Six Degrees of Separation Derivation

Andre Willers

27 Jan 2013

Synopsis :

We derive Reliability Percentages for Degrees of Separation from first principles of Small World Networks . Six Degrees has at most 98.85% reliability .

 

Discussion :

See Para 15 below for a real surprise .

 

1.Small-world Networks :

http://en.wikipedia.org/wiki/Small-world_network

"A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. Also known as degrees of separation ."

 

The major principle is that not all points(nodes) are interconnected . The nodes are clustered , with clusters connected by long connections

2.Examine an existing Small-world network in 2 dimensions . Sketch it on a piece of paper (2 dimensional) . The connections are shown as lines .

3.Define a Start node . The sender of the original message .

4.Then connect the Start Node with every other node , counting the number connections as Hops . This is computationally possible (not Hard) because not all nodes are connected .

5.Now , rewrite the Network Map in terms of Hops , Hops being the radius units from the Start node .

This gives a map of circular bands around the Start node as origin . The bands include all the nodes outside the Start node , with many duplicates .

The duplicates are important . Cluster n-hops are in them .

This is basically a Riemann-orbital system . See Appendix H .

6. The Trick !

We minimize the system by squeezing any wandering test path on the system to return to the Starter Node .

In effect , Riemann's theorem . We know that it returns to the origin if the real exponent = -1/2 .

(The only-if part exp=-1/2 is evident , but the theoreticians and red-pencil brigade have too much fun nit-picking)

7. The upperbound :

We can then calculate the Upperbound and Reliability percentage :

The actual Zeta = Sigma(x(i) / i^(1/2) , i=1 , 2, 3 ….

Where -1<= x(i) <=+1 at Beth(0) random for any i .

For the upper boundary , we set x(i) = 1 .

This gives :

Zeta (upper) = 1+(1/2)^0.5 + (1/3)^0.5 + (1/4)^0.5 + …

The Sum(n) = 1+(1/2)^0.5 + (1/3)^0.5 + (1/4)^0.5 + …+(1/n)^0.5 , the sum of Zeta(upper) to the n'th term . This is increasing .

The Diff(n) = (1/n)^0.5 – (1/(n+1))^0.5 is the difference between successive terms .This is decreasing

The Ratio(n) = Diff(n) / Sum(n) and is decreasing .

The Percentage Reliability PR(n) = (1-Ratio(n)) x 100

8.Table of Upperbound Reliability percentages for minimized Small-world Networks :

N denotes number of degrees (hops) in PR(n) below

PR(1) = 70.71%

PR(2) = 92.40%

PR(3) = 93.97%

PR(4) = 97.02%

PR(5) = 98.47%

PR(6) = 98.89%

PR(7) = 99.11%

 

9.What does this mean ?

See http://en.wikipedia.org/wiki/Six_degrees_of_separation .

It means "you pays your money and you takes your pick "

You get the most bang for your buck at about n=6 .

Small-world networks are very efficient at distributing information .

Remember , this is a very general derivation .

 

10 .What if just one node is left out ?

I suspect that the above will still hold .

It is a topological suspicion (the worst kind) . Even one little puncture causes a major shift in topological classifications .

In the class of "a little bit pregnant"

This has major consequences in the real world .

Plan on that there is always somebody on the "need-to-know" list being left out . This might be you .

 

11. What if many nodes are left out ?

Surprisingly , the system will default to 70% efficiency . Civil Service level . But remember that this is only the upperbound .

But is this then the Lowerbound for PR(2) and better ? I Think so , but some deeper analysis is called for .

No wonder civilization manages to struggle along .

Notice how close it is to 67% of Appendix H Infinite Probes .

 

12.The Family Paradox :

"My spouse does not understand me ."

Of course they don't . You are at PR(1) = 70% reliability as long as you communicate directly (eg language) . Just about bottommed out .

Add some more nodes . Usually this is children , grandparents , friends , hobbies , interests .

But different levels like emails , notes , flowers , presents , etc will work .

Even saving the Dodo .

Join the Dodo Liberation Army .

 

13. This is equally applicable to your work environment .

 

14. A surprising application .

How to turn your personal network into a Small-world network .

(Then you can use the system as described above )

14.1 Get long-range contacts . Pen-friends . Internet friends (but be careful) .

Even one will change the topology of your network .

You don't have to frantically network .

14.2 Virtual Topologies :

An imaginary friend .

https://www.facebook.com/MyImaginaryFriendLLC

These are available .

