# Fractal dimensions

Was: Why CAD is it needed in our classes?

``````We are fractal beings and doing fractal things are natural for us.
A sculptor refining a surface, can achieve in one minute what could
take computer a day to compute.
``````

Is there a difference between a fractal dimension, and “human
imperfection” when creating jewelry? What distinguishes the
acceptable fractal dimensions from other imperfections of a man made
object; or is imperfection just a poorly executed fractal dimension?

``````We are fractal beings and doing fractal things are natural for us.
A sculptor refining a surface, can achieve in one minute what
could take computer a day to compute. Is there a difference between
a fractal dimension, and "human imperfection" when creating
jewelry? What distinguishes the acceptable fractal dimensions from
other imperfections of a man made object; or is imperfection just a
poorly executed fractal dimension?
``````

People seriously want to go into fractals?

I’d have to dispute that we are not fractal beings, as fractals are
the most precise things in the universe.

A human being cannot ever reproduce that level of precision and
repetition, computers can do this with the precision needed to
execute the algorithms that produce the fractal sets.

However I would agree that a sculptor can rub his hand over a piece
of clay, and produce something really nice.

“A sculptor refining a surface”, can mean anything, a computer that
operates a mill can refine a flat surface to almost dead smooth, it
may take time, but a person can’t do it.

There must be an example in mind when the statement was made. I
wonder what it is?

Regards Charles A.

``````Is there a difference between a fractal dimension, and "human
imperfection" when creating jewelry? What distinguishes the
acceptable fractal dimensions from other imperfections of a man
made object; or is imperfection just a poorly executed fractal
dimension?
``````

Imperfection is in the eye of the beholder.

Fractals are something else again.

Elliot Nesterman

``````Is there a difference between a fractal dimension, and "human
imperfection" when creating jewelry? What distinguishes the
acceptable fractal dimensions from other imperfections of a man
made object; or is imperfection just a poorly executed fractal
dimension?
``````

Fractal is an object made of self-similar, but smaller objects.

Take a broccoli. Broccoli is fractal. We can take a bunch of
broccoli and start taking it apart. We quickly notice that all parts
regarding of size look alike. Not the same but similar. That is
necessary characteristic for fractal.

Is there an ugly broccoli ? No. Is there beautiful broccoli ? No.

Broccoli is simply interesting. One can study it’s shape for hours
and will keep finding variations. Nothing is the same, but similar,
and every element will have the same fractal dimension.

What is fractal dimension? Staying away from mathematics, draw a
square on the paper. We can easily compute the area and the perimeter
of this square.

Replace each side of the square with two lines joined at some angle
facing inwards.

What just happen is that area of the square become smaller, but the
perimeter became larger. We can continue this process forever and if
we do, we observe that length of the perimeter is infinite, but the
area is not. Area would become smaller and smaller tending to some
finite quantity. This is actually quite remarkable.

We have perimeter of infinite length encompassing finite area.

Going back to broccoli, is it possible to exactly measure the
surface of the broccoli ? Absolutely not, but we can measure the mass
and there by the volume by simply weighing it.

What about jewellery? Every article of jewellery has certain feel
about. Some call it soul, which is fine. Formally it is perception of
it’s fractal dimension. As long as “imperfections” have the same
fractal dimension, they will look like they belong.

To understan this study the work of Auguste Rodin. His surfaces
always rough and unfinished, but they belong. Take technique like
pave. If we examine cut by cut; stone by stone, - there is nothing
perfect about it. Nevertheless, some pave looks like it contribute to
the article of jewellery, and some looks like it been glued on top of
article by mistake. This is what is called “setter’s talent”, or we
can call it fractal intuition.

One more example. How does an artist make hair look like it has
life.

It is not possible to draw every single hair. What about sculptor
who works in marble ? Try to convey this feeling of wind playing with

Good luck if you decide to try. Nevertheless, it is accomplished by
thousands of artists on daily basis. A few strokes of pencil, or few
cuts with chisel and it is done.

On some level artistic talent is an ability to recognize and
reproduce fractal dimension of and object. That is why caricature
works. Caricature and it’s object do not look alike, but they have
the same fractal dimension.

Leonid Surpin

1 Like
``````"A sculptor refining a surface", can mean anything, a computer
that operates a mill can refine a flat surface to almost dead
smooth, it may take time, but a person can't do it.
``````

One must disagree. Long before the advent of computers and
computerized machinery mechanics had been manually lapping surface
plates to flatnesses of under 0.00001" using the three-plate lapping
method.

