this cool word Metachirality and you're like ooh Metachirality what does that mean,

and they're like well it's a very difficult and subtle mathematical concept that only

like one person understands, even we don't understand it with our mortal minds, and you're

like, pfff, if you don't understand something how would you know if it's hard to understand?

After all, regular ol' Chirality isn't a difficult concept even though it has a fancy

name.

It just means a thing that's not the same as its mirror image, and that's, like, most

things.

But y'know, it's faster to say something is “chiral” than something is “not the

same as its mirror image”, and somehow a “chiral pair” sounds more pairy and romantic

than “mirror image pair”.

So that's points for the word “chiral” and you figure maybe “metachiral” is a

cool word that's simpler than it sounds too.

So like, most animals appear mirror symmetric on the outside, they're not chiral, but

there's some exceptions, like snails which usually have their shell swirls to their right,

or those crabs with one big claw, and there's even some chiral people, of course we're

all chiral on the inside cuz organs.

Or maybe you think of chiral molecules, like, a lot of common molecules are symmetric but

some molecules are different from their mirror images and sometimes they taste different

or maybe one version is poison so the chirality matters.

Anyway then you get to thinking about the word “achiral”, which of course means,

not not-the-same-as-its-mirror-image, and that seems kind of silly.

So you start pondering the difference between the words chiral and asymmetric, or why achirality

is different from symmetry.

In fact you can make a diagram to inspect each combination.

Most things go in the asymmetric and chiral section of the diagram, things that have no

symmetry and are different than their mirror images.

And then there's mirror symmetric things that are both symmetric and achiral, like most

animals if you don't fuss too much about little details, and many common shapes like

squares and hearts and stars.

But it's not like mirror symmetry is the only kind of symmetry around.

Take rotational symmetry.

This rotationally symmetric swirlyflower is different from its mirror image so it is chiral,

but its still got symmetry, at least if I could draw better ok we're gonna use techniques,

ok so with this symmetric swirlyflower you can turn it 72 degrees clockwise and it still

looks the same.

So it goes in the symmetric and chiral part of the diagram.

Or like, a right handed square dance thingy, yknow the one?

Ok we'd better use more techniques, mm ok got it.

anyway a right handed squaredancey thingy is different from a left handed square dance

thingy but both still have 90-degree rotational symmetry, which is conveniently demonstrated

when you dance clockwise a quarter turn.

Or, well, for the right handed version you go clockwise, but for the left-handed version

you dance counterclockwise.

Or widdershins if you're into fancy words.

So that's fine, to make your left-handed square dance thingy go clockwise instead of

widdershins just dance backwards and you can see that both ways have symmetry under a 90-degree

clockwise rotation.

And also they both have symmetry when going widdershins 90 degrees, it really doesn't

matter and maybe you wouldn't even notice the difference except that dancing is made

of people and people want to walk forwards.

But the point is, the figure itself has the same symmetry whether it's the right hand

or left hand version and while that seems obvious you've found that its details like

that which often come in handy later.

In the swirlyflower it's even more obvious that both of the chiral pair have the same

symmetry.

You also notice that the two mirror image versions, layered together, would be a thing

with five fold mirror symmetry.

It's like they're two halves of an extra symmetric whole, and that's very pair-y

and romantic.

You figure that works for all kinds of rotational symmetries, that when you add them together

with their reflection they keep their original rotational symmetry, so nothing was lost,

but now they've added just as many mirrors as they had rotations, they're more than

the sum of their parts, and everyone lives happily ever after!

In fact, even if they're both the same chirality you can layer them into a twice-as-symmetric

version.

Maybe it works for other symmetries too?

So what other kinds of symmetry are there?

Maybe glide reflection symmetry, like an infinite set of footprints, which has no line of mirror

symmetry but if you combine a reflection and a translation then you get the same thing.

Over and over and over.

First of all, is it chiral?

The mirror image is almost the same but shifted, so does that count?

Though when you think about it, whenever you take the mirror image of a mirror symmetric

thing it also ends up shifted somewhere else, unless you reflect it exactly in the right

spot.

You can still make them match up without reflecting them again, and that's what counts.

Now if there were a first footstep, like does the right foot go first or the left foot,

then you'd be able to tell it apart from its mirror image, but the set of footprints

only has true glide reflection symmetry if the set of footprints is infinite in which

case there's no first foot cuz infinity is cool like that, and if I couldn't see

the ends of the strip I'd have no way of knowing whether it was flipped over or not.

It has no mirror symmetry but is still the same as its mirror image.

Which is kinda weird.

But like rotationally symmetric things you notice that you can layer it with itself to

get something with twice as much symmetry: now it has both glide reflection symmetry

and mirror symmetry.

And, well, it used to have translation symmetry too but now it has twice as much translation

symmetry which you figure probably counts for something even though twice infinity is

still infinity.

So you update your understanding of the word Chiral to make sure you don't confuse “achiral"

with “mirror symmetric", maybe you need another diagram.

And how weird that the weird case is in something as common as the symmetry of footsteps!

But maybe it should've been obvious that footsteps are achiral because it's not like

there's righty runners and lefty runners the way there's righty and lefty pitching

or batting in baseball.

Actually now that you think of it there's a lot of chiral sports.

And achiral sports that have glide reflections, and mirror symmetric sports, and maybe you

should make a diagram...

Anyway now that you've refreshed your understanding of chirality, you figure you're ready to

understand metachirality, just gotta get that last meta bit in there.

But that will have to wait until next time.

Meanwhile maybe try making some diagrams, I do like diagrams.

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Ok see you next time.