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A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
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 2-dimensional space, there is a line/axis of symmetry, in 3-dimensional space, there is a plane of symmetry
Generalizing from geometrical symmetry in the previous section, one can say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. [15] The set of operations that preserve a given property of the object form a group.
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the n th order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of (180°, 120°, 90°, 72°, 60°, 51 3 ⁄ 7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all ...
The smallest asymmetric regular graphs have ten vertices; there exist 10-vertex asymmetric graphs that are 4-regular and 5-regular. [2] [3] One of the five smallest asymmetric cubic graphs [4] is the twelve-vertex Frucht graph discovered in 1939. [5] According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric ...
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1 ⁄ 3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations.
The vertex-connectivity of a symmetric graph is always equal to the degree d. [3] In contrast, for vertex-transitive graphs in general, the vertex-connectivity is bounded below by 2(d + 1)/3. [2] A t-transitive graph of degree 3 or more has girth at least 2(t – 1). However, there are no finite t-transitive graphs of degree 3 or more for t ≥ 8.