Notice the interesting addition of Internet at the bottom of Maslow's hierarchy of needs .

A Facebook account for an imaginary friend with strict access control will bounce parental communication with a child from 70% to 90% .

 

15.Marital Problems :

Or even between couples who have communication problems . A virtual friend on Facebook where only the two partners have access will increase communication by a whopping 20% !

What a surprise !

Not exactly what I was expecting .

 

15. Upgrading to higher dimensional networks .

Project the many hops onto a two-dimensional space as above . Then crumple them . See Appendix C .

This collapses into the same Riemannian space as discussed above

 

16. Virtual Topologies :

 

 

Viva La Dodo Liberation Army !

Andre

 

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Appendix H

Infinite Probes 2

Andre Willers

30 April 2008

 

http://andreswhy.blogspot.com "Infinite Probes'

 

From discussing this with various recipients , there seems to be a need for a simpler explanation . I thought I had explained it in the simplest fashion possible . The subject matter is inherently complex .

 

But , here goes .

 

How much must you save ?

If you save too little , a random fluctuation can wipe you out .

If you save too much , you lose opportunity costs . If you are in competition , this loss can be enough to lose the competition (ie you die)

 

Intuitively , you can realize there is an optimum level of saving .

 

Methods exist of calculating this optimum in very specific instances (ie portfolios of shares ,eg Kelly criteria , or tactics in war eg MiniMax ) .

 

The General Case

We need to hold a Reserve in case Something goes wrong . But we do not know what thing goes wrong .

 

Infinite Probes tries to answer the general case . What is really , really surprising is that a answer is possible .

 

The Infinite bit comes from using the mathematical expansion of the Definition of Eulers Constant e = ( 1 + 1/2! + 1/ 3! + … 1/n! + …)

Where n!= n*(n-1)*(n-2)*(n-3)*…*(1)

 

This approaches a constant , widely used in mathematics and physics .

(e = 2.718…) .

 

All we need is a system that can be subdivided indefinitely (to infinity) .

 

First , we divide by 1

Then 2

Then 3

Then 4

 

And so forth till infinity .

 

What is important is not that we do know what these divisions are , only that they are possible . We also do not know which one element goes wrong .

 

The other critical insight is that it is the relation between elements that is important . (Permutations) (The failure of an element in total isolation cannot affect the whole system by definition .)

 

We can count the number of relationships where there is failure of one element .

It is n! , where n is number of divisions where only one failure .

 

Multiple failures are handled by summing :

 

Our Reserve(R ) is divided by n to infinity and summed .

 

TotReserve= R*( 1 + 1/2! + 1/ 3! + … 1/n! + …)

TotReserve= R * e

 

To find the boundaries of our Reserve , we set TotReserve = Cost

 

Then

 

R = 1/e * Cost

R ~ 0.37 * Cost

 

What does this mean?

This method measures the upper boundary of the reserve needed to survive failures in any element of the Cost-Universe . Ie , internal fluctuations .

 

This is the surprising bit . Any society that keeps at least 37% reserves , can only be destroyed by something outside it's envelope . It is internally stable , no matter what .

 

Empires like the Ancient Egyptians , Romans , Chinese are possible , as long as there is no climactic fluctuation , new inventions , diseases ,etc . Rare events . Hence the technological stasis of old civilizations . The two are synonymous .

 

This is true at any scale (except quantal , by definition.) .

 

Individuals too . Humans can be seen as empires of noospheres .

 

The upper boundary does not take any double-counting into account . It is true for any system whatsoever .

 

A truly remarkable precise result from such general axioms .

 

The Lower Boundaries .

This is where it gets interesting .

Remember , we are just counting the number of ways in which permutations of one element can fail . We then sum them to get the effect of the failures of other elements

 

The easiest is the business that just starts and is not selling anything . It fails on n elements on every term . It's floor capital must then be

R=Cost/(1+e)

R=0.27*Cost

 

This is the initial reserve to get off the ground .

This is true in any ecosystem . This is why it is so difficult to start a new business , or why a new species cannot succeed . Or why waves of pandemics are scarcer .

 

For the epidemically minded , this 10% difference is responsible for the demise of the Black Death ( smallpox outcompeted bubonic plaque variants for the CR5 access site.Ironically , the reason why we have only a limited HIV plague is the high competition for this site , probably some flu vectors . As one would expect , the incidence of HIV then becomes inversely proportional to connectivity (ie flights) .

