invariance of scale (split from Evolution not limited to life on earth?)

[Luc Turpin]

1 hour ago, joigus said:

It's a mathematical pattern rather than a process. I'm sure something like that is the reason behind @exchemist's excellent question.

Take. eg, principles of extremal time, action, length, etc. They appear everywhere in physics. It's more about a recurring mathematical theme than actually a particular process.

On invariance of scale, I am not saying that it is universal, but trying to determine if it is. 

Noted that it is a mathematical pattern rather than a process.

As for exchemist excellent question, he knows much more than I. In trying to understand, I probably referenced the wrong articles or did not understand them well enough. In my defence, I did say "seem" as in not entirely sure.

[studiot]

4 hours ago, Luc Turpin said:

So, self-similarity being synonym of scale invariance, both together forming a universal process in nature! correct?

Spatial pattern in objects and embedded behaviour in natural evolution! correct?

What about living organisms? spatial pattern, embedded behaviour or both?

Where did universal processes creep into the discussion ?

And what are universal processes ?

 

On 12/19/2023 at 3:23 PM, Luc Turpin said:

On 12/18/2023 at 2:31 PM, Phi for All said:

 

I wish to talk about invariance of scale.

Then we need to go back to school and do some geometry to separate out some fundamentals.

 

Self similarity is not the same as scale invariance.

the two concepts may concide  (and often do) but they are separate geometric concepts.

 

In preparation for more detail, please go and review our discussion about linear mathematics.

The point to revise is that over a short interval we can always pretend (and usually do) that a curved line is linear.

The more curved the line the shorter that interval has to be.

 

13 minutes ago, Luc Turpin said:

As for exchemist excellent question, he knows much more than I. In trying to understand, I probably referenced the wrong articles or did not understand them well enough. In my defence, I did say "seem" as in not entirely sure.

You don't need to defend yourself, no one is attacking you or blaming you.

This is a friendly discussion.

😀

Exchemist is talking about the science of mineralogy, and yes he certainly knows more than most.

[Luc Turpin]

8 minutes ago, studiot said:

Where did universal processes creep into the discussion ?

And what are universal processes ?

 

Then we need to go back to school and do some geometry to separate out some fundamentals.

 

Self similarity is not the same as scale invariance.

the two concepts may concide  (and often do) but they are separate geometric concepts.

 

In preparation for more detail, please go and review our discussion about linear mathematics.

The point to revise is that over a short interval we can always pretend (and usually do) that a curved line is linear.

The more curved the line the shorter that interval has to be.

 

You don't need to defend yourself, no one is attacking you or blaming you.

This is a friendly discussion.

😀

Exchemist is talking about the science of mineralogy, and yes he certainly knows more than most.

Universal was indirectly ascertained in my line of questioning without necessarily mentioning it

'self-similarity (synonym of scale invariance)' 'It's a mathematical pattern rather than a process.' from Joigus and probably misinterpreted his words.

Will gladly go back to school and will be reviewing our discussion about linear mathematics.

Got the point about pretending for a short while that curved is linear.

And yes, I defer to Exchemist on the science of mineralogy.

 

[studiot]

2 hours ago, Luc Turpin said:

Will gladly go back to school and will be reviewing our discussion about linear mathematics.

Please do because that thread introduced several utterly fundamental ideas in Mathematics, some of them so simple that people tend to pass over them too quickly.

 

In the mathematical world we have abstract idealisations.

These are often models of the material world.

We can pretend that the material world follows our mathematical model for some limited extent of the mathematical model.

(just like with the linear model of the curve)

So in that thread I introduced 'the world of shapes'

These are idealisations or perfect implementations of a mathematical idea for instance of a square.

The material world cannot match squares (or any other shape) perfectly at all scales.

If we make the square small enough the gaps between the atoms stop the square being complete.

But we can get pretty good squares from a block of concrete.

So nothing in the material world is truly scale invariant. It is a good match just for a range of scales.

 

So we come to self similarity.

Two figures  in mathematics are called 'similar' if their shape are the same, except for a scale factor.

For instance a triangle with angles 45, 45, 90 is similar to any other and all other triangles with these three angles.

But when we look at squares there is a difficulty.

For a square, not only do all the angles have to be 90, but all the sides have to be the same length for a figure to be a square !

We have just introduced a second parameter  -- Length, which was not needed for the triangle.

 

There is a no problem if we scale the side lengths by the same factor. So a square with all sides twice as long as another square is similar to the other square.

But now the area of the figure is scaled by a different factor as the larger square has four times the area of the smaller.

