The fourth dimension

The 4th Dimension

• What is the fourth dimension ?

• What do we know about it ?• How can we "see" it ?• The finite universe theory

What is the fourth dimension ?

In this part, we'll focus on the fourth SPATIAL dimension.

If the 4th dimension is time, we'll talk about spacetime.

What is the fourth dimension ?Easy algebraic construction :

2D : vector : u = (x,y)distance : scalar product :

x 2 y 2

u v xuxv yuyv

What is the fourth dimension ?Easy algebraic construction :

3D : vector : u = (x,y,z)distance : scalar product :

x 2 y 2 z2

u v xuxv yuyv zuzv

What is the fourth dimension ?Easy algebraic construction :

4D : vector : u = (x,y,z,w)distance : scalar product :

x 2 y 2 z2 w2

u v xuxv yuyv zuzv wuwv

What is the fourth dimension ?

Geometric point of view :Can we build a vector , orthogonal to each of the vectors ?

Impossible in !!!ℝ3

r w

r x ,

r y ,

r z

What we know about it :

• No difficulty in analyzing, describing, and cataloging the properties of all sorts of 4-d figures

• Equivalents of 3-d figures in 4-d

Sphere Hypersphere

5 Platonic solids 6 Polytopes :

• the tesseract (eight cubes, meeting three per edge)

• the 16-cell (16 tetrahedra, meeting four per edge)

• the 24-cell (24 octahedra, meeting three per edge)

• the 4-simplex (five tetrahedra, with three tetrahedra meeting at an edge)

• the 120-cell (120 dodecahedra, meeting three per edge)

• the 600-cell (600 tetrahedra, meeting five per edge).

How can we picture it ?

• Impossible to grasp 4D-objects in our 3D-space.

• What we CAN grasp: intersection of 4D-objects with 3D-spaces.

How can we picture it ?

• Projection :→ studies how 3D-sets and lesser dimension sets interact.→ examples (flatland, Plato's cave)http://www.youtube.com/watch?v=lwL_zi9JNkE&feature=fvw

→ allows us to visualize 3D-objects on 2D-surfaces (films, pictures,visual scope).→ analogy with 4D.

How can we picture it ?

Example : the hypercube

Defined by the formula:

H x1,x2,x3,x4 R4 /i 1,2,3,4 x i 0,1

How can we picture it ?

Example : the hypercube

projection of a cube on a 2D-plane

projection of a hypercube on a 3D-space

How can we picture it ?• Shadow :→ closely related to projection.→ 3D-objects cast a 2D shadow.

→ by analogy, a 4D-object lit in the 4th dimension would cast a 3D-shadow.

How can we picture it ?

Example : the hypercube 3D-shadow of a hypercube

A tesseract can be subdivided into smaller 4-d blocks in the same way that a cube can be divided into smaller cubes, or a square into smaller squares.

More stuff about the hypercube

A 4-d object needs to be rotated for us toappreciate its higher dimensionality.

The 3-sphere

• Aka the glome• Intersection with our 3D-space: sphere

The 3-sphere

Stereographic projection :

• The stereographic projection is a particular mapping that projects a sphere onto a plane.• The projection is defined on the entire sphere, except at one point — the projection point.• Where it is defined, the mapping is smooth and bijective. It preserves angles.

The 3-sphere

Stereographic projection :

The 3-sphere

Stereographic projection of the 3-sphere:

The 3-sphere

Stereographic projection of the 3-sphere:

• Projection in a 3D-space

• Red lines: paralells Blue lines: meridians Green lines: hypermeridians

• http://www.josleys.com/articles/ams_article/images/S3_01_s.mov

4th dimensional objects

• Klein Bottle • The 24-cell polytope or octacube

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