"the Lemoine hexagon is known as the first Lemoine circle. Lemoine hexagon In geometry, the Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. There are two definitions of the hexagon that differ based on the order in which the vertices are connected. The Lemoine hexagon can be drawn defined in two ways, first as a simple hexagon with vertices at the intersections as defined before. The second is a self-intersecting hexagon with the"
"Hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. A ""regular hexadecagon"" is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}. As 16 = 2 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any"
"Hectogon In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of any hectogon's interior angles is 17640 degrees. A ""regular hectogon"" is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}. One interior angle in a regular hectogon is 176°, meaning that one exterior angle would be 3°. The area of a regular hectogon is (with ) and its inradius is The circumradius of a regular hectogon is Because 100 = 2 × 5, the number of sides contains a ""repeated"" Fermat prime (the number"
"Heptagon In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ""sept-"" (an elision of """", a Latin-derived numerical prefix, rather than """", a Greek-derived numerical prefix) together with the Greek suffix ""-agon"" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}. The area (""A"") of a regular heptagon of side length ""a"" is given by: This can be seen by subdividing the unit-sided heptagon into seven triangular ""pie slices"" with vertices"
"is quasiregular, which alternates 2-edges and 6-edges. Hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or ""t""{3,6} (as a truncated triangular tiling). English mathematician John Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. The hexagonal tiling is the"
"Hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or ""t""{3,6} (as a truncated triangular tiling). English mathematician John Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. The hexagonal tiling is the densest way to arrange circles in two"
"and {100/49}, as well as 30 regular star figures with the same vertex configuration. Hectogon In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of any hectogon's interior angles is 17640 degrees. A ""regular hectogon"" is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}. One interior angle in a regular hectogon is 176°, meaning that one exterior angle would be 3°. The area of a regular hectogon is (with ) and its inradius is The circumradius of a regular hectogon is Because 100 ="
"Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has eight faces, it is an octahedron. However, the term ""octahedron"" is primarily used to refer to the ""regular octahedron"", which has eight triangular faces. Because of the ambiguity of the term ""octahedron"" and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long hexagonal prism. If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a"
"For ""p"" > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has eight faces, it is an octahedron. However, the term ""octahedron"" is primarily used to refer to the ""regular octahedron"", which has eight triangular faces. Because of the ambiguity of the term ""octahedron"" and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long"
"Hexacontagon In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon. The sum of any hexacontagon's interior angles is 10440 degrees. A ""regular hexacontagon"" is represented by Schläfli symbol {60} and also can be constructed as a truncated triacontagon, t{30}, or a twice-truncated pentadecagon, tt{15}. A truncated hexacontagon, t{60}, is a 120-gon, {120}. One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°. The area of a regular hexacontagon is (with ) and its inradius is The circumradius of a regular hexacontagon is This means that the trigonometric functions of π/60"
"based on a Petrie polygon projection of a 8-cube, with 28 of 1792 faces. The list enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection. A skew hexadecagon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an hexadecagon is not generally defined. A ""skew zig-zag hexadecagon"" has vertices alternating between two parallel planes. A regular skew hexadecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and"
"Heptahedron A heptahedron (plural: heptahedra) is a polyhedron having seven sides, or faces. A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven faces form a rudimentary projective plane. No heptahedra are regular. There are 34 topologically distinct ""convex"" heptahedra, excluding mirror images. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles"
