement health benefits, revenue dropped due to recession-influenced declining mail volume, prompting the postal service to look to other sources of revenue while cutting costs to reduce its budget deficit. In the early years of the North American colonies, many attempts were made to initiate a postal service.
These early attempts were of small scale and involved a colony, Massachusetts Bay Colony for example, setting up a location in Boston where one could post a letter back home to England. Other attempts focused on a dedicated postal service between two of the larger colonies, such as Massachusetts and Virginia, but the available services remained limited in scope and disjointed for many years. For example, informal independently-run postal routes operated in Boston as early as 1639, with a Boston to New York City service starting in 1672. A central postal organization came to the colonies in 1691, when Thomas Neale received a 21-year grant from the British Crown for a North American Postal Service. On February 17, 1691, a grant of letters patent from the joint sovereigns, William III and Mary II, empowered him: to erect and establish within the chief parts of their majesties' colonies and plantations in America, an office or offices for receiving and dispatching letters and pacquets, to receive and deliver the same under such rates and sums of money as the planters shall agree to give, to hold and enjoy the same for the term of twenty-one years.
The patent included the exclusive right to establish and collect a formal postal tax on official documents of all kinds. The tax was repealed a year later. Neale appointed Governor of New Jersey, as his deputy postmaster; the first postal service in America commenced in February 1692. Rates of postage were fixed and authorized, measures were taken to establish a post office in each town in Virginia. Massachusetts and the other colonies soon passed postal laws, a imperfect post office system was established. Neale's patent expired in 1710; the chief office was established in New York City, where letters were conveyed by regular packets across the Atlantic. Before the Revolution, there was only a trickle of business or governmental correspondence between the colonies. Most of the mail went forth to counting houses and government offices in London; the revolution made Philadelphia, the seat of the Continental Congress, the information hub of the new nation. News, new laws, political intelligence, military orders circulated with a new urgency, a postal system was necessary.
Journalists took the lead, securing post office legislation that allowed them to reach their subscribers at low cost, to exchange news from newspapers between the thirteen states. Overthrowing the London-oriented imperial postal service in 1774–1775, printers enlisted merchants and the new political leadership, created a new postal system; the United States Post Office was created on July 26, 1775, by decree of the Second Continental Congress. Benjamin Franklin headed it briefly. Before the Revolution, individuals like Benjamin Franklin and William Goddard were the colonial postmasters who managed the mails and were the general architects of a postal system that started out as an alternative to the Crown Post; the official post office was created in 1792 as the Post Office Department. It was based on the Constitutional authority empowering Congress "To establish post offices and post roads"; the 1792 law provided for a expanded postal network, served editors by charging newspapers an low rate.
The law guaranteed the sanctity of personal correspondence, provided the entire country with low-cost access to information on public affairs, while establishing a right to personal privacy. Rufus Easton was appointed by Thomas Jefferson first postmaster of St. Louis under the recommendation of Postmaster General Gideon Granger. Rufus Easton was the first postmaster and built the first post office west o
Brazil the Federative Republic of Brazil, is the largest country in both South America and Latin America. At 8.5 million square kilometers and with over 208 million people, Brazil is the world's fifth-largest country by area and the fifth most populous. Its capital is Brasília, its most populated city is São Paulo; the federation is composed of the union of the 26 states, the Federal District, the 5,570 municipalities. It is the largest country to have Portuguese as an official language and the only one in the Americas. Bounded by the Atlantic Ocean on the east, Brazil has a coastline of 7,491 kilometers, it borders all other South American countries except Ecuador and Chile and covers 47.3% of the continent's land area. Its Amazon River basin includes a vast tropical forest, home to diverse wildlife, a variety of ecological systems, extensive natural resources spanning numerous protected habitats; this unique environmental heritage makes Brazil one of 17 megadiverse countries, is the subject of significant global interest and debate regarding deforestation and environmental protection.
Brazil was inhabited by numerous tribal nations prior to the landing in 1500 of explorer Pedro Álvares Cabral, who claimed the area for the Portuguese Empire. Brazil remained a Portuguese colony until 1808, when the capital of the empire was transferred from Lisbon to Rio de Janeiro. In 1815, the colony was elevated to the rank of kingdom upon the formation of the United Kingdom of Portugal and the Algarves. Independence was achieved in 1822 with the creation of the Empire of Brazil, a unitary state governed under a constitutional monarchy and a parliamentary system; the ratification of the first constitution in 1824 led to the formation of a bicameral legislature, now called the National Congress. The country became a presidential republic in 1889 following a military coup d'état. An authoritarian military junta came to power in 1964 and ruled until 1985, after which civilian governance resumed. Brazil's current constitution, formulated in 1988, defines it as a democratic federal republic. Due to its rich culture and history, the country ranks thirteenth in the world by number of UNESCO World Heritage Sites.
