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Reflection symmetry

From Wikipedia, the free encyclopedia

 Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.
Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.

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

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.

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Transcription

This video is provided as supplementary material for courses taught at Howard Community College and in this video I'm going to talk about two kinds of symmetric, reflection symmetry and rotational symmetry. Let's start with reflection symmetry. Reflection symmetry is sometimes called line symmetry or mirror symmetry and here's what it's about. I've drawn a picture of a heart and I'm going to draw a dotted line right at the center of it. Now that the dotted line divides the image into two mirror images of each other, or two reflections. The line itself is called the line of symmetry. So anytime you can take an image and draw a line through it so that you've got a pair of mirror images, you've got what's called reflection symmetry. Another way of thinking of reflection symmetry would be, if you had, as I have here, a cut-out of that image. If I take this heart, this cut-out that I have, and fold it along the line of symmetry, both sides will match up perfectly. So that's another way of seeing what reflection symmetry is about. Now you can have more than one line of symmetry. Here I've got a diamond. If I draw a dotted line, a long one to connect two opposite angles, I'm going to create two mirror images. On the other hand, I could have drawn a dotted line, a shorter one, connecting the other two opposite angles, and that would also create a mirror image. If I do this with paper, I can fold this paper one way and both sites will match, or I can fold the paper the other way and both sides will match. So you can have more than one line of symmetry. If you had a pentagon, you'd have five lines of symmetry. So you could have a number of lines of symmetry . So that's what reflection symmetry is about. Then we've got something called rotational symmetry. So to demonstrate rotational symmetry I've drawn a triangle and I've cut out a triangle that's the same size and shape. So this is an equilateral triangle. If I take this paper triangle and rotate it 180 degrees., a third of the turn... I'm sorry 120 degrees is a third of a turn, I can match it up with the triangle that I'd drawn. I could rotate it another 120 degrees, another third, and it matches. I could rotate it again, and it will match. Now the smallest number of degrees I needed to rotate that triangle around until it matched was 120 degrees, so we say that this equilateral triangle has 120-degree rotational symmetry. If I have a square, as I have here, I would just have to rotate it 90 degrees until matches up again or a one quarter turn. So that has 90-degree rotational symmetry. So anytime you can take image, rotate it less than a full turn, and have it match up with itself, that's going to be an example of rotational symmetry. I said less than a full turn because if I took this heart that I started out with, I can rotate it 360 degrees and it will match up, because all it's done is gone full-circle. So you have to be able to rotate it less than 360 degrees to have rotational symmetry. Now there's a subset of rotational symmetry that's called point symmetry and point symmetry is a property of anything that has rotational symmetry and can be rotated 180 degrees. So, for instance, this parallelogram... if I start to rotate this I've gotta go all the way through whole half turn, or 180 degrees, before it matches up with itself again. A triangle would not have point symmetry because if I rotated it 180 degrees it's not go to match up with its original. Let me do that again. I'll rotate it 180 degrees and it doesn't match up. A square has point symmetry. I can rotate it less than 180 degrees, I can rotate it 90 degrees, but I CAN rotate it 180 degrees. So as long as I can rotate it 180 degrees and get back to exactly the same image I started with, then I've got point symmetry. So we've got reflection symmetry where we've got a line of symmetry -- you fold an image on itself. And then we've got rotational symmetry where you rotate image around, and within rotational symmetry, if you can rotate it 180 degrees and have it be symmetrical, then it's also called point symmetry. Okay, that's it. Take care. I'll see you next time.

Contents

Symmetric function

 A normal distribution bell curve is an example symmetric function
A normal distribution bell curve is an example symmetric function

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.

Symmetric geometrical shapes

2D shapes w/reflective symmetry
Isosceles trapezoid.svg
GeometricKite.svg
isosceles trapezoid and kite
Hexagon p2 symmetry.png
Hexagon d3 symmetry.png
Hexagons
Octagon p2 symmetry.png
Octagon d2 symmetry.png
octagons

Triangles with reflection symmetry are isosceles. Quadrilaterals with reflection symmetry are kites, (concave) deltoids, rhombuses,[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.

Mathematical equivalents

For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space.

In certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry; in such contexts in modern physics the term parity or P-symmetry is used for both.

Advanced types of reflection symmetry

For more general types of reflection there are correspondingly more general types of reflection symmetry. For example:

In nature

 Many animals, such as this spider crab Maja crispata, are bilaterally symmetric.
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

 Mirror symmetry is often used in architecture, as in the facade of Santa Maria Novella, Florence, 1470.
Mirror symmetry is often used in architecture, as in the facade of Santa Maria Novella, Florence, 1470.

Mirror symmetry is often used in architecture, as in the facade of Santa Maria Novella, Venice.[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

References

  1. ^ a b Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. p. 32. 
  2. ^ Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. W. W. Norton. pp. 394–395. ISBN 0-393-04002-X. 
  3. ^ Valentine, James W. "Bilateria". AccessScience. Retrieved 29 May 2013. 
  4. ^ "Bilateral symmetry". Natural History Museum. Retrieved 14 June 2014. 
  5. ^ Finnerty, John R. (2005). "Did internal transport, rather than directed locomotion, favor the evolution of bilateral symmetry in animals?" (PDF). BioEssays. 27: 1174–1180. doi:10.1002/bies.20299. PMID 16237677. 
  6. ^ "Bilateral (left/right) symmetry". Berkeley. Retrieved 14 June 2014. 
  7. ^ 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 
  8. ^ Johnson, Anthony (2008). Solving Stonehenge: The New Key to an Ancient Enigma. Thames & Hudson.
  9. ^ Waters, Suzanne. "Palladianism". Royal Institution of British Architects. Retrieved 29 October 2015. 

Bibliography

General

  • Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. is potty

Advanced

External links

This page was last edited on 10 April 2017, at 11:43.
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