Symmetry in the context of Equivariant map

Lyrics are words that make up a song, usually consisting of verses and choruses. The writer of lyrics is a lyricist. The words to an extended musical composition such as an opera are, however, usually known as a "libretto" and their writer, as a "librettist". Rap songs and grime contain rap lyrics (often with a variation of rhyming words) that are meant to be spoken rhythmically rather than sung. The meaning of lyrics can either be explicit or implicit. Some lyrics are abstract, almost unintelligible, and, in such cases, their explication emphasizes form, articulation, meter, and symmetry of expression.

Rhythm (from Greek ῥυθμός, rhythmos, "any regular recurring motion, symmetry") generally means a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recurrence or pattern in time can apply to a wide variety of cyclical natural phenomena having a periodicity or frequency of anything from microseconds to several seconds (as with the riff in a rock music song); to several minutes or hours, or, at the most extreme, even over many years.

The Oxford English Dictionary defines rhythm as "The measured flow of words or phrases in verse, forming various patterns of sound as determined by the relation of long and short or stressed and unstressed syllables in a metrical foot or line; an instance of this".

A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable and logical manner. There exists countless kinds of unclassified patterns, present in everyday nature, fashion, many artistic areas, as well as a connection with mathematics. A geometric pattern is a type of pattern formed of repeating geometric shapes and typically repeated like a wallpaper design.

Any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, rarely exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world.

Anhydrite, or anhydrous calcium sulfate, is a mineral with the chemical formula CaSO₄. It is in the orthorhombic crystal system, with three directions of perfect cleavage parallel to the three planes of symmetry. It is not isomorphous with the orthorhombic barium (baryte) and strontium (celestine) sulfates, as might be expected from the chemical formulas. Distinctly developed crystals are somewhat rare, the mineral usually presenting the form of cleavage masses. The Mohs hardness is 3.5, and the specific gravity is 2.9. The color is white, sometimes greyish, bluish, or purple. On the best developed of the three cleavages, the lustre is pearly; on other surfaces it is glassy. When exposed to water, anhydrite readily transforms to the more commonly occurring gypsum, (CaSO₄·2H₂O) by the absorption of water. This transformation is reversible, with gypsum or calcium sulfate hemihydrate forming anhydrite by heating to around 200 °C (400 °F) under normal atmospheric conditions. Anhydrite is commonly associated with calcite, halite, and sulfides such as galena, chalcopyrite, molybdenite, and pyrite in vein deposits.

In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-)  'many' and ἕδρον (-hedron)  'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term polyhedron is often used to refer implicitly to the whole structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices.

There are many definitions of polyhedra, not all of which are equivalent. Under any definition, polyhedra are typically understood to generalize two-dimensional polygons and to be the three-dimensional specialization of polytopes (a more general concept in any number of dimensions). Polyhedra have several general characteristics that include the number of faces, topological classification by Euler characteristic, duality, vertex figures, surface area, volume, interior lines, Dehn invariant, and symmetry. A symmetry of a polyhedron means that the polyhedron's appearance is unchanged by the transformation such as rotating and reflecting.

In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.

In two-dimensional space, there is a line/axis of symmetry, in three-dimensional space, there is a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric.

Axial symmetry is symmetry around an axis or line (geometry). An object is said to be axially symmetric if its appearance is unchanged if transformed around an axis. The main types of axial symmetry are reflection symmetry and rotational symmetry (including circular symmetry for plane figures and cylindrical symmetry for surfaces of revolution). For example, a baseball bat (without trademark or other design), or a plain white tea saucer, looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially symmetric.

Axial symmetry can also be discrete with a fixed angle of rotation, 360°/n for n-fold symmetry.

Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. The Hungarian biologist Aristid Lindenmayer and the French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.

In geometry, an apeirogon (from Ancient Greek ἄπειροv apeiron 'infinite, boundless' and γωνία gonia 'angle') or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal (from Greek τόξον  'arc') 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.

In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.

Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope acts transitively on its vertices, or that the vertices lie within a single symmetry orbit.

Obsessive–compulsive disorder (OCD) is a mental disorder in which an individual has intrusive thoughts (an obsession) and feels the need to perform certain behaviors (compulsions) repeatedly to relieve the distress caused by the obsession, to the extent where it impairs general function.

Obsessions are persistent unwanted thoughts, mental images, or urges that generate feelings of anxiety, disgust, or discomfort. Some common obsessions include fear of contamination, obsession with symmetry, the fear of acting blasphemously, sexual obsessions, and the fear of possibly harming others or themselves. Compulsions are repetitive actions performed in response to obsessions to reduce anxiety, such as washing, checking, counting, reassurance seeking, and situational avoidance.

In geometry and other fields, asymmetry is an absence or violation of symmetry in an object or process, such that some transformation (such as reflection in space) results in an observable difference. Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms. The absence of or violation of symmetry that are either expected or desired can have important consequences for a system.