"of Manchester's The Stone Roses. ""How Long's a Tear Take to Dry?"" reached number 12 in the UK Singles Chart in March 1999. Although not released on vinyl, it was given a dual-CD release in the UK. B-sides included a remix of ""How Long's a Tear Take to Dry?"" as well as acoustic versions of three other songs: ""Perfect 10"", ""Big Coin"", and ""Rotterdam"". On 18 March 1999 the band performed ""How Long's a Tear Take to Dry?"" live on the BBC's music programme, ""Top of the Pops"". The video, available on The Beautiful South's compilation DVD ""Munch"", is a"
"Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. This polyhedron can be formed from a dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb. The area ""A"" and the volume ""V"" of a truncated dodecahedron of edge length ""a"" are: Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2""φ"" − 2, centered at the origin, are"
"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"
"Chamfered dodecahedron The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron. It is also called a truncated rhombic triacontahedron, constructed as a truncation of the rhombic triacontahedron. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated. These 12 order-5 vertices can be truncated"
"of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons. In its flat state, the pentaflexagon looks much like the Chrysler logo: a regular pentagon divided from the center into five isosceles triangles, with angles 72-54-54. Because of its fivefold symmetry, the pentaflexagon cannot be folded in half. However, a complex series of flexes results in its transformation from displaying sides one and two on the front and back, to displaying its previously hidden sides three and four. By further dividing the 72-54-54 triangles of the pentaflexagon into 36-54-90 right triangles produces one variation of the 10-sided decaflexagon. The pentaflexagon"
"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"
"Tetrated dodecahedron The tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003 and named by Robert Austin. It has 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry. Topologically, as a near-miss"
"Dodecagram A dodecagram is a star polygon that has 12 vertices. There is one regular form: {12/5}. A regular dodecagram has the same vertex arrangement as a regular dodecagon, which may be regarded as {12/1}. The name ""dodecagram"" combines the numeral prefix ""dodeca-"" with the Greek suffix ""-gram"". The ""-gram"" suffix derives from ""γραμμῆς"" (""grammēs""), which denotes a line. A regular dodecagram can be seen as a quasitruncated hexagon, t{6/5}={12/5}. Other isogonal (vertex-transitive) variations with equally spaced vertices can be constructed with two edge lengths. There are four regular dodecagram star figures: {12/2}=2{6}, {12/3}=3{4}, {12/4}=4{3}, and {12/6}=6{2}. The first is"
"also be incorporated into star tessellations of the Euclidean plane. Dodecagram A dodecagram is a star polygon that has 12 vertices. There is one regular form: {12/5}. A regular dodecagram has the same vertex arrangement as a regular dodecagon, which may be regarded as {12/1}. The name ""dodecagram"" combines the numeral prefix ""dodeca-"" with the Greek suffix ""-gram"". The ""-gram"" suffix derives from ""γραμμῆς"" (""grammēs""), which denotes a line. A regular dodecagram can be seen as a quasitruncated hexagon, t{6/5}={12/5}. Other isogonal (vertex-transitive) variations with equally spaced vertices can be constructed with two edge lengths. There are four regular dodecagram"
"Rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long diagonal of each face is exactly times the length of the short diagonal, so that the acute angles on each face measure arccos(), or approximately 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on"
"Tetrahedrally diminished dodecahedron In geometry, a tetrahedrally diminished dodecahedron (also tetrahedrally stellated icosahedron or propello tetrahedron) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals). It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated, or from the pyritohedron, with 4 vertices diminished. Within its tetrahedral symmetry, it has geometric varied proportions. By Dorman Luke dual construction, a unique geometric proportion can be defined. The kite faces have edges of length ratio ~ 1:0.6325. Topologically, the"
"Truncated dodecadodecahedron In geometry, the truncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U. It is given a Schläfli symbol t{5/3,5}. It has 120 vertices and 54 faces: 30 squares, 12 decagons, and 12 decagrams. The central region of the polyhedron is connected to the exterior via 20 small triangular holes. The name ""truncated dodecadodecahedron"" is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by . For this reason,"
"Late Latin ""polygōnum"" (a noun), from Greek πολύγωνον (""polygōnon/polugōnon""), noun use of neuter of πολύγωνος (""polygōnos/polugōnos"", the masculine adjective), meaning ""many-angled"". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix ""-gon"", e.g. ""pentagon"", ""dodecagon"". The triangle, quadrilateral and nonagon are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon. Exceptions exist for side counts that are more easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for"
"even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,""n"" mirrors at each triangle face vertex. Disdyakis dodecahedron In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It superficially resembles an inflated rhombic dodecahedron—if one replaces"
"Disdyakis dodecahedron In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It superficially resembles an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. It is the net of a rhombic dodecahedral pyramid. It has O octahedral"
"a turn and one more edge. Pentakis snub dodecahedron The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry. Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles. The path between the valence-5 vertices is two"
"Regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}. If the edge length of a regular dodecahedron is ""a"", the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's"
