After this information let's turn back article about dimensions of objects on
the graph space.

As an
enter to the subject, ask yourself 'What is dimension of an
simple sphere?' If you are in those people that say
‘it has 3 dimension’. Please continue to read this
article.

Below
you can see mathematical expression of a sphere (curve sphere) that
has a center on the origin and has a radius 5.

0<a<Pi

0<b<2*Pi

fx(a,b) = 5*sin(a)*cos(b)

fy(a,b) = 5*cos(a)

fz(a,b) = 5*sin(a)*sin(b)

Now let's plot this curve
with 3DMath Explorer. If
we chose sample periods a and b as pi/15, we get
below pictures.

a(step)=pi/15

b(step)=pi/15

Maybe after looking these graphs, again you can
say these spheres have
3 dimensions. Yes vision of these spheres has 3
dimensions and the curve (curve sphere) can exists in a
3 dimensional coordinate system. But in reality is this
curve has 3 dimensions?

Let's
deal with the subject step by step (from simple to more complicated
samples - induction). For example let's take a simple function
curve. y=2*x+6 is a simple function curve and it exist on a plane
called XY plane. If you call above curve (the curve sphere) as a 3
dimensional curve, you may say this curve is 2 dimensional. Vision
of simple function curves became 2 dimensional, only if they draw in
a 2 dimensional coordinate system. But these curves also can be
drawn in 3 dimensional coordinate systems. In this case (when they
are drawn in a 3 dimensional coordinate system) their vision become
3 dimensional. Then should we say this curve has 3 dimensions? Could
a property of an object (whether it is in an imaginary word or in a
reel word) be change from place to place?

Let
us make some experiment on this sample curve with 3DMath Explorer
before continuing the subject. The y=2*x+6 simple function curve is
defined in a 3 dimensional coordinate systems as below.

-8<x<8

fx(x)= x

fy(x)= 2*x+6

fz(x)= 0

A similar curve that drawn in XZ Plane is
curve z =2*x+6 and its 3 dimensional coordinate systems defination is;

-8<x<8

fx(x)= x

fy(x)= 0

fz(x)= 2*x+6

A simple curve equation that has a 45-degree angle to all axis’s may be defined
as;

-8<m<8

fx(m) = m

fy(m) = m

fz(m) = m

Now let’s see all of this curves together on the same screen.

At the same time let’s define
XY, XZ and YZ planes that exist in this
graph as assistant plot elements. Their mathematical expressions in 3
dimensional coordinate systems are;

XY plane:

-8<x<8

-8<y<8

fx(x,y) = x

fy(x,y) = y

fz(x,y) = 0

XY plane:

-8<x<8

-8<z<8

fx(x,y) = x

fy(x,y) = 0

fz(x,y) = z

YZ plane:

-8<y<8

-8<z<8

fx(y,z) = 0

fy(y,z) = y

fz(y,z) = z

If we turn back the subject, what is the reason
of
wrong answers for questions about dimension of object in
mathematics? As you see, people mix up dimension of
objects with dimension of coordinate system they are
drawn in.

I hope we bring the problem to light. Now
questions that we should ask and try to find answers is ‘How
could we define dimension of a curve? Which criteria can we use?’
As you see
in above example we cannot define curve dimension with
coordinate system that they themselves defined, drawn and exist
in. We shouldn’t mix curve dimensions and coordinate
system dimensions.

Answer of this questions lies on properties of objects
like length, surface area and volume. Objects in math
space have or have not these properties with
their dimensions.

I think now we should study length, surface
area and volume properties. Height of a straight line is its length.
A rectangle has two lengths as width (wideness) and height.
Rectangle borders a area in a plane and has a surround/circle
length. A rectangle prism has three lengths as
width (wideness), height (length) and height (depth) properties. A
rectangle prism borders a volume in a space and
a total surface area with 6 surface that
it has.

Now
let’s group features (properties) that objects have with
their dimensions.

One Dimensional Objects

• Have a length.

• If they haven’t got a certain
start and stop points (or they have joined start and stop point (or
edge)) like circle or rectangle, then we mansion about their having
surround/circle length.

• Haven’t got surface arias and
volumes.

• Couldn’t be exist in real
word. They could only be in mind or be vision that projected to (or
reflect from) a paper and a computer display.

Two Dimensional Objects

• Have surface areas.

• could be mansion about having
surround/circle length. (might have surround/circle length)

• Haven’t got volume.

• Couldn’t be exist in real
word.

Three Dimensional
Objects

• Have volumes.

• be mansion about having width
(wideness), height (length) and height (depth, thickness)
properties.

• Have got a volume.

• Could be exist in real
word.

