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• When you hear the word symmetry,

• maybe you picture a simple geometric shape

• like a square or a triangle,

• or the complex pattern on a butterfly's wings.

• If you are artistically inclined,

• you might think of the subtle modulations of a Mozart concerto,

• or the effortless poise of a prima ballerina.

• When used in every day life,

• the word symmetry represents vague notions of

• beauty, harmony and balance.

• In math and science, symmetry has a different,

• and very specific, meaning.

• In this technical sense,

• a symmetry is the property of an object.

• Pretty much any type of object can have symmetry,

• from tangible things like butterflies,

• to abstract entities like geometric shapes.

• So, what does it mean for an object to be symmetric?

• Here's the definition:

• a symmetry is a transformation that leaves that object unchanged.

• Okay, that sounds a bit abstract, so let's unpack it.

• It will help to look at a particular example,

• like this equilateral triangle.

• If we rotate our triangle through 120 degrees,

• around an access through its center,

• we end up with a triangle that's identical to the original.

• In this case, the object is the triangle,

• and the transformation that leaves the object unchanged

• is rotation through 120 degrees.

• So we can say an equilateral triangle is symmetric

• with respect to rotations of 120 degrees around its center.

• If we rotated the triangle by, say, 90 degrees instead,

• the rotated triangle would look different to the original.

• In other words, an equilateral triangle is not symmetric

• with respect to rotations of 90 degrees around its center.

• But why do mathematicians and scientists care about symmetries?

• Turns out, they're essential in many fields of math and science.

• Let's take a close look at one example: symmetry in biology.

• You might have noticed that there's a very familiar kind of symmetry

• we haven't mentioned yet:

• the symmetry of the right and left sides of the human body.

• The transformation that gives this symmetry is reflection

• by an imaginary mirror that slices vertically through the body.

• Biologists call this bilateral symmetry.

• As with all symmetries found in living things,

• it's only approximate,

• but still a striking feature of the human body.

• We humans aren't the only bilaterally symmetric organisms.

• Many other animals, foxes, sharks, beetles,

• that butterfly we mentioned earlier,

• have this kind of symmetry,

• as do some plants like orchid flowers.

• Other organisms have different symmetries,

• ones that only become apparent

• when you rotate the organism around its center point.

• It's a lot like the rotational symmetry of the triangle we watched earlier.

• But when it occurs in animals,

• this kind of symmetry is known as radial symmetry.

• For instance, some sea urchins and starfish

• have pentaradial or five-fold symmetry,

• that is, symmetry with respect to rotations of 72 degrees around their center.

• This symmetry also appears in plants,

• as you can see for yourself by slicing through an apple horizontally.

• Some jellyfish are symmetric with respect to rotations of 90 degrees,

• while sea anemones are symmetric when you rotate them at any angle.

• Some corals, on the other hand, have no symmetry at all.

• They are completely asymmetric.

• But why do organisms exhibit these different symmetries?

• Does body symmetry tell us anything about an animal's lifestyle?

• Let's look at one particular group:

• bilaterally symmetric animals.

• In this camp, we have foxes, beetles, sharks, butterflies,

• and, of course, humans.

• The thing that unites bilaterally symmetric animals

• is that their bodies are designed around movement.

• If you want to pick one direction and move that way,

• it helps to have a front end

• where you can group your sensory organs--

• your eyes, ears and nose.

• It helps to have your mouth there too

• since you're more likely to run into food

• or enemies from this end.

• You're probably familiar with a name for a group of organs,

• plus a mouth, mounted on the front of an animal's body.

• Having a head leads naturally to the development of bilateral symmetry.

• And it also helps you build streamlined fins if you're a fish,

• aerodynamic wings if you're a bird,

• or well coordinated legs for running if you're a fox.

• But, what does this all have to do with evolution?

• Turns out, biologists can use these various body symmetries

• to figure out which animals are related to which.

• For instance, we saw that starfish and sea urchins have five-fold symmetry.

• But really what we should have said was

• adult starfish and sea urchins.

• In their larval stage, they're bilateral, just like us humans.

• For biologists, this is strong evidence

• that we're more closely related to starfish

• than we are, to say, corals,

• or other animals that don't exhibit bilateral symmetry

• at any stage in their development.

• One of the most fascinating and important problems in biology

• is reconstructing the tree of life,

• discovering when and how the different branches diverged.

• Thinking about something as simple as body symmetry

• can help us dig far into our evolutionary past

• and understand where we, as a species, have come from.

When you hear the word symmetry,

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# 【TED-Ed】The science of symmetry - Colm Kelleher

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許芷熒 posted on 2014/07/12
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