Hello, everybody.
I'd like to talk about a little problem that has intrigued people ever since we discovered mirrors.
Let's have a look at what I see when I look into the bathroom mirror in the morning.
Scary.
When I point up, Mirror Mathologer points up; when I point down, Mirror Mathologer points down.
When I raise my left hand, Mirror Mathologer raises his right hand.
When I raise my right hand, Mirror Mathologer raises his left hand.
Strange.
Also, with writing:
The tops of these characters are on top, both here and for the Mirror Mathologer.
And, the bottoms are on the bottom here and also for Mirror Mathologer.
But, this one here runs from left to right, and the other one runs from right to left.
What's going on here?
It appears that mirrors reflect left and right but, not up and down.
And in the setup everything's sort of equal.
So, if something gets flipped one way, it should also be flipped the other way, right?
Now, just to scramble your brain and a little bit further - have a look at this photo here of a mountain reflected in a mirror lake.
Here, up and down is being flipped, but not left and right.
Now if you find all this intriguing, you are in very good company.
I've linked in a video here of one my favorite scientists, Richard Feynman, talking about him being intrigued by this.
Richard Feynman: Genius. Nobel laureate. Really, really famous theoretical physicist.
This is a very slippery eel to pin down because...
when it comes to words like: - Flipping - Mirror Reflection - Mirror Images
...ordinary language is very "fuzzy".
For example, it doesn't distinguish between different phenomena...
when it comes to reflection, or mirror images, of 2D objects and 3D objects.
And, that's also part of the reason why most of the explanations of this puzzle (even the really, really good ones)...
kind of go off course a little bit, and maybe in the end confuse more than they explain.
So, what I thought I'd do today is try to provide a (there are many) really clean explanation.
So there's one very important observation which lets us sidestep this whole confusion (2D/3D).
and that is:
Whatever the explanation, if it works, it has to work for 2D things, like writing here on my t-shirt.
And, if it works for 2D things, it automatically works for 3D things.
Why? Because I can replace any 3D things (like myself) with a photo - which is a flat thing, and show it to the mirror.
Ok, so...
A clean explanation of the puzzle should just talk about 2D things.
I've prepared a 2D thing.
"CAT" written on a piece of paper.
What exactly is a mirror image of a 2D thing like this?
Well, let's have a look.
Look at this thing in the mirror. There's the mirror image.
Now, to really pin this thing down, let's press the piece of paper against the mirror.
Now, the original and the mirror image are facing the same spots.
So, corresponding points are touching each other.
So, if I were to make the piece of paper transparent (except for the writing)...
the back of this would actually be a mirror image, right?
And, I have actually arranged this so that it actually works.
Now things are transparent, and you can see...
This one object (this one object - this picture frame) has the original and the mirror image on both sides.
So it's really the same thing.
And this provides us with a short answer to our puzzle, which is this:
When we're saying the mirror exchanges left and right, but not up and down
What we're really doing is we're comparing the original to the mirror image.
So, we first look at the original.
And then, to be able to look at a mirror image, we make a flip...
to either show thing to the mirror, or here the physical object
So the flip is done by us in some way so that there's there's a flip in there.
It's not the mirror; the mirror reflects.
Whatever is here on the right comes back straight here.
Whatever is on the top comes back straight here.
Whatever is here comes back straight here.
The mirror doesn't exchange anything.
Ok. That's a short answer, but there's a bit more to it.
So, for example: What about this this mountain reflected in the lake?
So what I want to do now is to compare the original to the mirror image in all possible ways and see what I can say in general. OK?
Now I've prepared a very special object, a 2D object (it's just a blob) To take out any sort of bias we might have.
I'm not using a cat, or something else recognizable. OK?
Now, let's have a look at the mirror image. There it is.
But even this representation has a bit of a bias built in.
The way we kind of put these things next to each other will become very important in a little while.
But for the moment let's put this thing down in an arbitrary way, OK?
And now we are comparing original to mirror image.
And if you just kind of looked at this for the first time you may not even see that there is any similarity between these two things.
There is definitely no flipping happening.
But when we move things one on top of the other we get one of those strange Rorschach pictures
...with a nice symmetry axis here.
Pull them back apart again and you'll see that the mirror image differs from the original by a flip across this distinguished axis.
There you go.
And that will always be the case.
No matter how we juxtapose original and mirror image, there's always going to be this distinguished axis.
