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All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges. For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between two-thirds and 1 for any convex shape.
Symmetry is one class of patterns in nature whereby there is near-repetition of the pattern element, either by reflection or rotation. While sponges and placozoans represent two groups of animals which do not show any symmetry (i.e. are asymmetrical), the body plans of most multicellular organisms exhibit, and are defined by, some form of ...
Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids. [30] Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. [31]
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically. In biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. [21]
The triangles with reflection symmetry are isosceles, the quadrilaterals with this symmetry are kites and isosceles trapezoids. [11] For each line or plane of reflection, the symmetry group is isomorphic with C s (see point groups in three dimensions for more), one of the three types of order two (involutions), hence algebraically isomorphic to ...
Let l, m, n be integers greater than or equal to 2. A triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians).
An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel.
7 heptominoes (coloured green) have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection. 4 heptominoes (coloured blue) have point symmetry, also known as rotational symmetry of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
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