 

A cessation of airplane flights will then lead to a flare-up of diseases like these .

Not exactly what anybody has in mind . )

 

When we find that we really need the spread of infectious vectors to stay healthy , then we know we have really screwed ourselves .

 

These are the two main boundaries .

The literature is full of other limits the series can approach . Keep a clear head on what the physical significance is .

 

Fat

I cannot leave the subject without the thing closest to human hearts : appearance .

Fat and fitness .

Sadly , the present fad for leanness is just that . The period of superfluous food is coming to an end .

Rich individuals can afford to be lean because the reserves are in the monetary wealth Women have to bear children individually , so they cannot store the needed reserves externally . Hence their fat storage is close to the theoretical optimum even in Western societies (33%) . In other societies the percentage is about 37-40% .

 

Human males have been bred (Mk III humans) for muscle and little fat (8% in a superbly fit male) . He does not have reserves to withstand even garrison duty (even little diseases will lay him low .) Note the frequent references to diseases laying whole armies low .

 

Note what is left out : the camp-followers . They survived The women and babyfat children . Every army seeded the invaded area with women and children .

 

The bred soldier has to eat a high-carb food frequently : not meat or fat , his body cannot store it . This is the definition of a wheat-eating legionary .

 

Ho ! Ho! Ho!

The Atkinson diet .

No wonder it does not make sense in evolutionary terms .

Mesomorphic humans have been bred not to transform expensive proteins and fats into bodymass .

 

The soldier-class were kept on a carbohydrate leash , which could only be supplied by farming .

 

 

The Smell of Horses .

 

Horses exude pheromones that promote body-leanness in humans . This has an obvious advantage to horses . Horses are breeding jockeys .

 

The time-span is enough : at least 8 000 years . (400 generations)

 

Because pregnant women cannot ride horses , there was a selection pressure to breed horses who have a pheromone that block female dominance pheromones , especially since females have to weigh more because of fat-reserve considerations .

 

Outside a farming environment , horses will sculpt their riders as much as the riders are sculpting them .

 

Small Mongolian ponies , small Mongolians .

 

This is why alpha-males like horses and horse-dominated societies were able to conquer and keep matriarchies .

 

Note the effect of the pheromones on women riders . Androgeny .

On males it becomes extreme blockage of oestrogen . It seems like a surge of male hormones , but it is just an imbalance . (If too much male hormones , the men just kill their horses )

 

This is why the auto-mobile had such a big sociological effect . No horses , so the men became more effeminate .

 

Want to be Lean and Mean ?

Sniff Horse sweat pheromone .

Perfumiers take note .

 

Dogs

 

The other leg of the human-horse-dog triumvirate .

Dogs accept female pack-leaders and have evolutionary reasons for blocking horse inhibitions of human female pheromones .

 

While the males are away , the females look after and rely on the dogs .

(The reason why Mongols ride from yurt to yurt: they are too scared of the dogs.)

 

With dogs around , the male testosterone activity is ameliorated . This is a well known effect , especially if horses are around .

 

Hence the female love of lap-dogs . They are actually quite ferocious , and exude large amounts of pheromones that soothes the savage male breast .

 

Your attention is drawn to the Pekinese lapdog , which has had a disproportionately large effect on human history .

 

If this sounds convoluted , it is because this is exactly how this type of bio-system operates : by inhibitions of inhibitions of inhibitions ,etc .

 

Andre

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Appendix IV A

Orders of Randomness 2

Andre Willers

15 Aug 2008

 

See http://andreswhy.blogspot.com : "Orders of Randomness"

 

I have been requested to expand a little on orders of Randomness and what it means .

Please note that human endeavours at this date use only randomness of the order of flipping a coin ( Beth(0) )

 

Aleph is the first letter of the Hebrew Alphabet . It was used by Cantor to denote

Classes of Infinity (ie Aleph(0) for Rational numbers , Aleph(1) for Irrational Numbers , etc

 

Beth is the second letter of the Hebrew Alfabet . It means "House"

 

I will first repeat the derivation of Orders of Randomness from http://andreswhy.blogspot.com : "Orders of Randomness" because it is so important .

 

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Start Quote:

First , simple Randomness .

Flip of a coin .

Heads or Tails . 0 or 1

Flip an unbiased coin an infinite number of times ,write it down below each other and do it again .

All possible 0 and 1's

 

An example : Beth(0)

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Flips(1) 0,1,1,1,1,… etc

Flips(2) 0,1,1,1,0,… etc

.