"a sequence of expanded hypercubes: Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers"
"Flexagon In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be ""flexed"" or folded in certain ways to reveal faces besides the two that were originally on the back and front. Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon. In hexaflexagon theory (that is, concerning flexagons with"
"regular skew polygon around the perimeter is called the petrie polygon. Heptagon In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ""sept-"" (an elision of """", a Latin-derived numerical prefix, rather than """", a Greek-derived numerical prefix) together with the Greek suffix ""-agon"" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}. The area (""A"") of a regular heptagon of side length ""a"" is given by: This can be seen by"
"Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a"
"Hessian polyhedron In geometry, the Hessian polyhedron is a regular complex polyhedron {3}{3}, , in formula_1. It has 27 vertices, 72 {} edges, and 27 {3} faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the ""Hessian configuration"" formula_2 or (912), 9 points lying by threes on twelve lines, with four lines through each point. Its complex reflection group is [3][3] or , order 648, also called a Hessian group. It has 27 copies of , order 24, at each vertex. It has 24 order-3 reflections. Its Coxeter number is 12, with degrees of the fundamental"
"Hendecagon In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name ""hendecagon"", from Greek ""hendeka"" ""eleven"" and ""gon–"" ""corner"", is often preferred to the hybrid ""undecagon"", whose first part is formed from Latin ""undecim"" ""eleven"".) A ""regular hendecagon"" is represented by Schläfli symbol {11}. A regular hendecagon has internal angles of 147.27 degrees (=147 formula_1 degrees). The area of a regular hendecagon with side length ""a"" is given by As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction"
"Hexadecahedron A hexadecahedron (or hexakaidecahedron) is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism. There are 387,591,510,244 topologically distinct ""convex"" hexadecahedra, excluding mirror images, having at least 10 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) There are 302,404 self-dual hexadecahedron, 1476"
"triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version. While the most commonly seen hexaflexagons have either three or six faces, variations exist with four, five, seven, twelve, twenty-four, and forty-eight faces. In these more recently discovered flexagons, each square or equilateral triangular face of a conventional flexagon is further divided into two right triangles, permitting additional flexing modes. The division of the square faces of tetraflexagons into right isosceles triangles yields the octaflexagons, and the division of the triangular faces"
"Hexomino A hexomino (or 6-omino) is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different ""free"" hexominoes. When reflections are considered distinct, there are 60 ""one-sided"" hexominoes. When rotations are also considered distinct, there are 216 ""fixed"" hexominoes. The figure shows all possible free hexominoes, coloured according to their symmetry groups: If reflections of a hexomino are considered distinct, as they are"
"Enneacontahexagon In geometry, an enneacontahexagon (or enneacontakaihexagon) or 96-gon is a ninety-six-sided polygon. The sum of any enneacontahexagons's interior angles is 16920 degrees. The ""regular enneacontahexagon"" is represented by Schläfli symbol {96} and can also be constructed as a truncated tetracontaoctagon, t{48}, or a twice-truncated icositetragon, tt{24}, or a thrice-truncated dodecagon, ttt{12}, or a fourfold-truncated hexagon, tttt{6}, or a fivefold-truncated triangle, }. One interior angle in a regular enneacontahexagon is 176°, meaning that one exterior angle would be 3°. The area of a regular enneacontahexagon is: (with ""t"" = edge length) The enneacontahexagon appeared in Archimedes' polygon approximation of pi,"
"Truncated hexagonal tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of ""t""{6,3}. Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille). There are 3 regular and 8 semiregular tilings in the plane. There is"
"between edges or faces.) An example of each type is depicted below, along with the number of sides on each of the faces. The images are ordered by descending number of six-sided faces (if any), followed by descending number of five-sided faces (if any), and so on. Heptahedron A heptahedron (plural: heptahedra) is a polyhedron having seven sides, or faces. A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven faces form a rudimentary projective plane. No heptahedra are"
