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.
Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.
In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical.
In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation, if, when applied to the object, this operation preserves some property of the object.[1] The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).
The symmetric function of a two-dimensional figure is a line such that, for each perpendicular constructed, if the perpendicular intersects the figure at a distance 'd' from the axis along the perpendicular, then there exists another intersection of the shape and the perpendicular, at the same distance 'd' from the axis, in the opposite direction along the perpendicular.
Another way to think about the symmetric function is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror images.[1]
Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry.
Triangles with reflection symmetry are isosceles. Quadrilaterals with reflection symmetry are kites, (concave) deltoids, rhombi,[2] and isosceles trapezoids. 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 2/3 and 1 for any convex shape.
Advanced types of reflection symmetryEdit
For more general types of reflection there are correspondingly more general types of reflection symmetry. For example:
Animals that are bilaterally symmetric have reflection symmetry in the sagittal plane, which divides the body vertically into left and right halves, with one of each sense organ and limb pair on either side. Most animals are bilaterally symmetric, likely because this supports forward movement and streamlining.[3][4][5][6]
^Valentine, James W. "Bilateria". AccessScience. Retrieved 29 May 2013.
^"Bilateral symmetry". Natural History Museum. Retrieved 14 June 2014.
^Finnerty, John R. (2005). "Did internal transport, rather than directed locomotion, favor the evolution of bilateral symmetry in animals?" (PDF). BioEssays. 27 (11): 1174–1180. doi:10.1002/bies.20299. PMID 16237677.
^"Bilateral (left/right) symmetry". Berkeley. Retrieved 14 June 2014.
^Tavernor, Robert (1998). On Alberti and the Art of Building. Yale University Press. pp. 102–106. ISBN978-0-300-07615-8. More accurate surveys indicate that the facade lacks a precise symmetry, but there can be little doubt that Alberti intended the composition of number and geometry to be regarded as perfect. The facade fits within a square of 60 Florentine braccia
^Johnson, Anthony (2008). Solving Stonehenge: The New Key to an Ancient Enigma. Thames & Hudson.
^Waters, Suzanne. "Palladianism". Royal Institution of British Architects. Retrieved 29 October 2015.
BibliographyEdit
GeneralEdit
Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson.
Wikimedia Commons has media related to Reflection symmetry.
Mapping with symmetry - source in Delphi
Reflection Symmetry Examples from Math Is Fun
October 15, 2023
reflection, symmetry, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, octob. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Reflection symmetry news newspapers books scholar JSTOR October 2015 Learn how and when to remove this template message Mirror symmetry redirects here For other uses see Mirror symmetry disambiguation 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 Figures with the axes of symmetry drawn in The figure with no axes is asymmetric In 2D there is a line axis of symmetry in 3D a plane of symmetry An object or figure which is indistinguishable from its transformed image is called mirror symmetric In conclusion a line of symmetry splits the shape in half and those halves should be identical Contents 1 Symmetric function 2 Symmetric geometrical shapes 3 Advanced types of reflection symmetry 4 In nature 5 In architecture 6 See also 7 References 8 Bibliography 8 1 General 8 2 Advanced 9 External linksSymmetric function Edit nbsp A normal distribution bell curve is an example symmetric functionIn formal terms a mathematical object is symmetric with respect to a given operation such as reflection rotation or translation if when applied to the object this operation preserves some property of the object 1 The set of operations that preserve a given property of the object form a group Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations and vice versa The symmetric function of a two dimensional figure is a line such that for each perpendicular constructed if the perpendicular intersects the figure at a distance d from the axis along the perpendicular then there exists another intersection of the shape and the perpendicular at the same distance d from the axis in the opposite direction along the perpendicular Another way to think about the symmetric function is that if the shape were to be folded in half over the axis the two halves would be identical the two halves are each other s mirror images 1 Thus a square has four axes of symmetry because there are four different ways to fold it and have the edges all match A circle has infinitely many axes of symmetry Symmetric geometrical shapes Edit2D shapes w reflective symmetry nbsp nbsp isosceles trapezoid and kite nbsp nbsp Hexagons nbsp nbsp octagonsTriangles with reflection symmetry are isosceles Quadrilaterals with reflection symmetry are kites concave deltoids rhombi 2 and isosceles trapezoids 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 2 3 and 1 for any convex shape Advanced types of reflection symmetry EditFor more general types of reflection there are correspondingly more general types of reflection symmetry For example with respect to a non isometric affine involution an oblique reflection in a line plane etc with respect to circle inversion In nature Edit nbsp Many animals such as this spider crab Maja crispata are bilaterally symmetric Main article Bilateral symmetry Animals that are bilaterally symmetric have reflection symmetry in the sagittal plane which divides the body vertically into left and right halves with one of each sense organ and limb pair on either side Most animals are bilaterally symmetric likely because this supports forward movement and streamlining 3 4 5 6 In architecture Edit nbsp Mirror symmetry is often used in architecture as in the facade of Santa Maria Novella Florence 1470 Main article Mathematics and architecture Mirror symmetry is often used in architecture as in the facade of Santa Maria Novella Florence 7 It is also found in the design of ancient structures such as Stonehenge 8 Symmetry was a core element in some styles of architecture such as Palladianism 9 See also EditPatterns in nature Point reflection symmetry Coxeter group theory of Reflection groups in Euclidean space Rotational symmetry different type of symmetry References Edit a b Stewart Ian 2001 What Shape is a Snowflake Magical Numbers in Nature Weidenfeld amp Nicolson p 32 Gullberg Jan 1997 Mathematics From the Birth of Numbers W W Norton pp 394 395 ISBN 0 393 04002 X Valentine James W Bilateria AccessScience Retrieved 29 May 2013 Bilateral symmetry Natural History Museum Retrieved 14 June 2014 Finnerty John R 2005 Did internal transport rather than directed locomotion favor the evolution of bilateral symmetry in animals PDF BioEssays 27 11 1174 1180 doi 10 1002 bies 20299 PMID 16237677 Bilateral left right symmetry Berkeley Retrieved 14 June 2014 Tavernor Robert 1998 On Alberti and the Art of Building Yale University Press pp 102 106 ISBN 978 0 300 07615 8 More accurate surveys indicate that the facade lacks a precise symmetry but there can be little doubt that Alberti intended the composition of number and geometry to be regarded as perfect The facade fits within a square of 60 Florentine braccia Johnson Anthony 2008 Solving Stonehenge The New Key to an Ancient Enigma Thames amp Hudson Waters Suzanne Palladianism Royal Institution of British Architects Retrieved 29 October 2015 Bibliography EditGeneral Edit Stewart Ian 2001 What Shape is a Snowflake Magical Numbers in Nature Weidenfeld amp Nicolson Advanced Edit Weyl Hermann 1982 1952 Symmetry Princeton Princeton University Press ISBN 0 691 02374 3 External links Edit nbsp Wikimedia Commons has media related to Reflection symmetry Mapping with symmetry source in Delphi Reflection Symmetry Examples from Math Is Fun Retrieved from https en wikipedia org w index php title Reflection symmetry amp oldid 1148200161, wikipedia, wiki, book, books, library,