The last items must be take your attention.
The real world is 4 dimensional with time dimension. In
this world, objects exist in a 3 dimensional space
with a cross-section of dimension time that we call
it as ‘now’. To exist in 3 dimensional real world
space, a object must have a volume. 1 and 2 dimensional objects
couldn’t be existed in real world, as they haven’t got
volume.

Henceforth we can make this classification:
objects that only have a height as length are 1 dimensional,
objects that have surface areas but don’t have volume are 2
dimensional and objects that have volume are 3
dimensional
.

After making this classification, if
we turn back to the question about dimension of
sphere, however this sphere graph is a 3 dimensional
graphs as it drawn in a 3 dimensional coordinate system,
in fact the sphere is a 2 dimensional object. Because this sphere
has a surface area but it do not have a
volume.

Now, even when I have been writing this
article, I am hearing objections. For example you may say ‘How does
a sphere not has a volume? Isn’t there a formula about sphere
volume’. Yes, of course there is a volume formula of sphere. But
when we mansion about this formula, we also use a statement as
‘volume of a filled sphere’
.

A simple sphere
curve borders a volume with its surface area in
the space (in math. space). But in fact it
is only a cover, a shell. As nothing that doesn’t
have a thickness (height, depth) could have a volume and this cure
sphere doesn’t have a thickness, it also couldn’t have a
volume.

(Here I want to make explanation in parenthesis not
to get headache by your possible questions that come to your head
letter on: I said before no objects that haven’t volumes could have
exist in real world. Now I am saying this 2 dimensional sphere
(curve sphere) do not have a volume. So this sphere couldn’t be
exist in real world. Then how an object that in reality do not exist
could border a volume in the space. Yes, a 2 dimensional object
couldn’t exist in real world and as a result of this couldn’t border
anything. But when the space in question is the math. space (graph
space),
1 and 2 dimensional objects could be exist in
this space. In fact they are real in this
imaginary world (in math. space). So 1 dimensional objects like
circle and rectangle could border an area and 2 dimensional objects like
this 2 dimensional sphere could border a volume in this
space.)

To add a third dimension to 2
dimensional sphere object above, we must give it thickness (depth) dimension. We
can do this by making radius 5 as a variant.
0

0<a<Pi and a(step)=0.3

Pi/5<b<2*Pi and b(step)=0.3

4<r<5 and
r(step)=1

fx(a,b,r) = r*sin(a)*cos(b)

fy(a,b,r) = r*cos(a)

fz(a,b,r) = r*sin(a)*sin(b)

In the below you can see inside of this 3D
sphere and another
sphere that stays outside of this sphere. (But from
this picture we can not understand whether this white sphere
has a thickness or not.)

Now we let’s see striped view of
the graph instead of the polygon view. (We notice
that the white sphere does not have a thickness.
So we say this sphere is 2 dimensional.)

0<a<Pi and a(step)=0.3

Pi/5<b<2*Pi and b(step)=0.3

4<r<5 and
r(step)=1

fx(a,b,r) = r*sin(a)*cos(b)

fy(a,b,r) = r*cos(a)

fz(a,b,r) = r*sin(a)*sin(b)

You must notice another thing in this equation.
Variants a, b and r in the equation are loop variants for 3DMath
Explorer. Moving from this fact we can say ‘Dimension number
of a curve that drawn with 3DMath Explorer are equal to number of
variant that it has’
.

Lastly let’s examine dimension of basic elements point, line, plane and
curve that exist in graph space

Point

• Does not have a dimension
(they are only coordinates),

• Could be defined in all
coordinate systems.

Line

• has only one dimension,

• require (need) a coordinate
system (or a plane) that at least has two axis’s (or two dimension)
to be drawn,

• Has a length.

Plane

• has two dimension,

• requires (need) a coordinate
system (or a plane) that at least has three axis’s (or two
dimension) to be drawn. (Exception is the plane may be smooth and be
just on the very same two-dimensional coordinate system. In fact
two-dimensional coordinate systems also are planes may define as a
two-dimensional curve and exist in a three dimensional coordinate
system (in a graph space) or in itself, in a two-dimensional-plane
coordinate system.)

• Has surface area and might
have surround/circle length.

Curve

• May be 1,2,3 and 4 (and n)
dimensional.

• required (needed) axis number
to be drawn change with their dimensions

• If they are one dimensional
have a length, if they are two dimensional have a surface aria and
if they three dimensional have a volume.

As a human being and a subject like table and
chair, we live in a 4 dimensional word. With a time dimension as a
4th dimension, we exist in this real coordinate system. But we are 3 dimensional objects just have a life and spend
it in a 4 dimensional word.

Thank you for your interest and bother to read
such a long article. If I manage to tell you something
really interesting, that makes me happy.

I wish you a good
life in this 4 dimensional word.

Written by Dursun TEBER, TEBER.biz, Ltd. (09.10.2002, Istanbul/Turkey)