@normonics
' Intro to Applied Complexity #ACS101 #SpringA2021 Highlights
Session 5: Fractals and Scaling
(Thank you Benoit B. Mandelbrot)
thread/
Hysteresis is a basic form of memory.
It matters where you come from.
1/n
There are tools that people bring to bear that become selective mechanisms over the possible objects of study and so we get this hugely biased sample of what we consider normal/regular/typical.
2/n
Our tools are set up for this -> so we study this -> so we start treating this as general or typical,
whereas other kinds of objects or geometries are common or more so.
3/n
With algorithms the structure of the problem leads to different kinds of scaling behaviour of the solution you might try.
Some problems scale in a more well-behaved way.
Some scale more drastically from input size depending on the input.
4/n
Characteristic scale:
as you lower the resolution you lose the pattern - as you raise the resolution you don't find anything new.
there is a clean identity of scale where it exists.
5/n
Fractals don't have a characteristic scale.
They are self-similar across all.
6/n
No discussion of fractals is complete without doing a Mandlebrot zoom (on Zoom)
Zoom squared
7/n
Multiple scales of behaviour are typical of complex systems but not always.
8/n
How long is the coast of Britain? (PICUTRE)
It depends on the ruler.
(Effectively the whole big island that is not Ireland is Britain btw Joe/Americans.)
9/n
The size of the ruler depends on the measure you get.
(I see you Wittgenstein)
10/n
A Euclidean object converges to a finite value at higher and higher resolutions.
But many, many other things do not.
11/n
The assumption of smoothness is common in mathematics.
Rough texture is more difficult to deal with using traditional mathematical tools.
12/n
Calculus depends on the assumption you can find the tangent of a curve.
13/n
The lung alveoli branching pattern creates a massive amount of surface area.
So much if you could peel it out - it would be about the size of a tennis court
14/n
Romanesco Broccoli is delicious.
"Freaking awesome, better than regular broccoli"
"It's like eating math"
10.2 out of 5.1 stars
(Joe gets excited every time.)
15/n
Self-similarity is not only in fractals, it does not depend on being one.
e.g. a line is self-similar, a square is self-similar
16/n
We define a fractal by measuring its dimension.
A fractal quite literally has a fractional dimension - such as between 0 & 1 or 1 & 2.
Euclidean shapes have an integer of dimension.
17/n
(The mysterious belch did indeed show up on the recording.)
18/n
A fractal is something between a point & a line.
This is defined by the relationship of the dimension.
If the fractal dimension is greater that the topological dimension (lowest dimension to cut into two +1).
19/n
There are 2 forms of growth (organic & mechanic)
Growth by duplication.
Growth by hypertrophy.
20/n
(Made me think of the seemingly related idea of r/k selection distinctions)
21/n
Too often when we're thinking about growing human-made systems we are only thinking in terms of hypertrophy.
22/n
Shipping is the reverse of this e.g. with container ships.
Large ships for a while were thought wouldn't work because of drag forces in the water but this is not the case because surface area grows by the square while volume grows by the cube.
23/n
You can make ships gigantic without having to worry too much about a diseconomy of scale. In some sense the bigger you make it the more efficient it is.
24/n
But if your ship is too large you can't even dock.
25/n
@chrismanfrank relates the case of @synthesischool
As you scale like say if a company is growing 15% a week - things breakdown in an unexpected way.
The relationships for example: for every new person you add everyone has to have a relationship with everyone else.
26/n
You're always surprised by what breaks.
27/n
So, if everyone needs to know everybody then you are scaling by the square & quickly that's not going to work.
There is going to be time where it might make sense to limit your own growth even though you could grow more, just to keep yourself from overscaling too quickly.
28/n
Maybe it makes sense to grow but sometimes it's better to do things slowly so the subtle forces of coevolution etc. can take place under the surface.
29/n
Growing to a size in 3 months vs next week are two different things.
30/n
Some say there are rules of thumb, but typically we scale too fast in start-ups, everything breaks, and then there's a bunch of fighting.
31/n
There is an ever-present challenge in/of formalisation.
We have some intuitive notion of what things are like, we formalise it, it seems like it works, then we find something that breaks it - this goes on & on.
32/n
Formalisation turns out to have a lot of blindspots.
33/n
Another special vs general case.
Euclidean shapes are a special case because they have integer dimensions.
All other case are general because how many numbers are there between 1 & 2?
(A LOT)
34/n
Chaotic (strange) attractors - Lorenz systems
Sort of 2 dimensions with a bit of 3 dimensionality.
Looks like a butterfly hence the 'flaps its wings' expression.
35/n
Often when you have one metric for a system that plays nicely in this way others do too.
Which shows some lawful underlying dynamic at play.
Wave frequencies hitting the shore or their size, double pendulums, etc.
36/n
Strange attractors may be mathematically exotic but in the real world are not exotic at all.
(resisting the joke)
37/n
Mathematics seems like a deductive activity, but in general it is a hugely creative process.
38/n
Mathematics itself can't be formalised.
39/n
Power laws:
Functional relationships that something is being raised to.
40/n
Allometry:
You can only scale so far before there is conflicts in the structure of the system.
Things scale at different rates and when those relationships breakdown then the system doesn't work anymore.
41/n
As an animal get bigger its mass scales to the cube while the cross section of say its limbs scale by the square.
42/n
(Would you rather fight a horse-sized duck or 100 duck-sized horses?)
43/n
You run into all sorts of new problems as you scale.
44/n
At a certain point in time it has to have a paradigm shift or it doesn't work anymore:
Different materials, different, tools, etc. - just different
45/n
Something has got to give something has got to be different.
A whale can grow so large compared to elephant because it is in (and has to be in) the ocean.
46/n
Instead maybe try a smallish system then, run a new smallish system in a similar way e.g. cells
Sometimes duplication allows things to evolve differently.
47/n
Consider the 50 states.
Or better yet consider the near 200 countries
Or even better again consider all the individual towns and cities spread across them.
48/n
To close with an apparent paraphrasing of Dr. Dwanye Beck
Do you want 600lbs of lover?
It depends is it four 150lb lovers or one 600lb lover?
49/n
Put another way:
Do you want a 100kg of boulder in one blow or 1kg pebbles spread across a long span of time?
50/thread
Case in point with this thread length
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