level to the shape of gigantic spiral galaxies. In all
cases these forms are associated with evolutionary
growth, should it be a new layer of atoms in a crystal,
a living organism, a tropical storm, or a galaxy.Fibonacci Petals on flowers: On many plants,
the number of petals is a Fibonacci number. Here are
some examples:3 petals: lily, iris (often lilies have 6 petals formed
from two sets of 3)
5 petals: buttercup, wild rose, larkspur, columbine
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55 or 89 petals: michaelmas daisies, asteraceae
family.Some species are very precise about the number of
petals they have, but others have petals that are very
near those above, with the average being a Fibonacci
number.Lesson 6. Dimensions
by Dragon SingingOne very important concept for a Wizard to under-
stand is the physical concept of dimensions. While
whole volumes of mathematics and physics have been
devoted to the study and understanding of dimen-
sions, the easiest way to think about them is as direc-
tions of possible movement.
Mathematicians (who are their own kinds of Wiz-
ards, make no mistake) begin their thoughts about
dimensions by imagining a zero-dimension object,
which would be a Point. You know how people talk
about “a point in space”? Well, that’s exactly what it
is: a location, like an infinitely tiny dot. A Point has
zero height, zero width, zero length. It can’t move any-
where.
If you have two Points, you can draw a Line be-
tween them. A Line is a one-dimensional object. A
Point on a Line can only move forward and backward
along the line: not side to side, not up and down...
When you think about it, we’re already in terri-
tory that’s pretty mysterious: because a Point has no
width, length or height, no matter how short a Line is,
there is always space for an infinite number of Points
between any two Points. So how does anyone ever
get anywhere? You start from New York and head for
California (or, for that matter, from one side of the room
to the other), and there are an infinite number of points
between. How do you ever cross them all? But you
do...
But wait: it gets better.
If you add another Point (one which isn’t exactly
lined up with the other two, which would just extend
the Line), you add another direction of possible move-
ment, creating a two-dimensional object known as a
Plane, like the surface of a sheet of paper. Just
the surface, mind you: no thickness, none at
all. A Point on a Plane can move left, right,
forward, backward...any direction except
up and down.
Add another dimension, and we’re al-
most in the situation we humans live in.
Three dimensional objects have vol-
ume, and are known as Solids. A solid
can be a ball or a cube or anything that has
height, width, depth and thickness.
Now we’re getting somewhere! But the day-to-
day world as we know it doesn’t exist just in three
dimensions. You don’t just move up, down, sideways,
forward and backward: you also move forward through
Time, which is the fourth dimension. I won’t go into
why Time only goes forward here: there’s a reason,
but it’s complicated, and if you’re interested, you
should go investigate and find out (researching ar-
cane knowledge is one of the most basic skills a Wiz-
ard must master!)
But there you have it: the Universe as we know it
exists in four dimensions: three of space, plus Time.
And the Universe, as we have learned, is expanding...
which means that there must be a fifth dimension for it
to expand into!
There are many physicists who now believe that
our Universe may have extra dimensions as well, which
have been compressed into subatomic spaces so small
that we are never aware of them. The more we under-
stand about our Universe, the more we come to see
that there may very well be alternate layers of reality:
other Universes, other versions of reality.Moëbius Strip
There are very strange and fascinating things you
can do with the concept of dimensions. Take, for ex-
ample, the Moëbius Strip. It’s easy to make: just take a
strip of paper, put in a half-twist and tape the ends
together. No big deal, right?
Wrong! Take a pencil, put it down
at any point on the paper strip, and
draw a line along one side. How
many sides does the strip have?
That’s right! Only one. The Moëbius Strip is a two-
dimensional, one-sided object. It’s all one sur-
face. But wait, there’s more!
Cut the strip down the
middle. What happens?Now cut the strip
down the middle again.
Surprise!Course Five: Spectrum, Part 2 275
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