Tuesday, June 03, 2014

Psyllium Fractals and other Psillies .

Psyllium Fractals.

Andre Willers
3 June 2014
Synopsis :
We approximate the Haussdorff (fractal) dimension of psyllium in water .
Discussion :
1.Fractal dimensions and what they mean :

2.Examples to give you an idea :
The first number is the fractal dimension in 3 space dimensions .

Balls of crumpled paper
When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the ISO 216 A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made.[46] Creases will form at all size scales (seeUniversality (dynamical systems)).

Broccoli DSC00862.png
Surface of human brain
Cerebellum NIH.png
Lung surface
Thorax Lung 3d (2).jpg
The alveoli of a lung form a fractal surface close to 3.[41]

2. Reality check : Crumpled balls of paper :
See Appendix AA . Theory gives a fractal dimension of 2.44 , compared to 2.5 above .
This is important , as we already suspect that the fractal dimension of psyllium must be close to that of a crumpled piece of paper . They are similar .
3.Catalysts :
Bio-catalysts have Fractal dimensions about 2.14 to 2.44
(Lysozyme , ribonuclease A , superoxide dismutase)
4. Diesel exhausts :
Fractal dimension about 2.35
This is a quite mature technology , judging from the complexity .

5. Psyllium Fractal dimension :
Well , we know it can’t be higher than 2.5 (the Crumped Paper model) .
I couldn’t find data on the internet (either proprietal or nobody could be bothered)
The closest  I could get was that psyllium absorbs more than 10 times and less than 20 times .
Average it at 15 times  gives a Haussdorff dimension of (15)^(1/3) ~ 2.47 .
This is close enough to fractal dimension of 2.5 of a ball of crumpled paper .
6. What does it mean ?
It tells you that psyllium is close to an optimal catalyst .
It brings molecules close together so interaction can occur .
7. Do you want to live dangerously ?
Combine psyllium(2.47)  and broccoli (2.66) as catalysts in microwaved reactor chambers .
8. Or replace platinum catalysts with broccoli or psyllium (crumpled paper would do in a pinch)
9. Aluminium foil should crumple up finely enough to mimic a good catalyst .
10. Notice the really big Haussdorff dimensions of human lungs .
Mammals pay for the complexity via diseases of the respiratory system .
The triumph of function over process . Typical .
Bird’s lungs are more efficient , without the complexity .
11. How smart are broccoli ?
The complexity difference = 2.79 – 2.66 = 0.13   … about 5 %
Examples of cranial capacity:
·         Orangutans: 275–500 cc (16.8–30.5 cu in)
·         Chimpanzees: 275–500 cc (16.8–30.5 cu in)
·         Gorillas: 340–752 cc (20.7–45.9 cu in)
·         Modern HumanAustralian Aborigines: 1,199 cubic centimetres (73.2 cu in) [17]
·         Modern HumanJapanese: 1,318 cubic centimetres (80.4 cu in) [18]
·         Modern HumanAzande: 1,345 cubic centimetres (82.1 cu in) [19]
·         Modern HumanItalian: 1,411 cubic centimetres (86.1 cu in) [20]
·         Modern HumanNez Perce: 1,483 cubic centimetres (90.5 cu in) [21]
·         Modern HumanAleut: 1,518 cubic centimetres (92.6 cu in) [22]
·         Neanderthals: 1,500–1,800 cc (92–110 cu in)

Broccoli : not limited by a skull .

I am not joking .
The biggest problem with AI’s is to find a complex enough substrate ,
With a big heat-sink .
Broccoli is ideal for human purposes .
Cheap and vulnerable .
Just tack on genes for electron transmission
Spray-on smartness .
This can be done , if anyone was insane enough to make a vegetable smarter than humans .
This means it will certainly be done .
A few tweaks to add transistors , and leave the rest to evolution .
Oh well .
Hope they keep some humans to till the old homestead .
Roll on the Singularity .
It is Green !
It is Broccoli !

Appendix AA
The fractal dimension :
mu=0.097421589 . mu gives the ratio of crumpled ball space to volume of paper mass .
The paper mass has two sides in a real-world application .
Approximate the surface area by 2^0.5
Fractal dimension = (1/0.0974 x 2^0.5 ) ^ (1/3)
=14.42 ^(1/3)
Wednesday, February 29, 2012
Crumpling paper and Space-Time
Crumpling Paper and Space-Time
Andre Willers
23 Feb 2012

“The moving finger writes , and having writ , crumples it in random ruins.”
With apologies to Omar Khayyam .

Crumpled paper gives a good approximation of spacetime as a membrane with clumpy masses .
“Empty” spaces not occupied by the membrane gives an impression of dark matter .
We derive an expression to give this ratio using Infinite Descent and Beth(0) Random Walk .

Discussion :
1.The Crumpled paper :
Consider a paper disk of radius r and thickness d .
It's volume is then Vp=pi * r^2 * d
Draw a line from the center to the edge , in steps of length d , over the edge , then back to the center Let nu=r/d , a measure of the thickness of the paper . Note that it is a pure number .

The number of steps in the line is then n0=(2r/d)+1
But the number of steps to the edge of the original paper disk is n1=r/d=(n0-1)/2

Vp=(pi*d^3*(n0-1)^2 )/4

Crumple it up in a way that is as random as flipping a coin (ie Beth(0) )

The Trick : The line we have drawn up above breaks up into random vectors by rotating through a third dimension = crumpling into a ball .

