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 .
|
2.5 |
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)).
|
|
Measured
|
2.66
|
 |
||
|
2.79
|
 |
|||
|
2.97
|
Lung surface
|
.jpg) |
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:
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
Available from some deep-sea bacteria : see http://footnote1.com/electricity-conducting-bacteria-form-living-wires-on-ocean-floor/
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 !
Regards
Andre
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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)
=2.44
---
Wednesday, February 29, 2012
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 .
Synopsis:
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
r=d*(n0-1)/2
n0=2*nu+1
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
Mu=4*4/3*(n0^3/2/(n0-1)^2)
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
nu=150/0,1
nu=1500
mu=4/3*(3001^3/2)/(1500^2)
mu=0.097421589
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.
Andre
xxxxxxxxxxxxxxxxxxxxxxxxxxxx
Andre Willers
23 Feb 2012
“The moving finger writes , and having writ , crumples it in random ruins.”
With apologies to Omar Khayyam .
Synopsis:
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
r=d*(n0-1)/2
n0=2*nu+1
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
Mu=4*4/3*(n0^3/2/(n0-1)^2)
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
nu=150/0,1
nu=1500
mu=4/3*(3001^3/2)/(1500^2)
mu=0.097421589
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.
Andre
xxxxxxxxxxxxxxxxxxxxxxxxxxxx
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