We take this curved 12 dimensional space and transform it into a flat four dimensional space.

We take this curved 12 dimensional space, and transform it into a flat, four dimensional space.

Often curves in two dimensional (plane curves) or three dimensional (space curves) Euclidean space are of interest.

It s truly a three dimensional space.

I couldn't imagine three dimensional space.

It's truly a three dimensional space.

One idea is that our three dimensional universe may be embedded in a high dimensional space, just as you can imagine on these sheets of paper.

This is still a two dimensional space.

When n 2, the set of all such locations is called two dimensional space or bi dimensional space, and usually is thought of as a Euclidean space.

In fact, they live in a 12 dimensional space.

So, a one dimensional space is just a line.

We won't go into three dimensional space in geometry class.

And here we're dealing with a two dimensional Euclidean space.

This is going to map within this two dimensional space.

Every b means all the bs in three dimensional space.

Now, take into account the fact that we live in a three dimensional universe, OK, a three dimensional space.

Kismet has an underlying, three dimensional emotional space, a vector space, of where it is emotionally.

do the linear combinations of the columns fill three dimensional space?

combinations may very well fill out the whole nine dimensional space.

So it's a deep think about it in a big three dimensional space, very deep three dimensional space with herds of tuna, whales, all kinds of deep sea marine life

Each node in this tree represents a region of the two dimensional space.

And now it starts moving in three dimensional space so up or down.

So they all split, and give a nice pattern in six dimensional charge space.

So if you mapped two dimensional space, you're dealing with a vector valued function.

and each one would be a vector in nine dimensional space and we would

So it's a deep think about it in a big three dimensional space, very deep three dimensional space with herds of tuna, whales, all kinds of deep sea marine life like we've seen here before.

And that four dimensional space consists of X, Y, Z and then the yaw angle.

For a simplified notion of a wormhole, visualize space as a two dimensional (2D) surface.

And that four dimensional space consists of X, Y, Z, and then the yaw angle.

A two dimensional space is defined by two perpendicular lines, which describe a flat plane

But the whole study of linear algebra is abstracting these ideas into multi dimensional space.

And of course, if space had 9 dimensions rather than three, there'd be plenty of space for other three dimensional surfaces that appeared,

I create mathematical objects, symmetrical objects, using Galois' language, in very high dimensional spaces.

And the power of Galois' mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two dimensional, three dimensional, all the way through to the four or five or infinite dimensional space.

And the power of Galois' mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two dimensional, three dimensional, all the way through to the four or five or infinite dimensional space.

So the electromagnetic and weak forces are described by this pattern of particle charges in two dimensional space.

Time and space, according to Einstein's theories of relativity, are woven together, forming a four dimensional fabric called

And these patterns of points that I've shown for you actually represent symmetries of this high dimensional object that would be warping and moving and dancing over the space time that we experience.

However, if we work in eight dimensional charge space, then we can assign unique new charges to each particle.

It makes a two dimensional image look more three dimensional.

Three dimensional shapes and the space they create in the middle creates a new shape, the answer to the sum.

And a three dimensional space adds a third perpendicular line, which gives us height and the world we're familiar with.

So it's essentially a way of measuring how much space something takes up on kind of a two dimensional surface.

length of things in two or three dimensional space, but it becomes very abstract when we get to n dimensions.

nine dimensional space, and we would be trying to find the combination that hit the correct right hand side b.