We thus have a continuous line of random steps of known number of steps .
In 3 dimensions , the mean square distance from the center then is known
R = d * (n0)^0.5 …. See true for all dimensions as long as all are of Beth(0) order of randomness.

Volume of crumpled ball Vb=4/3*pi*R^3
The Ratio Vb/Vp = mu then gives the ratio of crumpled ball space to volume of paper mass .

Mu={4/3*pi*d^3 *n0^(3/2)} / (pi*d^3*(n0-1)^2 )/4
Notice the d^3 term and pi cancels out . This has profound physical implications .
This simplifies to

Expressed as thickness of paper , nu , which is a pure number independent of metric chosen .
mu=4*4/3(2*nu+1)^3/2 / (2*nu)^2
mu=4/3*(2nu+1)^3/2 / nu^2
This gives a quartic equation in nu , which can be solved exactly algebraically .
(mu)^2*(nu)*4 – 2^7/3^2 *(nu)^3 – 2^6/3^2 (nu)^2 – 2^5/3^2 * (nu)^1 - 2^4/3^2 =0

Test it on A4 paper:
A4 paper has thickness d~0,1 mm and r~150 mm
mu= 1- 0.90257841
This means that the crumpled A4 paper ball encloses about 90% empty space .
This agrees with experimental results . See NewScientist.

Note that the force applied does not matter . As long as the paper is untorn , mu will be the same .

How many times can it be folded ?
Solving the above (see below) gives mu=1 for about nu=14.7 to 14.8 .
This means there are no empty spaces left to fold into .

This can get complicated , so I will keep it simple .
Take a piece of paper and fold it . You then have a new piece of paper .The test-circle of same r will have double the thickness .
Ie , nu will double .
Between 7 and 8 folds , nu will hit the ceiling of mu=1 , regardless of the starting value of nu .
This is the maximum number of paper folds , as confirmed from other sources .

Physical interpretations :
Take an m-dimensional space . Randomness of order Beth(0) applies equally to all . The underlying equalizer . Collapse it to three dimensions and let the third one approach single Planck lengths .
Then we can use the above paper approximation . Notice how d cancels out except for an addition of 1 in final ratio .

What does it mean ?
See the physical universe as a brane (ie sheet of paper) in a multiverse . Crumpling it means it has mass and singularities . Both are aspects of the same thing .
An estimate of the number of singularities can be made from edges and points in crumpled paper .

Can we crumple the paper to a ball that is just paper ?
That is a particle .
The answer is “Yes” .

Such crumpling means that mu=1 (no empty space in any dimension )
This gives an quartic equation in nu that solves to four values , other dimensions than three denoted by i=(-1)^0.5

See http://www.1728.org/quartic.htm for a calculator
nu1= 14.722181 (this makes the physical particle universe possible . Mass .
Nu2= - 0.004167 + i*0.49558 (Rotation :Spin :charge and magnetism)
nu3 = - 0.004167 - i*0.49558 (Rotation :Spin :charge and magnetism) notice the minus sign .
Nu4= - 0.49164542 (quantum effects as the particles dither. Inertia?)

What does a negative nu mean ?
nu=r/d . A negative nu means one of r or d must be negative .
1.If r is negative , it can be interpreted as curled up dimensions , inside the “outside” dimensions as defined by i . See http:andreswhy.blogspot.com “ The inside of zero” Aug 2009
2.If d is negative , it can be interpreted as quantum effects . A particle does not “occupy” all the space . Likes hopscotch .
3.But notice the the two are interrelated .The notorious observer effect . Where we place the minus sign between r or d .

There should be relationships between nu2 , nu3 and nu4 . Various rotations between macro- and micro dimensions .
This means the contraption is not symmetrical But we already know that ,

Physical constants :
Things like charge , mass , etc should be derivable from these basics . Hint:use lots of crumpled paper .

There is hope . The fact that it is quartic equation , which is always solvable , means that the Universe can be understood . Complicated and perverse , but as long as you stick to Beth(0) randomness , it can be understood . For higher orders of randomness , good luck .

Dark Matter :
I nearly forgot . Using Planck units , we can define the ratio of thickness of the brane as
nu=c*PlanckTime/(1*Planck Time)
nu=c = 3*10^8
This gives a
Mu=4/3*(2c+1)^3/2 / c^2
Simplifying (c is very large) . This gives the approximation
mu=4/3* 2^1.5 / c^0.5
mu=2.1773242 * 10^ (-4)
mu = 1-0.999783357
This means that 99.9783357 % of the universe can be interpreted as being “Dark Matter”.
Ie with attractive and repulsive qualities . Basically empty space .
May you have joy of that .

An interesting aside :Creative artists .
How many pieces of paper does an artist need to crumple up and throw away before he finds something acceptable ?
Something acceptable would translate to mu=1 . Thus , we can say 7-8 truly random foldings should give a result .
The same holds for cryptanalysis or any attempt to find an unknown .
Algorithm :
Try 8 times , crumple , then put it aside and try again later .
There is a quantum connection , strange as it might seem .

And what about a nice little Crumpling App for smartphones ?
But the randomness should be from truly random tables , not pseudo-random generators .

Randomly yours.

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