 

But since the two squares are still similar we observe that the similarity property can involve more than one scale factor being applied appropriately.

So when we talk about self similarity between two shapes we need to specify what property is being scaled and therefore self similar.

 

It gets yet more complicated as we could apply different scales along two or more of the coordinate axes.

This will change the spae of the similarity so it is not called a similarity but an affinity.

 

Fractals can also be made from self affinities.

Fractals that have nothing to do with shape in the material world can also be made by

Self organisation, though not all such organisation leads to fractal geometry.

Self replication

probabilistic trees,  though not all such trees are fractal.

 

Fractals are so named because their apparent 'dimension' is not a whole number.

Our material world is firmly 3 dimensional plus time so nothing material can actually be fractal.

So when we say that a coastline length is fractal, we are saying that there is a fractal abstract model which is self similar down to infinitesimal sizes that matches the given coastline to some finite limiting size.
Coastlines are interesting because this limiting size of self similar is actually very large.

Yes there are smaller and smaller bays and inlets and peninsulas, but the smaller ones are not an exact shape copy of the larger ones.

This is the difference between a material world fractal and a mathematical world one like the Koch snowflake, where every reduction of scale brings an exact copy of shape.

[Luc Turpin]

13 hours ago, studiot said:

Please do because that thread introduced several utterly fundamental ideas in Mathematics, some of them so simple that people tend to pass over them too quickly.

 

In the mathematical world we have abstract idealisations.

These are often models of the material world.

We can pretend that the material world follows our mathematical model for some limited extent of the mathematical model.

(just like with the linear model of the curve)

So in that thread I introduced 'the world of shapes'

These are idealisations or perfect implementations of a mathematical idea for instance of a square.

The material world cannot match squares (or any other shape) perfectly at all scales.

If we make the square small enough the gaps between the atoms stop the square being complete.

But we can get pretty good squares from a block of concrete.

So nothing in the material world is truly scale invariant. It is a good match just for a range of scales.

 

So we come to self similarity.

Two figures  in mathematics are called 'similar' if their shape are the same, except for a scale factor.

For instance a triangle with angles 45, 45, 90 is similar to any other and all other triangles with these three angles.

But when we look at squares there is a difficulty.

For a square, not only do all the angles have to be 90, but all the sides have to be the same length for a figure to be a square !

We have just introduced a second parameter  -- Length, which was not needed for the triangle.

 

There is a no problem if we scale the side lengths by the same factor. So a square with all sides twice as long as another square is similar to the other square.

But now the area of the figure is scaled by a different factor as the larger square has four times the area of the smaller.

 

But since the two squares are still similar we observe that the similarity property can involve more than one scale factor being applied appropriately.

So when we talk about self similarity between two shapes we need to specify what property is being scaled and therefore self similar.

 

It gets yet more complicated as we could apply different scales along two or more of the coordinate axes.

This will change the spae of the similarity so it is not called a similarity but an affinity.

 

Fractals can also be made from self affinities.

Fractals that have nothing to do with shape in the material world can also be made by

Self organisation, though not all such organisation leads to fractal geometry.

Self replication

probabilistic trees,  though not all such trees are fractal.

 

Fractals are so named because their apparent 'dimension' is not a whole number.

Our material world is firmly 3 dimensional plus time so nothing material can actually be fractal.

So when we say that a coastline length is fractal, we are saying that there is a fractal abstract model which is self similar down to infinitesimal sizes that matches the given coastline to some finite limiting size.
Coastlines are interesting because this limiting size of self similar is actually very large.

Yes there are smaller and smaller bays and inlets and peninsulas, but the smaller ones are not an exact shape copy of the larger ones.

This is the difference between a material world fractal and a mathematical world one like the Koch snowflake, where every reduction of scale brings an exact copy of shape.

Very well explained; I understand better invariance of scale, self similarity, the need to specify what property is being scaled, from similarity to affinity.

And the main point for me is that peninsulas are not exact shape copies of larger ones and that this is the difference between a material world fractal and a mathematical world like the koch snowflake.

For me, this is a "revelation" (Joigus - we are losing the patient) moment 😉

[joigus]

17 hours ago, studiot said:

Where did universal processes creep into the discussion ?

 

Sorry, because I was guilty of it.

Quote

• Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.

https://en.wikipedia.org/wiki/Scale_invariance

Quote

In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system.

https://en.wikipedia.org/wiki/Universality_(dynamical_systems)

Universality is frequently mentioned in association to scale invariance, not because it always happens, but because it seems to always happen in connection to critical phenomena.