So whenever we're comparing original and mirror image, there is a distinguished axis across which we flip to get from one to the other.
It's always going to be there.
And that, of course, explains most of the puzzle.
But we still have to somehow explain: Why do we get a vertical axis, and not some other axis?
But actually...
Usually when you look at something via mirror, you see the thing and a mirror image at the same time.
And whenever you do this, this axis is going to be apparent.
So: The bathroom mirror surface is a bit different from that, because there you only see the mirror image
So let's have a look at the usual set up.
You look at something by a mirror and you see the mirror image and the original.
Where the axis is then depends on the relative position of the mirror and original.
So in this case you basically got the mountain and Mirror Lake setup.
The mirror axis is going to be down here.
And we actually can do the flipping here physically, like so.
When I change the position of the mirror, the mirror axis will change.
[It] will always be the bottom part of this mirror frame here.
And, we can do the flipping.
The flipping happening.
And we move things a little bit further.
There is a flip like that happening.
Now, let's have a look at a special case: Bathroom Mirror.
[In a] Bathroom mirror, we only see the mirror image.
Actually, I have to fess up here. I've got a special guest again.
The blob was really not a blob. It was Professor Farnsworth, one of my favorite characters from Futurama.
I'm a huge Futurama fan.
So this is what Professor Farnsworth sees in the mirror in the morning. Ok?
Now that's the only thing he sees. Now, he makes a statement about "left and right but not up and down"
How does the original come into the picture?
Well, he (or we) supply it. We somehow put it in, virtually, in our minds.
How do we put it in?
Well, if I give you your original on a piece of paper, like so, would you put it next to that one like that?
Probably not. The way you put it next to the mirror image is going to be like this.
There's a certain bias in which we put the mirror image and original next to each other.
It's our bias. It's what we do. We provide the second image in a certain way.
And it's kind of natural. This guy here is basically a person standing up in front of us with a natural up going up.
So when we put the original next to that? Well, we will also put the "up" up. Because that's how people stand in space.
So the explanation for all of this?
Whenever you compare original to mirror image, there's always going to be a flip axis.
And for the special case of the bathroom mirror:
We arrange things in a certain way.
So we arrange for the missing picture to be compared to the original in a certain way, and that then ensures that we get a vertical axis.
Now, often you find explanations of this puzzle culminating in the statements:
Mirrors don't exchange left and right; they don't exchange up and down. What they do do, is to exchange front and back.
And that's a true statement, but it's actually confusing in this context.
As I said before, you only need really to talk about 2D images.
That's essential bits. As soon as you kind of drag in the 3D bits, things get confusing.
So explanation for the puzzle is finished now. What comes now is a footnote.
Left and right doesn't get flipped? Well that's reflected by these two arrows here going in the same direction.
Up and down dont get flipped? That's corresponds by these two red arrows going in the same direction.
But what happens to the green arrow - the green arrow that's kind of pointing towards the mirror?
The reflection is actually pointing in the opposite direction.
Ok, so, in terms of 3D objects (like myself)?
The 3D mirror image is actually only inverted in the direction of the of the green arrow.
That's how I turn myself into my mirror image.
Also, us being roughly self-similar across this mirror plane here probably contributes to the overall puzzlement that we experience at the very beginning.
Now I'm really crazy about mirrors and there would be a lot more things that I could say.
What I want to do is...
I want to just leave you with a couple of fun things.
And for those I've asked my friend, Professor Farnsworth, to model.
And here he is standing in front of the straight mirror.
Now what I want you to do is...
I want you to get a spoon like this and then you look at it in various ways.
So for example, if you look into the spoon something strange happens
All of the sudden you see yourself standing on your head.
If you look at the spoon like this you'll actually just see a normal mirror image (a little bit distorted).
Now for something really spooky, just take the spoon and bring it really close to one of your eyes.
Now, you can try and explain those things in the comments. I'm look forward to your answers.
Also, if you want something else crazy to experiment with:
You need a mirror like this. I cut this out of a garbage bin.
And what a mirror like this does is it actually shows you like other people see you.
So, not flipped.
So that this thing doesn't show your normal mirror image. It shows you the way other people see you.
A little bit distorted in this one. But basically, that's what it is.
So, all very nice effects.
Hopefully you've enjoyed this, and hopefully you'll be able to provide some explanations here.
That's it for today.