- This mirror is convex and concave at the same time.
So what's it like to look into this mirror?
We all know what it's like to look into a convex mirror,
like this security mirror.
It creates a sort of fisheye effect.
And if you've ever looked into a makeup mirror,
you'll know what it's like to see your reflection
from a concave surface.
It magnifies your face.
Or you know, just look at the front and back of a spoon,
though it's interesting that this concave mirror
magnifies your face, while this one
turns your face upside down.
We'll need to figure that out.
What I'm actually showing you here is what it's like
to be a camera looking into concave and convex mirrors.
But for this video, I want you to experience
what it's like to be a person looking into
all these different types of mirrors,
so I stuck my phone to the inside
of this Arnold Schwarzenegger mask.
So this is what it's like to be Arnold Schwarzenegger
looking into a concave mirror and this is what it's like
to be Arnold Schwarzenegger looking into a convex mirror.
But what's it like to be Arnold Schwarzenegger
looking into a mirror that's convex
and concave at the same time?
What do I even mean by that?
Well, I'm talking about a mirror that has a saddle point,
and just from the name, you probably already have an idea
what I'm talking about.
I don't have a saddle for illustration purposes,
but I do have a Pringle, and a Pringle is a saddle,
from a mathematical point of view
and only a mathematical point of view.
This is the saddle point of the Pringle
and if you travel left or right away from that point,
the height of the Pringle goes down.
In other words, the saddle point is a maxima,
but if I rotate the Pringle and do it again,
traveling left or right away from the saddle point,
the height of the Pringle goes up.
In other words, this point is a minima.
So depending on which plane through the Pringle you choose,
the saddle point is either a maxima or a minima,
but when you take the shape as a whole,
it's neither of those things.
But for the purposes of experiencing a mirror of this shape,
well, it's concave in this direction
and convex in this direction.
I was trying to figure out how the hell
do you make a saddle-shaped mirror,
and then I came across this video
on the AlphaPhoenix channel about putting
a mirrored surface on 3D prints, so I asked him
if he'd be interested in working on it together
and this is what he came up with.
Actually, it doesn't involve 3D printing at all.
It's really clever actually, and if you wanna find out
how it's done, I recommend you watch his video
on the subject, but actually check out
the whole AlphaPhoenix channel, because it's incredible.
The link is in the card and the description there.
So this is what it's like to be Arnold Schwarzenegger
looking into a saddle-shaped mirror.
As I rotate the mirror, the direction of stretch
also rotates, but the image itself
never ends up inverted.
I found that quite surprising initially,
because the mirror's concave in one direction,
and concave mirrors flip images, but then I realized
this mirror is acting like a makeup mirror.
The concave axis is magnifying the image
in that direction instead of flipping it.
So how can I get this thing to act like a spoon
and invert the image in the concave axis?
Well, it's all to do with the location of the focal point.
If the head is between the mirror and the focal point
of the mirror, then the head is magnified.
If the focal point is between the mirror and the head,
then the head is flipped.
With this mirror, the focal point is somewhere out here,
and if I tried to film from outside the focal point,
it just wouldn't work.
It's too small.
You wouldn't be able to see any of Arnold Schwarzenegger.
So what I need to do is bring the focal point
closer to the mirror.
In other words, I need to make the curvature
of the mirror more extreme, and fortunately,
this mirror is adjustable.
But then we run into a slight issue.
When you try to do that with, say, a playing card,
you actually don't get a saddle point.
You get either a trough in this direction
or a trough in this direction,
and you can flip between the two.
If you try hard enough, you can end up
with this situation where two corners
are pointing up and two corners are pointing down,
but then you've got this flat region in the middle.
An acrylic sheet is more elastic than a playing card,
so this isn't an issue for shallow curvatures,
but when I try to increase the curvature,
you end up with one axis more curved than the other
and you have this slight flat region in the middle.
Andrew Dubinsky and Michael Barson also offered
to make me a saddle-shaped mirror and I'm hugely grateful
for what they've produced.
They used a different method.
The traditional way to make a saddle point mathematically
is to construct a surface called a hyperbolic paraboloid.
The equation for that is Z equals X times Y,
which makes intuitive sense.
If X and Y are both positive or both negative,
then Z is positive, which pulls these two corners up,
but if X is positive and y is negative or vice versa,
then Z is negative, which pulls these two corners down.
But one thing that's really surprising
about a hyperbolic paraboloid is that
it can be constructed using only straight lines.
You can see that here in these two sweeps.
Look, you've got straight lines in this plane
and straight lines in this plane.
With that in mind, when Andrew and Michael
came across a bamboo placemat, they realized
they might have the perfect beginnings
of a hyperbolic paraboloid mirror.
You see, the strips of bamboo constrained
those straight lines, but with a twist,
we can get that saddle point.
You can see here how they created a cardboard frame
to lock it in the saddle-point configuration.
Then by heating sheets of PETG above
the plastic transition temperature,
they could lay that over the top and then clamp it down
with a similarly-manufactured top plate.
Once cooled, the sheet of PETG would be smoother
than the surface it cooled on.
Then that became the surface that the next sheet
is molded to.
Each step makes the surface smoother and smoother
than the original bamboo, and this is what we have.
So there's definitely a trade off
between these two methods of making a saddle-shaped mirror.
You can achieve a greater curvature with this one,
but it's a little more wobbly.