Elliot Nesterman

Hi Leonid,

``````Broccoli is simply interesting. One can study it's shape for hours
and will keep finding variations. Nothing is the same, but
similar, and every element will have the same fractal dimension.
``````
``````     What is fractal dimension? Staying away from mathematics,
draw a square on the paper. We can easily compute the area and the
perimeter of this square.  etc.  <snip>
``````

If you mean this broccoli :- http://www.ganoksin.com/gnkurl/ep7zwt

Then this is not an example of a fractal, no even an approximate.

If you mean this broccoli :- http://www.ganoksin.com/gnkurl/ep7zwu

Then this is an example of a fractal, although only an approximate
of a Fibonacci set. And I could argue very successfully that a
Romanesco broccoli is very beautiful. If you can’t see the beauty in
this vegetable, then I think you are just arguing for the sake of
arguing. If that’s the case nothing can be achieved.

Fractals are mathematical sets, suggesting that people stay away
from mathematics and attempt to plot fractals is pointless.

If people want to play around with fractals here is a link that may
be of use :- http://www.ganoksin.com/gnkurl/ep7zwv

Regards Charles A.

Ah Leonid…

``````Going back to broccoli, is it possible to exactly measure the
surface of the broccoli ? Absolutely not [snip]
``````

As usual you are woefully uninformed on current technology. We have
been able to exactly measure the surface of broccoli and just about
everything else since the 1980’s. One technology is called “confocal
laser scanning microscopy”. Measurements are obtained through
optical sectioning. This was patented in 1957, but not fully fleshed
out until years later. I suggest you read up on this. It is highly
useful for biological applications, but one can also see how it
might be useful in 3D jewelry applications…say CAD for
instance…

There are In fact many other methods for measuring the surface of
opaque objects, amazingly, one can even measure the surface of
transparent objects using a polarization Re ectance Model. but then
we slide into physics. Very useful stuff physics.

Lisa, (planting the orchard. We now have 80 fruit trees. If the
weather doesn’t kill all of them, there will be quite a harvest next
year!) Topanga, CA USA

My father was responsible for milling telescoping parts to use in
space satellite’s. and you can’t use oil in space. The parts were
made within One Millionth of an inch. less than a human hair. and it
was as smooth as you can imagine. He won awards from the government
for these parts.

Russ

``````"A sculptor refining a surface", can mean anything, a computer that
operates a mill can refine a flat surface to almost dead smooth, it
may take time, but a person can't do it. One must disagree. Long
before the advent of computers and computerized machinery mechanics
had been manually lapping surface plates to flatnesses of under
0.00001" using the three-plate lapping method.
``````

Okay that’s a good example, and my example was hasty due to the
heated nature of this discussion.

However I can salvage my example, by extending the size of a surface
to be made almost dead smooth to the point where a human cannot
extend himself to accomplish the task.

Regards Charles A.

``````As long as "imperfections" have the same fractal dimension, they
will look like they belong. On some level artistic talent is an
ability to recognize and reproduce fractal dimension of and object.
``````

Does an intellectual understanding of the concepts of fractal
dimension play a part in the creation of objects of art? I would
think that most people instinctively “recognize” correct fractal
dimension and are attracted to it. “Artistic” people can produce
objects of correct fractal dimension even thought they may have
never heard of a fractal. My very basic understanding of fractal
dimension tells me that my appreciation for symmetry is related to
fractal dimension. I cannot say that I have “artistic” talent but
can my understanding of fractal dimension be applied in the design
of an object of art?

``````However I can salvage my example, by extending the size of a
surface to be made almost dead smooth to the point where a human
cannot extend himself to accomplish the task.
``````

That’s a distinction of quantity, not quality. But even then,
computers and CNC are not necessary. It’s still a purely mechanical
problem with a purely mechanical solution.

Elliot Nesterman

``````As usual you are woefully uninformed on current technology. We
have been able to exactly measure the surface of broccoli and just
about everything else since the 1980's.
``````

Why do I do that to myself ? Measurement of anything is an
approximation depending on chosen scale, so it has meaning only if
scale is established. Let me give you an example. Area of a circle is
2 * Pi * radius squared. Seems simple enough. However Pi is a
transcendental number, which means that it exact value is unknown.
The number of digits to the right of decimal points (mantissa) goes
on forever.

