So you’re tired of symmetry being all about reflections and rotations or the most boring
of all: translations, which is just like, puttin stuff places.
Puttin stuff places.
It’s math!
It’s funny because “puttin stuff places” is your english translation of the word translation
so to translate "translate" you metatranslate, just like how when you have data about data
that’s metadata or in science if you do a study on a set of studies its a metastudy
or if you dream that you’re dreaming it’s dream-ception.
Anyway so you’re really into this word Metachirality which you figure has gotta mean, like, chirality
that itself has chirality cuz maybe that is a thing?
But if chiral means something is not the same as its mirror image, what does it mean for
something to not be the same as its mirror image in a not-the-same-as-its-mirror-image
kind of way?
I mean in school they made you learn about just four kinds of symmetry transformation
thing: reflection, rotation, good ol' translation, and everyone’s least favourite symmetry,
the glide reflection.
Well maybe they're not that bad, but you’ve always been a little suspicious of them, y’know?
Are they one thing?
Or are they two things?
How can you be both a reflection and a translation without being two things or is the whole point
that puttin stuff places all flippy-like is its own special thing?
And then you wonder why do you never hear about other combinations?
So you try putting a reflection and a rotation together, which you name a reflectotation.
And a reflectotation is definitely different from either a reflection or a rotation alone.
Although you notice the reflectotation looks an awful lot like a glide reflection which
is interesting because, ooh, you can make a diagram, see there’s reflections and rotations
and translations and we can make sure we get all combinations, like a glide reflection
is a reflection translation, and between reflection and rotation is our reflectotation which is
a glide reflection, or maybe a glide reflection is a reflectotation?
Anyway they’re the same, so what about a glide rotation?
It can’t just be another one that’s the same as glide reflection because it never
flips, both feet are the same chirality.
but it looks…
well, you think maybe you could get that with just one rotation from a different spot, yeah,
so the glide’s not doin much for ya except moving the rotation point.
And what do you get when you translate, reflect, and rotate?
Just another glide reflection?
Well you guess there’s just only one way to put stuff places all flippy-like so unfortunately
those combinations don’t get you anything new, it’s still four kinds of thing whatever
you call them, and what you want is a new kind of transformation that will put some
chirality in your chirality.
Of course this assumes it’s always true that these combinations get you a rotation
or glide reflection.
It certainly seems like a rotation plus translation will always just move the rotation point around
on the plane…
it seems like you can take any two left feet anywhere on the plane and there will be a
single rotation between them.
Anywhere on the plane.
Unless…
what if you’re not on the plane?
What if we travel into the third dimension?
“But wait”, says 2-dimensional you.
“Isn’t the third dimension time?”
Maybe sometimes but we’ll make time for time another time.
Anyway so you take your thing and rotated thing and instead of translating it along
the plane you bring it out into the third dimension.
Well that wasn’t very dramatic.
And now we have to get it to stay there…
There you go, and why not do it again, and again and again until it comes back around,
and seen from the top it’s like if it were flat it would have 72-degree rotational symmetry
but in 3d it doesn’t, it’s like... a spiral staircase that climbs itself.
cuz its feet… um anyway so you decide to keep adding to it because it’s turning into
a spirally sort of thing and you like spirals, you were born to like spirals, it’s in your
DNA.
And you like it because this 3d glide rotation is definitely not one of the old four kinds
of transformation, it’s something new, and it looks like an actual symmetry: that every
time you transform this whole thing by a 72-degree rotation plus a little translation, it stays
the same.
Well, if you pretend its infinite.
But infinity is totally your bestie so you got this.
So you’re feeling pretty good about discovering this new kind of symmetry until you find out
that not only does it already have a name but you even already discovered that name
back when you were first forced to learn about symmetry in math class and said: screw symmetry.
Screw symmetry, y'know that translating rotating feeling like screwing a screw, except to be
a proper symmetry it’s an infinite screw so when you do the translation and rotation
thing it doesn’t go past its own edge or get to the end.
Its spirally like a spring or like that swirly pasta but only if you pretend its an infininoodle
which it is coming out of the noodle machine before they cut it up to fit it in a box,
right?
And now that you think of it there’s a lot of spirally helix things out there and unlike
your foot staircase most of them have this smooth continuous symmetry like, y’know
the barber pole effect, which you figure happens because things with screw symmetry look the
same if you turn it a little and move it forward a little at the same time, until you realize
your drill bit isn’t infinite but anyway if you only turn it without moving forward
then under symmetry its like its moving backwards a little.
You wonder if it would look like it were barberpoling the other way if you turned the drill bit
around around, and strangely enough it still barberpole effects back towards the drill,
and the symmetry still wants you to move forward and clockwise.
It looks the same when turned around, which you notice is the case for a lot of these
things, which means they have 2-way rotational symmetry, which explains why, like, a nut
turns clockwise onto a screw but if you flip the nut over then it… still turns clockwise
onto the screw.
That’s different from like, if you have a clockwise pointing arrow and you flip it
over now it’s a counterclockwise pointing arrow.
A thing turning in place looks clockwise from one side and counterclockwise from the other.
Even your helixy things that look a little different when turned around, like this swirlynoodle
which has a direction to its flappybits, like petting a cat, but pasta.
It still has clockwise screw symmetry whether its moving head first or tail first.
Good kitty.
Your springs, on the other hand, all have counterclockwise screw symmetry.
You notice it’s like the spiral of your spiral notebook, but that one’s right handed.
They’re like mirror images kinda.
Which reminds you of how when you have a chiral thing with 72 degree rotational symmetry to
the right, the mirror image also has 72 degree rotational symmetry to the right, even though
the reflection of the symmetry is a 72 degree rotation to the left.
Or to look at it a different way, if the original has symmetry 72-degrees to the right it also
has the mirror image symmetry of 72-degrees to the left, which is another way of saying
its the same symmetry.
The figure might not be the same as its mirror image, left foots turn to right foots or whatever,
but there’s only one kind of 72-degree rotational symmetry because it’s the same as its mirror
image symmetry.
Well hold on now, normal chiral rotational symmetry is the same symmetry as its mirror
image?
That sounds… not very chiral.
Wouldn’t it be more chiral if a chiral symmetry were not the same as its mirror image?
That sounds meta.
That sounds… metachiral.
So you take another look at your spirally helix things.
The screw symmetry has you move your screw a little bit forward with a right-hand twist.
But mirror screws don’t let you move a bit forward with a right hand twist, you need
to use forward and left.
Well that’s not so complicated, right?
One spirally foot thing has a right handed 72-degree screw symmetry, and its mirror image
doesn’t have a right handed 72-degree screw symmetry.
The symmetry has its own chiral partner, a left handed 72-degree screw symmetry.
Yeah, some kinds of chiral symmetry come in chiral pairs.
That's metachirality.
You didn’t even need the fourth dimension or some fancy crystal lattice!
You’ve been eating metachirality all along!
It’s been in your candy canes and telephone cords and slinkys!
Sure maybe its useful if you're trying to understand crystal lattices or wanna know
which way ‘round spacetime goes but its been all around you this whole time but only
now can you recognize the fundamental difference between a pastakitty and springfingers.
It’s this season’s fashion!
Ok maybe its time to end this video… just gonna… do that now.
bye.