- 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. If you did, don't forget to hit Subscribe, and the algorithm thinks
you'll enjoy this video next. (logo whoops) (funky music)
