- You've all seen the pattern of light
produced when the sun shines through a glass of water,
but how would you need to change the shape of the glass
so that the pattern of light
became a recognizable image like this?
This piece of acrylic is wobbly in just the right way
to produce the image of a lion,
and look how the image warps and changes
when I change the orientation of the acrylic.
I've also got this tiny mirror
that projects the word FIRE from its reflection,
and the word gets jumbled up
as it gets too close to the wall.
This is what it's like
to look through the acrylic at a checkerboard.
There's a hint of the lion shape there and the words,
but you really couldn't discern from looking at the surface
what you might expect the image itself to be.
Here's one that shows the face of Alan Turing.
Here's a brain.
These different images are achieved
by manipulating the shape of a lens,
and it's a massive mathematical challenge.
You've definitely seen accidental versions
of this sort of image
caused by undulations in transparent materials.
The most obvious example is ripples in a pool.
See how the bright lines move around
as the waves on the surface of the pool change shape?
And you can understand those bright spots
in terms of how a lens works,
so you have parallel rays of light coming from the sun
shining on the surface of the pool,
and this curved part of the wave focuses the light,
and if the curvature is just right,
those rays will focus exactly onto the bottom of the pool,
and you'll have an incredibly bright line.
If the surface is either too curved or not curved enough,
the bright lines won't be as sharp,
a bit like how you get a blurry image in a photo
if it isn't focused correctly.
Other places you might see this sort of thing,
a glass of water,
sun shining through that weird glass ornament
at your Nan's house,
looking inside an empty mug.
This video about Penrose unilluminable room
link in the description,
or check out the inside of your wedding ring on a sunny day.
You'll often see a sharp edge like this,
and that's where the light rays change direction.
What I mean is
if you get a laser pointer and sweep it over the surface,
you could watch as the reflected point of light moves
and see that at some point it changes direction.
There's an inflection point.
All those inflection points together form the curve,
and these types of curves are called caustics.
In fact, you may have heard of these images
referred to as caustics,
even though they don't contain
any of those inflection points.
That's probably because some of the early work
with these sorts of lenses
was attempting to produce bright lines,
like you see in this one of the brain.
This one really is made of caustics.
Actually, caustics are related
to those curves you can make with pegs and strings.
This one, in particular,
is like looking at the inside of a mug,
but anyway, the question is
how can you create a lens,
either a glass lens or a mirror lens,
that produces the image that you want?
Suppose you're starting with a picture of Alan Turing,
and you want to be able
to project that picture of Alan Turing onto a wall
by shining light through a weird, wobbly lens.
The more you think about it,
the more you can see the challenge there.
Like, here's a dark region on the picture of Alan Turing,
so I want to divert light rays away from there,
but then, those light rays are going to scatter
to other parts of the image,
and maybe I don't want those parts of the image
to be lighter,
so we want to shape the lens
so that we redistribute the light
in exactly the way that we want to,
and brilliantly, this relates to an area of mathematics
called optimal transport.
A good example of an optimal transport problem is
I want to create a sand sculpture, like this, for example,
and what I start with is a level tray of sand,
so to make my sculpture, I need to move sand around,
but I want to do it optimally.
I want to move the fewest grains of sand
the shortest distance.
As a side note,
the optimal transport problem really came into its own
in World War II
when generals were thinking
about moving dirt and rubble around
while using the least amount of energy to do it,
and they hired mathematicians to figure that out,
but anyway, once I have my sculpture,
there are shallower parts of the sand
and deeper parts of the sand,
and you can see how that would be analogous
to the creation of an image using wobbly lenses.
I start off with a flat, even coverage of light
from my point source,
and I want to move that light around
to create bright regions and dark regions,
like the deep regions and shallow regions of the sand,
and because I want my lens to be as simple as possible,
I want the light to be moved around as little as possible,
just like with the grains of sand,
and so it's an optimization problem.
It's an optimal transport problem.
A research scientist called Matthew Brand
did a lot of the work applying the optimal transport problem
to these wobbly lenses,
and he really helped me out with this video.
He also helped me out
with an upcoming video about handmade holograms.
