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From the
th
4
Dimension
An introductory lecture that describes the
mathematics behind Field Dynamics
• What do you see?
• Creating a 4D geometry
• Thinking outside the 4D box
• Field dynamics
What do you see?
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?
The communication line
ct’ = r – ct
–t
t=0
ct’
ct
r
You don’t directly measure
t’
the time of an event, its
speed, or its distance from
you. You only record the
time t’ it reaches you.
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
b
a–b
c
a
2
c
=
2
a
+
2
b
The Pythagorean Theorem uses area = base x height
It doesn’t make sense when a coordinate is temporal.
Creating a 4D geometry
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
y
R = (x, y)
x
Creating a 4D geometry
R1 + R2 = (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
aR = a(x, y) = (ax, ay)
aR
R
R2
R
R1
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
Rays can be rotated,
added, and lengthening
in any order. They
satisfy the five rules of
arithmetic. This
produces the general
operation
(a + ib)(x, y)
iR = (–y, x)
y
R = (x, y)
x
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.
Creating a 4D geometry
….now we rotate time and create the 4th
geometric coordinate.
x4
R = (r, x4) = ct’
x42  r 2
r
ct’ = r – ct = r + i2ct = r + ix4
x4 = ict
Creating a 4D geometry
x4
R = (r, x4) = ct’
x r
2
4
2
ct '  r  ct  r  ix 4
 r 
2
x42


r
x4


i
2
2
2
2
 r  x4
r  x4 
r
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.
Thinking outside the 4D box
b)
x2 = 0
x1 = 1
x2 = 1
x1 = 0
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.
Thinking outside the 4D box
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.
In 3D, the normal
to a surface is
Ai  eijk B j Ck
In 4D, the normal
to a volume is
Ai  eijkl B j Ck Dl
Field dynamics
The ordinary 4D geometry
discussed in this talk is the
foundation of field dynamics.
Field dynamics
The Problem-Solving Process in
Field 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.
THE
END