So what is this hyperbolic geometry that corals and sea slugs embody?

Before hyperbolic geometry, mathematicians knew about two kinds of space Euclidean space, and spherical space.

The frilly crenulated forms that you see in corals, and kelps, and sponges and nudibranchs, is a form of geometry known as hyperbolic geometry.

The discovery of hyperbolic space ushered in the field of mathematics that is called non Euclidean geometry.

Margaret Wertheim leads a project to re create the creatures of the coral reefs using a crochet technique invented by a mathematician celebrating the amazements of the reef, and deep diving into the hyperbolic geometry underlying coral creation.

So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry.

Even mathematicians, who in some sense are the freest of all thinkers, literally couldn't see not only the sea slugs around them, but the lettuce on their plate because lettuces, and all those curly vegetables, they also are embodiments of hyperbolic geometry.

Create and study hyperbolic 3 manifolds

So what is this impossible hyperbolic structure?

CaRMetal interactive geometry

Now, that is only a little bit hyperbolic.

Interactive geometry viewing program

Extract geometry from the images.

RG Hanging cable takes on the form of a hyperbolic cosine.

They have a very rich, intertwined geometry.

It turns out that the natural world is full of hyperbolic wonders.

We're just seeing the coarse scale geometry here.

The Greeks invented a theoretical model of geometry.

Africans can't possibly really be using fractal geometry, right?

So in Manghetu sculpture, you'd see this recursive geometry.

This is exploring geometry and the rhythm of shape.

And so, too, we've discovered that there is an infinite taxonomy of crochet hyperbolic creatures.

We use a modified inkjet printer, and we print geometry.

Geometry, fantastic. You know, quantum mechanics these are cool things.

The saddle shaped structure is also known as a hyperbolic paraboloid surface, or an anticlastic surface.

This is the result of computational geometry, gesture recognition, and machine learning.

SP ... These four things. Prospect theory, hyperbolic discounting, status quo bias, base rate bias. They're all well documented.

And primary visual cortex sees just simple geometry, just the simplest of shapes.

If geometry is the science of space, what is the science of time?

And that's exactly what happened when Daina Taimina in 1997, showed that you could crochet models in hyperbolic space.

But it's reassuring that at the heart of this mathematics is pure, beautiful geometry.

This sort of geometry revolutionized mathematics when it was first discovered in the 19th century.

And then I went in, and measured all the geometry because I figured, hey, biomimicry.

I saw my geometry teacher in high school, Mr. Rucell's wry smile under his handlebar mustache.

And where I was bad was algebra. And I was never allowed to take geometry or trig.

What physics tells us is that everything comes down to geometry and the interactions of elementary particles.

The first ingredient is materials and geometry, and that needs to be tightly coupled with the energy source.

And the power of mathematics is often to change one thing into another, to change geometry into language.

And a surprising aspect of it is that very often, the rules of this geometry are extremely short.

Gigantic mistake I'm finding a lot of kids who need to skip algebra, go right to geometry and trig.

The toroidal geometry made for a very efficient building every beam in this building is the same radius, all laminated wood.

Together with quantum mechanics, the geometry of this shape could describe everything about how the universe works at the tiniest scales.

Talking about geometry and everything, if you take the circle and you split it in half can you run S Curve music?

The problem, as we see it, has to do with a single, simple word space, or a single, simple phrase real world geometry.

So, very interesting parallels between the biological and the engineering worlds. And interestingly, this turns out the stomatopod saddle turns out to be the first described biological hyperbolic paraboloid spring.