# How would you define symmetry?

One thing mathematicians do that does not get much publicized is to find ways of defining concepts that have been around for ages but which have not been definitively specified. If you have ever taken a course in calculus then you have come across the delta-epsilon definition of continuity, which gives students all kinds of difficulties, but which is really a very compact way of defining the concept that was not discovered until the 19th century.

The defnition of symmetry is easier to understand, but still quite ingenious and I thought I would share it.

Consider the following:

The letter M

The letter N

The equation x^2 + xy + y^2

The description “A red lamp on a blue table next to a blue lamp on a red table”

Now see if you can come up with a definition of symmetry that covers all these cases. Kind of tough?

The definition of symmetry has 3 components: An object or set of objects, a property and an operation.

We say that a property of a set of objects is symmetic with respect to an operation if the property remains the same after the operation is performed.

For example, for the letter M, you could reflect the two different sides with respect to a verical mirror and the letter occupies the same space. So the letter M has reflective symmetry. For the letter N, you get similar results if you rotate it 180 degrees, so it has rotational symmetry.

The equation retains the same value if the symbols x and y are swapped and the description has the same meaning if the words red and blue are swapped.

Okay, I have gotten that out of my system. Any thoughts?(other than why in the world I would feel compelled to post this)

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## 16 Answers

That is why symmetry has multiple defintions/considerations, because there are so many different cases. Is it the reflexive property? Or is it rotational? I’m not sure there is a concise definition. I’d be curious to know how many definitions of symmetry there are in the dictionary.

It’s when it’s not asymmetric. ;)

@miasmom , The point is the unifying aspect of talking about things that remain the same after an operation is performed. If you look at the online Merriam-Webster definition of symmetry, the third and fourth definitions give the modern point of view.

http://www.merriam-webster.com/dictionary/symmetry

@LostInParadise ,

“x^2 + xy + y^2” is technically not an equation. I believe could be classified as a statement or formula, and it is symmetrical as you describe.

“x^2 + xy + y^2 = y^2 + xy + x^2” is an equation and it is symmetrical.

An equation is a statement of equality.

You are correct. I realized my mistake afterwards.

oooOOO | OOOooo <——this——> oooOOO | oooOOO

As an architectural designer, I tend to think of symmetry in terms of pairing, duplicating or patterning. That is, the repetition of multiples is inherent to symmetry. The modules needn’t be always identical but, rather, can often imply a sort of ghosted framework or gridwork from where modules can be added or deleted. Also, the grid, at least in its most essential – as an x-y intersection with 4 quadrants – is visibly or invisibly always characteristic of symmetry. With no axes with which to mirror or rotate, there is no symmetry. Generically, I also think of symmetry as mostly a duality. That is, if two mirrored objects are accurately copying one another, they are considered symmetrical. There is no singular symmetry. So, symmetry only is possible when there are multiples to compare. Similarly, symmetry sometimes requires asymmetry in order to contrast and embolden its own existence.

@TheRocketPig

This is asymmetry———->oooOOO | oooOOO

Akin to looking in the mirror and seeing a reflection of the back of your head.

@SeventhSense it’s translational symmetry. Asymmetry would be oooOOO | OooOoO.

Don’t limit yourself just to reflectional symmetry.

when i lie in my bed and my body feels perfectly balanced and i don’t feel the need to indulge my OCD…..

i’ve heard for the past 10 years in random places that beauty is defined by symmetry. maybe that is a start. metaphorical balance and zen vs numerical equality and zerio balanced ratios? Who knows. Is anything truly symmetrical that is not man made? nope.

Snowflakes are symmetrical Would you say a sphere is symmetrical? A drop of water assumes a symmetrical shape after it falls a short distance. The little splash that the drop of water makes when it hits a body of water is also symmetrical.

Symmetry is interesting observation that you almost get two mirror pieces when you slice an average mathematician from crown straight down through the bottom of the torso, from head on, or from directly behind.

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