"Octadecagon An octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. A ""regular octadecagon"" has a Schläfli symbol {18} and can be constructed as a quasiregular truncated enneagon, t{9}, which alternates two types of edges. As 18 = 2 × 3, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. The ""regular"
"Great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer. The great dodecahedron can be constructed analogously to the"
"Excavated dodecahedron In geometry, the excavated dodecahedron is a star polyhedron having 60 equilateral triangular faces. Its exterior surface represents the Efg stellation of the icosahedron. It appears in Magnus Wenninger's book ""Polyhedron Models"" as model 28, the ""third stellation of icosahedron"". It has the same external form as a certain facetting of the dodecahedron having 20 self-intersecting hexagons as faces. The non-convex hexagon face can be broken up into four equilateral triangles, three of which are the same size. A true excavated dodecahedron has the three congruent equilateral triangles as true faces of the polyhedron, while the interior equilateral"
"a dodecahedron. Regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}. If the edge length of a regular dodecahedron is ""a"", the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the"
"Decagon In geometry, a decagon is a ten-sided polygon or 10-gon. A ""regular decagon"" has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges. The area of a regular decagon of side length ""a"" is given by: In terms of the apothem ""r"" (see also inscribed figure), the area is: In terms of the circumradius ""R"", the area is: An alternative formula is formula_4 where ""d"" is the distance between parallel"
"Elongated dodecahedron In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism. Along with the rhombic dodecahedron, it is a space-filling polyhedron. This is related to the rhombic dodecahedral honeycomb with an elongation of zero. Projected normal to the elongation direction, the honeycomb looks like a square tiling with the rhombi projected into squares. The expanded"
"Johnson solid, the four triangles corresponding to the face planes of a tetrahedron are always equilateral, while the pentagons and the other triangles only have reflection symmetry. The 12 pentagons and 16 triangles are colored in this net by their locations within the tetrahedral symmetry. Tetrated dodecahedron The tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003 and named by Robert Austin. It has 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs"
"Pentakis snub dodecahedron The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry. Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles. The path between the valence-5 vertices is two edges in a row, and then"
"side edges of a dodecagonal antiprism with the same D, [2,24] symmetry, order 48. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossed-antiprism, s{2,24/7} also have regular skew dodecagons. The regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including: Icositetragon In geometry, an icositetragon (or icosikaitetragon or tetracosagon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees. The ""regular icositetragon"" is represented by Schläfli symbol {24} and can also be constructed as a truncated dodecagon, t{12}, or a twice-truncated hexagon, tt{6}, or thrice-truncated triangle, ttt{3}. One"
"as two hexagons, and {12/3} is reduced to 3{4} as three squares, {12/4} is reduced to 4{3} as four triangles, and {12/6} is reduced to 6{2} as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t{6}={12}. A quasitruncated hexagon, inverted as {6/5}, is a dodecagram: t{6/5}={12/5}. In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines. A cross is a dodecagon, as is the logo for"
"distinct rectangular faces have edges (""a"", ""b""), (""a"", ""c""), and (""b"", ""c""); so the respective face diagonals have lengths formula_1 formula_2 and formula_3 Thus each face diagonal of a cube with side length ""a"" is formula_4. A regular dodecahedron has 60 face diagonals (and 100 space diagonals). Face diagonal In geometry, a face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to a space diagonal passing through the interior of the polyhedron. A cuboid has twelve face diagonals (two on each of the six faces), and it has four space diagonals. The cuboid's face"
"Nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ""nonagon"" is a prefix hybrid formation, from Latin (""nonus"", ""ninth"" + ""gonon""), used equivalently, attested already in the 16th century in French ""nonogone"" and in English from the 17th century. The name ""enneagon"" comes from Greek ""enneagonon"" (εννεα, ""nine"" + γωνον (from γωνία = ""corner"")), and is arguably more correct, though less common than ""nonagon"". A ""regular nonagon"" is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length ""a"" is given by"
"an edge length of one unit has the properties: The ""regular dodecahedron "" has two special orthogonal projections, centered, on vertices and pentagonal faces, correspond to the A and H Coxeter planes. In perspective projection, viewed above a pentagonal face, the regular dodecahedron can be seen as a linear-edged Schlegel diagram, or stereographic projection as a spherical polyhedron. These projections are also used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3-dimensions. The regular dodecahedron can also be represented as a spherical tiling. The following Cartesian coordinates define the 20 vertices"
"Tetradecagon In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. A ""regular tetradecagon"" has Schläfli symbol {14} and can be constructed as a quasiregular truncated heptagon, t{7}, which alternates two types of edges. The area of a regular tetradecagon of side length ""a"" is given by As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis with use of the angle trisector, or with a marked ruler, as it in the following two examples showned. The animation below gives an approximation of about 0.05°"
"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"
"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"