2 dimensions into 3 dimensions

river said:
What would a " dent " mean ?
It actually proves what I said. There is no surface.

It actually proves what I said. There is no surface.

Disagree

In order to have a " dent " in a surface implies a soild surface in the first place .

Fractals are only present in life forms .

Show that my post # 102 is wrong .
I have no idea where you picked this up but it is entirely untrue.

Have a look at weather patterns, ocean currents, mixing chemicals, etc.

Here's a bunch of em. Some are life-based, many are not:

https://www.wired.com/2010/09/fractal-patterns-in-nature/

From River Posts 95, 97, &117
Fractals are only found in life forms
Incorrect: Various objects in nature approximate fractals. Coast lines are one example.

True fractals are mathematical entities which do not appear in nature. Examples are the Mandelbrot Set & graphics generated via use of successive approximations to the solution of cubic equations.

BTW: The graphic generated via successive approximations to the three cube roots of a number (2 are complex) has an interesting property when its various regions are colored using three colors.

The borders between regions are all triple border points like the center of a circle divided into three slices. None are like the linear borders from the center to the circumference.​

Fractals are only found in life forms

Incorrect: Various objects in nature approximate fractals. Coast lines are one example.

True fractals are mathematical entities which do not appear in nature. Examples are the Mandelbrot Set & graphics generated via use of successive approximations to the solution of cubic equations.

BTW: The graphic generated via successive approximations to the three cube roots of a number (2 are complex) has an interesting property when its various regions are colored using three colors.

The borders between regions are all triple border points like the center of a circle divided into three slices. None are like the linear borders from the center to the circumference.

Approximate fractals is not an argument to my point of life fractals .

Fractals , real fractals are from life , the further one looks into life fractals , the repetition continues . The form repeats no matter how small we go . We take the smaller and it will be exactly the same as the larger form .

The further we get into a focus on a coastline the less we can build a dupicate of the FULL coastline.

But in life the focus small or large does does not affect the completeness of the whole .

Approximate fractals is not an argument to my point of life fractals .

Fractals , real fractals are from life ,[/QUOTE]
This is not true. There are all sorts of fractals in nature that are not alive.

And more to the point:

The form repeats no matter how small we go . We take the smaller and it will be exactly the same as the larger form .
Just as with any real world fractals, this is not true for life. Of course it has a lower limit!!

The further we get into a focus on a coastline the less we can build a dupicate of the FULL coastline.
Just like in life.

But in life the focus small or large does does not affect the completeness of the whole .
Not true.

river's obsession with 'life fractals' as the only real fractals was dealt with in #96, #98, #100, #103. Not only has he not budged an inch, but subsequently other members go on to basically repeat what I wrote, yet without any reference to above posts i.e. as though bringing it up for the first time. Typical SF pattern. Maybe even a fractal-like pattern.
A fork into pedantic arguing over surfaces are/are not 'really' 2D entities also took off. Despite, apart from the in hindsight unfortunate detour into fractals territory, being briefly but adequately covered in #2. Such are things at SF. Like many threads going on for sometimes 10's of pages, a simple agreement over definitions early on should have avoided such repetitious time wasting. But, to repeat, such is SF.

Quantum Quack Post # 1
If we have a 2 dimensional plane and roll it into a cylinder, is it now a 3 dimensional object or is it a 2 dimensional object.

Care to discuss?
My Post #6 (slightly edited)
A solid cylinder would be a 3D object, although plane geometry would be applicable to the surface.

Plane geometry is applicable to this surface, although plane topology might not be.

Note that the Pythagorean formula can be used for right triangles.

If you apply differential geometry to this surface, the metric is the same as that for a plane.?

Similar remarks apply to any surface which can be cut & flattened without crinkling.

There is a more formal way to express the above remark. I do not think that crinkling is a scientific term in the above context.

My above Post pretty much tells the entire tale.

I find it incredible that there are 127 Posts to this thread, most of which I have not read.

Mea culpa for not pointing out that the discussion of fractals is a digression with little or no relevance the question asked in Post 1.

Quantum Quack Post # 1
My Post #6 (slightly edited)

My above Post pretty much tells the entire tale.

I find it incredible that there are 127 Posts to this thread, most of which I have not read.

Mea culpa for not pointing out that the discussion of fractals is a digression with little or no relevance the question asked in Post 1.

You are right Dinosaur , I went off on a tangent that I never should have .

My apologies to all .

Anyway what 2dimemsional concept can become 3dimensional physical object ?

Assuming that is what QQ goal is ?

Continuing the digression from the initial Post to this Thread

From River Post 125
Fractals , real fractals are from life , the further one looks into life fractals , the repetition continues . The form repeats no matter how small we go . We take the smaller and it will be exactly the same as the larger form .
False. In the world of our senses (or life as you express it), There is a lower limit to fractals at or before the level of a single atom or single molecule.

In the abstract world of mathematics there are fractals with the properties you describe.

There is no lower limit to the fractal nature of these mathematical abstractions.

The Mandelbrot Set is one example of such a fractal.

Other examples can be generated via graphics generated via the mathematics of successive approximations to the nth roots of positive integers​

BTW: Fatou & Julia (both French, I think) laid the foundations for the modern study of fractals prior to the existence of modern computers capable of generating images using computer monitors & printers.

Mandelbrot got more credit than he deserved due to having a computer capable of doing the calculations & displaying images on a monitor or printing them.

I believe David Bohm called it the enfolding and unfolding of spacetime

As to the fractal properties of spacetime read "causal dynamical triangulation" (CDT)
Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent.

This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.
https://en.wikipedia.org/wiki/Causal_dynamical_triangulation