From the 4th DimensionAn introductory lecture that describes the

mathematics behind Field Dynamics

• What do you see?

• Creating a 4D geometry

• Thinking outside the 4D box

• Field dynamics

Look at the stars. Where do they appear to be? They

all appear to lie on a surface. You can’t distinguish

between objects nearby and far away.

The signals that reach your eye come from very different times – some come from

many centuries ago.

What do you see?

In absence of relative spatial information, like color and texture, you don’t know

where an object is.

What do you see?

Sometimes perception fools you…

What do you see?

What do you see?

An event occurs at time –t.

Age is measured in the negative t direction and time t moves in the positive t direction.

An event reaches your eye at time t’. The signal from the event travels towards you at speed c.

The relative time between the event and when you see it is t + t’. The distance traveled is

r = c (t + t’).

What do you see?

You don’t directly measure the time of an event, its speed, or its

distance from you. You only record the time t’ it

reaches you.

The communication line

ct’ = r – ct

What do you see?

ct’ ct

r

t’ t = 0 – t

Creating a 4D geometry

Goal:

To Create an Ordinary 4D Geometry

Question:

What is an Ordinary 4D geometry?

Answer:

A Geometry that Bases its Length

on the Pythagorean Theorem

Creating a 4D geometry

a

b

c

a – b

c2 = a2 + b2

The Pythagorean Theorem uses area = base x heightIt doesn’t make sense when a coordinate is temporal.

The 4th dimension of an ordinary 4D geometry is created using the communication line.

The time t on the communication line is rotated 90 degrees to create a 4th perpendicular coordinate.

To do this, we first need to review complex numbers.

Creating a 4D geometry

Historical perspective

Geometry once consisted of only zero and positive

numbers. The construction of geometric shapes

requires only line segments.

When the coordinate system was introduced the

need arose for rays. Rays accompanied an

acceptance of negative numbers and complex

numbers.

Creating a 4D geometry

The Ray

R = (x, y)

x

y

Creating a 4D geometry

R1 + R2 = (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

aR = a(x, y) = (ax, ay)

Rays can be added and lengthened in any order.

They obey the five rules of ordinary arithmetic

(associative rules of addition and multiplication, commutative rules of addition and multiplication, distributive rule).

Thus, rays can be manipulated like numbers.

This is the foundation of real vector algebra.

Creating a 4D geometry

R1

R2

R

R

aR

(a + ib)(x, y) i means rotate 90 degrees

It’s standard to write (x, 0) as simply x and (0, y) = i(y, 0) as simply iy so a ray (x, y) can be viewed as

(x, y) = (x, 0) + (0, y) = x + iy

This is the foundation of complex algebra and this is what allows the operation i to be regarded as a number.

Rays can be rotated, added, and lengthening in any order. They satisfy the five rules of arithmetic. This produces the general operation

Creating a 4D geometry

x

yR = (x, y)

iR = (–y, x)

….now we rotate time and create the 4th geometric coordinate.

ct’ = r – ct = r + i2ct = r + ix4

x4 = ict

Creating a 4D geometry

22

4rx

R = (r, x4) = ct’

r

x4

224 rx

24

2

4

24

2

24

2

4'

xr

xi

xr

rxr

ixrctrct

This development showed that physical reality can be represented by an ordinary 4D geometry.

We saw why the Pythagorean Theorem can be used with a temporal coordinate, and found how

geometric time x4, conventional time t, and the measurement t’ are related to each other.

Creating a 4D geometry

R = (r, x4) = ct’

r

x4

Thinking outside the 4D box

x2 = 0 x1 = 1 x2 = 1 x1 = 0

b)

Thinking outside the 4D box

Thinking outside the 4D box

The faces of a 4D cube are 3D cubes. The faces consist of

8 3D cubes – a positive and

negative cube for each axis.

In 3D, the normal

to a surface is

In 4D, the normal

to a volume is

Thinking outside the 4D box

kjijki CBeA

lkjijkli DCBeA

The two most common 3D vector operations are the dot product and the cross product. The dot product works in 4D, too. Here’s how the right-hand rule and the cross product extend to 4D.

The ordinary 4D geometry

discussed in this talk is the

foundation of field dynamics.

Field dynamics

The Problem-Solving Process inField Dynamics

• Formulation

Set-up: A constitutive topology is set up.

– particles, boundary conditions, types of interactions (electro-mechanical)

– order-reduction (irreversible processes)

Transition: The system is drawn (free-body diagram).

• Solution

Equation: Governing equations are listed.

Answer: Equations are solved.

Knowledge: Insight is gained.

Field dynamics

T H E

E N D