That should be out next month,
so make sure you're subscribed
if you want to catch that one.
The mathematics of the optimum transport problem
are beyond the scope of this video,
but I wanted to show you one part
that's really interesting: power diagrams.
This power diagram represents this distribution of light.
In a power diagram,
each region has an equal amount of power,
so brighter regions are smaller,
and darker regions are larger,
but if we want this distribution of light instead,
we need to update the power diagram.
Look, as it changes shape,
you can see the dark regions
that represent Einstein's eyes get larger,
and the brighter regions get smaller.
Optimal transport involves a lot more than that,
but once you have your solution,
once you know that you want light from here to go there,
light from here to go there and so on,
you code that into the curvature of the lens,
so suppose I've decided that the light that lands here
needs to end up over here?
Well, I curve the surface of the lens appropriately
at that point where the ray of light is leaving the lens.
You do that all over the surface, and you have your lens.
It's more complicated than that,
but that's the simplified version.
A computer scientist by the name of Mark Pauly
has contributed a lot
to the understanding of how these things work.
He very kindly sent me the lion lens
from his company, Rayform.
He also sent me the mirror,
which would normally be part of a ring
with a custom message.
Link in the description
if you want to get one of those for yourself.
They're not paying me to say that, by the way.
I'm just really grateful for their help with this video.
I really like this image of the Olympic rings.
What's nice about it is it's in color,
and what's nice about it being in color
is that when you look at the lens,
you can see where the light comes from,
and in fact, with a simple image like this,
you can kind of make sense of it.
Like, to get the black background behind the rings,
all the light that would normally pass through there
needs to be sent to one of the rings,
so the colors of the rings
stretch out to the very edges of the transparency.
When you look at the image
as it appears back on the lens in this way,
it's called a pullback,
and this isn't the only place you'll find pullbacks.
This thing is called the sensory homunculus,
and it shows how sensitive to touch
different parts of the body are
by changing the size of those parts of the body,
and it's also a pullback.
One thing I really like about these lenses
is the transition between two recognizable things
that really don't go together,
like this kind of pattern you recognize
from all those times
you've seen light shining through wobbly glass
or light reflecting off a pool or whatever.
This kind of light movement is familiar.
Then it unexpectedly transitions into a recognizable image,
and that's a really fun moment.
I wanna talk about the economics of printers
because it's really interesting and really annoying.
Printer ink is extortionately expensive
because of the business model that the manufacturers use.
They sell the printers at a loss
to entice you into buying one,
and then you're locked into buying their printer ink,
which is sold at an inflated price.
Eventually, you end up paying more.
It's a really annoying business model,
and it has a name, the razor and blades model.
It's called that because it was made famous
by the manufacturers of shaving products.
The handle is sold at a loss,
and the blade cartridges have a huge markup.
Wouldn't it be nice
if you could just pay the right price for things?
Splash out on something that's well built
and then pay pennies for the consumables that go with it
instead of this weird pseudo-subscription model?
That's true as much for printers as it is for razors,
which is why I'm happy to have Henson
as a sponsor of this video.
They don't make printers, but they do make safety razors.
You pay an honest price, once,
for this beautifully engineered safety razor,
and then you can buy your own quality blades for pennies.
The Henson AL13 is what you get
when a family-run aerospace machine shop
pivots to making its own product.
It's CNC machined to aerospace standards,
and because it's all metal, the blades are well supported,
unlike with the multi-blade plastic cartridge razors,
and it's just nice to own something that's well made
and that you will make good use of.
Never mind the fact that the total cost of ownership
would easily beat cartridge razors after just a few months.
I must admit,
I've never actually used a safety razor before,
so I was a little bit apprehensive,
but as you can see, I tried it just now,
and of course, there's nothing to be scared of,
and actually, I'm really happy with the results.
Go to hensonshaving.com/stevemould,
and use the promo code Steve Mould at checkout
to get 100 free blades with your purchase of a razor.
Just make sure both products are in the basket
when you apply the code.
The link is also in the description,
so get yourself a precision-engineered safety razor today.
I hope you enjoyed this video.
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