Brazil is considered an advanced emerging economy. It has the ninth largest GDP in the world by nominal, eight and PPP measures, it is one of the world's major breadbaskets, being the largest producer of coffee for the last 150 years. It is classified as an upper-middle income economy by the World Bank and a newly industrialized country, with the largest share of global wealth in Latin America. Brazil is a regional power and sometimes considered a great or a middle power in international affairs. On account of its international recognition and influence, the country is subsequently classified as an emerging power and a potential superpower by several analysts. Brazil is a founding member of the United Nations, the G20, BRICS, Union of South American Nations, Organization of American States, Organization of Ibero-American States and the Community of Portuguese Language Countries, it is that the word "Brazil" comes from the Portuguese word for brazilwood, a tree that once grew plentifully along the Brazilian coast.
In Portuguese, brazilwood is called pau-brasil, with the word brasil given the etymology "red like an ember", formed from brasa and the suffix -il. As brazilwood produces a deep red dye, it was valued by the European textile industry and was the earliest commercially exploited product from Brazil. Throughout the 16th century, massive amounts of brazilwood were harvested by indigenous peoples along the Brazilian coast, who sold the timber to European traders in return for assorted European consumer goods; the official Portuguese name of the land, in original Portuguese records, was the "Land of the Holy Cross", but European sailors and merchants called it the "Land of Brazil" because of the brazilwood trade. The popular appellation eclipsed and supplanted the official Portuguese name; some early sailors called it the "Land of Parrots". In the Guarani language, an official language of Paraguay, Brazil is called "Pindorama"; this was the name the indigenous population gave to the region, meaning "land of the palm trees".
Some of the earliest human remains found in the Americas, Luzia Woman, were found in the area of Pedro Leopoldo, Minas Gerais and provide evidence of human habitation going back at least 11,000 years. The earliest pottery found in the Western Hemisphere was excavated in the Amazon basin of Brazil and radiocarbon dated to 8,000 years ago; the pottery was found near Santarém and provides evidence that the tropical forest region supported a complex prehistoric culture. The Marajoara culture flourished on Marajó in the Amazon delta from 800 CE to 1400 CE, developing sophisticated pottery, social stratification, large populations, mound building, complex social formations such as chiefdoms. Around the time of the Portuguese arrival, the territory of current day Brazil had an estimated indigenous population of 7 million people semi-nomadic who subsisted on hunting, fishing and migrant agriculture; the indigenous population of Brazil comprised several large indigenous ethnic groups. The Tupí people were subdivided into the Tupiniquins and Tupinambás, there were many subdivisions of the other gro
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two sides of equal length, sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, the faces of bipyramids and certain Catalan solids; the mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from earlier times, appear in architecture and design, for instance in the pediments and gables of buildings; the two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height and perimeter, can be calculated by simple formulas from the lengths of the legs and base; every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base.
The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs. Euclid defined an isosceles triangle as a triangle with two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides; the difference between these two definitions is that the modern version makes equilateral triangles a special case of isosceles triangles. A triangle, not isosceles is called scalene. "Isosceles" is a compound word, made from the Greek roots "isos" and "skelos". The same word is used, for instance, for isosceles trapezoids, trapezoids with two equal sides, for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has two equal sides, the equal sides are called legs and the third side is called the base; the angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles.
The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle is acute, right or obtuse depends only on the angle at its apex. In Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right an isosceles triangle is obtuse, right or acute if and only if its apex angle is obtuse, right or acute. In Edwin Abbott's book Flatland, this classification of shapes was used as a satire of social hierarchy: isosceles triangles represented the working class, with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles; as well as the isosceles right triangle, several other specific shapes of isosceles triangles have been studied. These include the Calabi triangle, the golden triangle and golden gnomon, the 80-80-20 triangle appearing in the Langley’s Adventitious Angles puzzle, the 30-30-120 triangle of the triakis triangular tiling.
Five Catalan solids, the triakis tetrahedron, triakis octahedron, tetrakis hexahedron, pentakis dodecahedron, triakis icosahedron, each have isosceles-triangle faces, as do infinitely many pyramids and bipyramids. For any isosceles triangle, the following six line segments coincide: the altitude, a line segment from the apex perpendicular to the base, the angle bisector from the apex to the base, the median from the apex to the midpoint of the base, the perpendicular bisector of the base within the triangle, the segment within the triangle of the unique axis of symmetry of the triangle, the segment within the triangle of the Euler line of the triangle, their common length is the height h of the triangle. If the triangle has equal sides of length a and base of length b, the general triangle formulas for the lengths of these segments all simplify to h = 1 2 4 a 2 − b 2; this formula can be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles.
The Euler line of any triangle goes through the triangle's orthocenter, its centroid, its circumcenter. In an isosceles triangle with two equal sides, these three points are distinct, all lie on the symmetry axis of the triangle, from which it follows that the Euler line coincides with the axis of symmetry; the incenter of the triangle lies on the Euler line, something, not true for other triangles. If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles; the area T of an isosceles triangle can be derived from the formula for its height, from the general formula for the area of a triangle as half the product of base and height: T =