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In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere (), the frieze groups and wallpaper groups of the Euclidean plane (), and their analogues on the hyperbolic plane (). == Definition of the notation == The following types of Euclidean transformation can occur in a group described by orbifold notation: * reflection through a line (or plane) * translation by a vector * rotation of finite order around a point * infinite rotation around a line in 3-space * glide-reflection, i.e. reflection followed by translation. All translations which occur are assumed to form a discrete subgroup of the group symmetries being described. Each group is denoted in orbifold notation by a finite string made up from the following symbols: * positive ''integers'' * the ''infinity'' symbol, * the ''asterisk'', * * the symbol ''o'' (a solid circle in older documents), which is called a ''wonder'' and also a ''handle'' because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation. * the symbol (an open circle in older documents), which is called a ''miracle'' and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations. Each symbol corresponds to a distinct transformation: * an integer ''n'' to the left of an asterisk indicates a rotation of order ''n'' around a gyration point * an integer ''n'' to the right of an asterisk indicates a transformation of order 2''n'' which rotates around a kaleidoscopic point and reflects through a line (or plane) * an indicates a glide reflection * the symbol indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way. * the exceptional symbol ''o'' indicates that there are precisely two linearly independent translations. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「orbifold notation」の詳細全文を読む スポンサード リンク
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