So Pi is only has meaning with predetermined scale. It can be 3.14
or 3.14158, or whatever precision is required. But it is only
approximation. So, not only surface of broccoli cannot be precisely
known, even area of the circle also depends on chosen scale. Since no
matter how small we make our scale, it can be made even smaller by
dividing it in two. Since division by two never stops, we say that
such values are infinite. This has been true since beginning of time
and will remain to be true, regardless of technology.

Leonid Surpin

Hi Leonid,

``````Why do I do that to myself ?
``````

Funny, I wonder the same thing…

OK, you’ve talked about all measurements being an approximation,
being scale and precision dependent.

All physical measurements are approximations. Assuming they’ve been
made at a level of accuracy fit for purpose, they’re close enough
for any practical purpose.

The only ‘perfect’ measurements are either theoretical, or CAD. I’m
sure you don’t want to concede that CAD is more perfect than
physical reality.

So, having thrown sand all over the notion of physical measurement,
now what? If you want to make real, physical objects, at some point,
you have to say that the frog has hopped far enough for any
practical purpose, and get started working.

Regards,
Brian.

``````Does an intellectual understanding of the concepts of fractal
dimension play a part in the creation of objects of art? I would
think that most people instinctively "recognize" correct fractal
dimension and are attracted to it. "Artistic" people can produce
objects of correct fractal dimension even thought they may have
never heard of a fractal.
``````

That is absolutely true. Fractal is just a word. Before it was
called artistic intuition.

How many times one looks at other person’s smile and intuitively
knows that teeth are fake. Nothing wrong with them, except of feeling
that they are fake. How do we know that? I live on east coast. I went
to California on hiking trip and noticed right away that mountains
look different. How did I make that determination ? Did I memorized
every nook and cranny of east cost mountains ? Of course not.

So what is this general property. It is difference in fractal
dimension, that is what we see.

Leonid Surpin

``````So, not only surface of broccoli cannot be precisely known, even
area of the circle also depends on chosen scale.
``````

There’s a margin for error in any measurement system, but I think
that even with that margin for error, a 3D laser scanner can do a far
better job than a human being ever could. when it comes to measuring
the dimensions of the humble broccoli.

Regards Charles A.

``````OK, you've talked about all measurements being an approximation,
being scale and precision dependent.
``````
``````Yes. And your point is?
``````

It was a response to a statement that we have technology that can
measure anything precisely. My modest efforts were to show that
nothing can be further from the truth. At best we can only
approximate.

Leonid Surpin

``````There's a margin for error in any measurement system, but I think
that even with that margin for error, a 3D laser scanner can do a
far better job than a human being ever could. when it comes to
measuring the dimensions of the humble broccoli.
``````

The property of fractal objects is an infinitely long perimeter
encompassing finite area; an infinitely large area contains finite
volume; and so on. That is the nature of fractal universe.

This is an area where mathematics and philosophy merges.

I cannot explain it better than I have. If you want to understand,

Leonid Surpin

Hi Leonid

``````It was a response to a statement that we have technology that can
measure anything precisely. My modest efforts were to show that
nothing can be further from the truth. At best we can only
approximate.
``````

OK, fine. But this is getting into angels tapdancing on pins
territory again. Yes, physical measurement is imperfect by
definition. On the other hand, we can measure things down to
angstroms without too much fuss, if we choose to.

At what point does reductio meet absurdum? And what, exactly, does
that have to do with trying to make jewelry, either by hand, or by
machine?

The unaided human eye can’t perceive offsets smaller than about
0.0005", which is easily within the range obtainable by pretty basic
machines. So arguing about the theoretical nature of the 21st
decimal place doesn’t seem to have any real point in a discussion
that hinges on physical reality.

Regards,
Brian

This has been a pleasant and interesting discussion. Thank you Brian
and Leonid. I’ve enjoyed it.

``````I went to California on hiking trip and noticed right away that
mountains look different. How did I make that determination ? Did
I memorized every nook and cranny of east cost mountains ? Of
course not. So what is this general property. It is difference in
fractal dimension, that is what we see.
``````

That’s generally called gestalt.

Fractal dimension is a mathematical concept, a ratio of the change
in detail with change of scale. I encourage you to research it, along
with fractals themselves. A fascinating topic, but not particularly
applicable here.

Al Balmer
Pine City, NY