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  2. Symmetry (geometry) - Wikipedia

    en.wikipedia.org/wiki/Symmetry_(geometry)

    Symmetry (geometry) 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 ...

  3. Hermann–Mauguin notation - Wikipedia

    en.wikipedia.org/wiki/Hermann–Mauguin_notation

    The direction of a symmetry element corresponds to its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction, then they are denoted as a fraction ⁠ n / m ⁠ or n /m. If two or more axes have the same direction, the axis with higher symmetry is shown.

  4. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    Rotation matrix. In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system.

  5. Rotational symmetry - Wikipedia

    en.wikipedia.org/wiki/Rotational_symmetry

    Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth 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 ...

  6. Parabola - Wikipedia

    en.wikipedia.org/wiki/Parabola

    The point E is an arbitrary point on the parabola. The focus is F, the vertex is A (the origin), and the line FA is the axis of symmetry. The line EC is parallel to the axis of symmetry, intersects the x axis at D and intersects the directrix at C. The point B is the midpoint of the line segment FC.

  7. Symmetry operation - Wikipedia

    en.wikipedia.org/wiki/Symmetry_operation

    Symmetry operation. 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 ...

  8. Crystallographic point group - Wikipedia

    en.wikipedia.org/wiki/Crystallographic_point_group

    Crystallographic point group. In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions.

  9. Kite (geometry) - Wikipedia

    en.wikipedia.org/wiki/Kite_(geometry)

    Kite (geometry) A kite, showing its pairs of equal-length sides and its inscribed circle. In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, [1] but the word deltoid may ...

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