It's interesting to see how even very minor wobbles
in the surface can distort the reflection so much.
I mean, I should've known, that given that I made
two videos on the subject, one about caustics
and one about Chinese magic mirrors.
Link in the card and the description for them.
So now that the focal point is between
Arnold Schwarzenegger and the mirror,
look what happens when I rotate the mirror.
The head ends up on its side at one point,
which is quite surprising.
When the mirror is positioned like this,
with that concave axis at 45 degrees,
then you're flipping the image in this direction.
Look what happens when I do that in Photoshop.
Here's the flip, and that quite intuitively
puts the image on its side.
The mirror's convex in this direction,
so the image is squished in this direction.
Shrinking the image in this direction
is what skews the image.
That focal point actually does something brilliant
to the rays of light hitting the mirror.
Andrew and Michael used a laser line
and a fog machine to illustrate this.
See, there's a focal point, but as you rotate
the laser line around, the focal point
kind of untwists itself as it moves
from the concave plane to the convex plane.
How cool is that?
By the way, it's fiendishly difficult to keep the subject
in the mirror when you're watching a mirrored version
of your phone filming a mirror that behaves differently
when you tilt it, depending on how it's oriented,
and it's constantly overheating,
because you've stuffed the head to stop it collapsing.
Which leads me to the third saddle-shaped mirror,
which I commissioned animator peak Pete McPartland to make.
With this, there's no fiddling about with masks
and cameras and everything.
You just put a 3D model of a head in there
and stick the camera inside the head,
and I can easily adjust the parameters of the mirror.
So there's lots to play with here,
but first, as this is rendered digitally,
we can explore a little bit of the mathematics of the curve.
It's really cool to look at a contour map of a saddle point.
Look, as I cut through the mirror,
these two curves are hyperbolas, and look,
when you get to this point, they touch
and you end up with a cross, and actually,
that's true of all saddle points.
Every saddle point has contour lines that cross.
Interestingly, a hyperbolic paraboloid can also
be formed with the equation Z equals X squared
minus Y squared.
Look, you can see there's the X squared curve
and there it is sweeping over the minus Y squared curve.
So this is what happens when you approach the mirror
but don't go past the focal point.
Your face basically stretches out.
Then if you carry on past the focal point,
your face unstretches again.
That's counterintuitive to me, but actually,
if we rotate the mirror through 45 degrees
so that the concave axis is vertical,
we can see that the face starts flipped top to bottom,
and when you pass through the focal point,
it comes back 'round to the right way up.
And so actually what you're seeing in this orientation
is the face flipped horizontally and then unflipping
when you go through the focal point.
You also see copies of the face creeping in from the sides.
It might be more correct to say that those side faces
replace the middle face, as opposed
to the middle face flipping, and that's a brand new sentence
that's never been said before.
I used an iPhone to take a scan
of my face so I could put that in the render.
It's not a great quality model, but it's interesting
that you can use an iPhone's face ID tech in this way.
Here's what happens when you move your head up and down,
and when you move your head side to side,
and for completeness and because we can,
here's what happens when the mirror slowly flattens
and then goes the other way.
Probably the ultimate way to make a physical
saddle-shaped mirror would be with CNC.
There are some significant complications with that.
A lot of it's to do with the time and expense
of finishing the surface, though because the saddle shape
is made of straight lines, you could potentially
make this thing using wire EDM.
Link in the card and description to my video about wire EDM.
But potentially you could have a really nice surface
after the cut that doesn't need a huge amount of finishing.
If that ever happens, I'll make a follow-up video.
I will watch any movie starring Arnold Schwarzenegger.
"Terminator," "Terminator 2," "Twins."
Here's the thing, though.
I've never seen "Conan the Barbarian,"
even though it's on Netflix, and that's because
it's not on Netflix in the UK.
Brilliantly, if I search for, "Conan the," Netflix shows me
this show called "Barbarians."
It knows what I want.
It's just not giving it to me.
But it is available in the U.S.,
so if I turn on a U.S. location in Surfshark VPN,
the sponsor of this video, and then look,
I go back to Netflix, and there's "Conan the Barbarian!"
Yay!
A VPN encrypts all the web traffic leaving your device,
meaning that nobody on your Wi-Fi network,
nobody at your internet service provider,
nobody in your government, if that's something
you need to worry about, is able to see
the websites that you visit.
Typically, they wouldn't be able to see the content
of the websites you visit, because most websites use HTTPS,
but they would still be able to see the domain names
that you visit if you didn't use a VPN.
If that's important to you, then that's another
use case scenario.
To flip the Netflix use case on its head,
I was in France recently and it was a real pain in the arse
to do any kind of life admin, because any website
that I visited while I was there wanted to redirect me
to the French version, but by switching Surfshark VPN
to a UK IP address, everything worked as if I was at home.
That's just a couple of use cases.
What else to say?
Once Surfshark account can be used
on an unlimited number of devices.
There's a 30-day money back guarantee,
which means you can try absolutely risk free.
If you're interested, then there's a great promotion
on this one.
If you go to surfshark.deals/stevemould,
you'll get three extra months absolutely free.
The link is also in the description,
so check out Surfshark today.
I hope you enjoyed this video.
If you did, don't forget to hit subscribe,
and the algorithm thinks you'll enjoy this video next.
(funky jazzy music)