"{96/37}, {96/41}, {96/43}, and {96/47}, as well as 32 compound star figures with the same vertex configuration. Enneacontahexagon In geometry, an enneacontahexagon (or enneacontakaihexagon) or 96-gon is a ninety-six-sided polygon. The sum of any enneacontahexagons's interior angles is 16920 degrees. The ""regular enneacontahexagon"" is represented by Schläfli symbol {96} and can also be constructed as a truncated tetracontaoctagon, t{48}, or a twice-truncated icositetragon, tt{24}, or a thrice-truncated dodecagon, ttt{12}, or a fourfold-truncated hexagon, tttt{6}, or a fivefold-truncated triangle, }. One interior angle in a regular enneacontahexagon is 176°, meaning that one exterior angle would be 3°. The area of a"
"Hexacontatetragon In geometry, a hexacontatetragon (or hexacontakaitetragon) or 64-gon is a sixty-four-sided polygon. (In Greek, the prefix hexaconta- means 60 and tetra- means 4.) The sum of any hexacontatetragon's interior angles is 11160 degrees. The ""regular hexacontatetragon"" can be constructed as a truncated triacontadigon, t{32}, a twice-truncated hexadecagon, tt{16}, a thrice-truncated octagon, ttt{8}, a fourfold-truncated square, tttt{4}, and a fivefold-truncated digon, }. One interior angle in a regular hexacontatetragon is 174°, meaning that one exterior angle would be 5°. The area of a regular hexacontatetragon is (with ) and its inradius is The circumradius of a regular hexacontatetragon is Since"
"are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal hexecontahedron. The faces are irregular pentagons with two long edges and three short edges. The ratio of edge lengths is about 1:1.7498525667362. The faces have four obtuse angles of about 118.13662 degrees each, and one acute angle (between the two long edges) of about 67.45351 degrees. The dihedral angle is about 153.178732558 degrees between all faces. Note that the face"
"of the Virgin"" has 16 sides, elaborating on an eight-sided tower in a previous painting by Pietro Perugino. Hexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra. In the Philippines, in local carnivals (peryahan), Ferris Wheels with maximum of 16 seats or gondolas An octagonal star can be seen as a concave hexadecagon: Hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. A ""regular hexadecagon"" is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a"
"as 44 compound star figures with the same vertex configuration. 120-gon In geometry, a 120-gon is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees. Alternative names include dodecacontagon and hecatonicosagon. A ""regular 120-gon"" is represented by Schläfli symbol {120} and also can be constructed as a truncated hexacontagon, t{60}, or a twice-truncated triacontagon, tt{30}, or a thrice-truncated pentadecagon, ttt{15}. One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°. The area of a regular 120-gon is (with ) and its inradius is The circumradius of a"
"hexagonal symmetry in the plane. The K complete graph is often drawn as a ""regular enneagon"" with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex. They Might Be Giants have a song entitled ""Nonagon"" on their children's album ""Here Come the 123s"". It refers to both an attendee at a party at which ""everybody in the party is a many-sided polygon"" and a dance they perform at this party. Slipknot's logo is also a version of a nonagon, being a nine-pointed star made of three triangles. King"
"is one, and the winding number around the central heptagon is two, so adding these two numbers together, each point of the plane is surrounded three times; that is, the density of the tiling is 3. In the Euclidean plane, a heptagram of type {7/2} would have angles of 3/7 at its vertices, but in the hyperbolic plane heptagrams can have the sharper vertex angle 2/7 that is needed to make exactly seven other heptagrams meet up at the center of each heptagram of the tiling. It has the same vertex arrangement as the regular order-7 triangular tiling, {3,7}. The"
"""r = 1 m"", the absolute error of the 1st side would be ""approximately -1.7 mm"" The ""regular heptagon"" belongs to the D point group (Schoenflies notation), order 28. The symmetry elements are: a 7-fold proper rotation axis C, a 7-fold improper rotation axis,S, 7 vertical mirror planes, σ, 7 2-fold rotation axes, C, in the plane of the heptagon and a horizontal mirror plane, σ, also in the heptagon's plane. The regular heptagon's side ""a"", shorter diagonal ""b"", and longer diagonal ""c"", with ""a""<""b""<""c"", satisfy and hence and Thus –""b""/""c"", ""c""/""a"", and ""a""/""b"" all satisfy the cubic equation formula_19"
"Hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and because its faces cannot be equilateral triangles. It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The hexagonal"