Flips(infinity) 0,0,0,0,0,0,…etc

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This describes all possible states in a delineated binary universe .

"delineated binary" means a two sided coin which cannot land on it's side .

 

Now draw a diagonal line from the top left of Flips(1) to Flips(infinity) .

At every intersection of this diagonal line with a horizontal line , change the value .

The Diagonal Line of (0,1)'s is then not in the collection of all possible random

Horizontal coin-Flips(x) .

 

This means the Diagonal Line is of a stronger order of randomness .

This is also the standard proof of an Irrational Number .

 

This is the standard proof of aleph numbers .

Irrational numbers ,etc

Since any number can be written in binary (0,1) , we can infer that the order of randomness is the same as aleph numbers .

 

This means we can use number theory in Randomness systems .

Very important .

 

Google Cantor (or Kantor)

 

Define coin-flip Randomness as Beth(0) , analogous to Aleph(0)

Then we have at least Beth(1) , randomness an order stronger than flipping a coin .

Then we can theorize Beth(Omega) <->Aleph(Omega) .

 

End Quote

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Cardinal Numbers .

 

The cardinal number is the index x of Aleph(x) .

Cantor proved that

Aleph(n+1) = 2 ^ Aleph( n )

 

Where n is the cardinal number of the infinity .

 

Tying them together :

He also proved that

P(A) = 2^ n

Where A is any set , P(A) is the PowerSet of A and n is the cardinal number of set A

Thus , Cardinal Number of P(A) =(n+1)

 

The PowerSet of A = the Set of all subsets of A .

This sounds fancy , but it is simply all the different ways you can combine the elements of set A . All the ways you can chop up A .

You can see it easily in a finite binomial expansion (1+1)^n = P(A) = 2^n

 

See http://andreswhy.blogspot.com : "Infinite Probes"

There we also chop and dice , using infinite series .

 

Can you see how it all ties together ?

 

Why 2 ?

 

This derives from the Delineation Axiom . Remember , we can only talk about something if it is distinct and identifiable from something else . This gives a minimum of 2 states : part or non-part .

 

That is why the Zeta-function below is described on a 2-dimensional plane , or pesky problems like Primes always boil down to 2 dimensions of some sort .

 

This is why the irrational numbers play such an important part in physics .

Z=a+ib describes a 2-dimensional plane useful for delineated systems without feedback systems

 

Its in the axiom of Delineation , dummy .

 

But we know that Russell proved that A+~A<Universum .

The difference can be described as the Beth sequences . Since they are derivatives of summation-sequences(see below) , they define arrows usually seen as the time-arrows .

 

These need not to be described a-la-dunne's serial time , as different Beth levels address the problem adequately without multiplying hypotheses .

 

 

 

Self-referencing systems and Beth sequences .

 

A Proper Self-referencing system is of one cardinal Beth number higher than the system it derives from .

Self-referencing systems (feedback systems) can always be described as sequences of Beth systems . Ie as Beth(x) <-> Beth(y) . The formal proof is a bit long for inclusion here .

 

The easiest way to see it is in Bayesian systems . If Beth(x) systems are included , Bayesian systems become orders of magnitude more effective .

 

Life , civilization and markets are such . See below .

 

Conservation Laws :

By definition , these can always be written in a form of

SomeExpression = 0

 

Random (Beth(0) Walk in Euclidean 2-dimensions

 

This is a powerful unifying principle derived from the Delineation Axiom .

 

In Random Walk the Distance from the Center is = d * (n)^0.5 . This is a property of Euclidean systems .

(Where d = step , n=number of random beth(0) steps)

 

Immediately we can say that the only hope of the Walker returning to the center after an infinity of Beth(0) steps is if d ~ 1/(n)^0.5 . This is the Riemann Hypothesis .

 

Now , see a Universum of 2-dimensional descriptors z=a+ib

 

Sum all of them . Add together all the possible things that can be thus described .

 

This can be done as follows :

From z=a+ib Raise both sides to the e

e^(z) = e^(a) . e^i(b)

Raise both sides to the ln(j) power where j is real integers.

j^(z) = j^(a) . e^(b/ln(j))

 

Now , sum them :

Zeta=Sum of j^(z) for j=1 to infinity

 

Now we extract all possible statements that embody some Conservation Law . Beth(1)

 

This means that Zeta is zero for the set of extracted statements if and only if (b/ln(j)) is of the order of Beth(0) and a=(-1/2)

 

Tensors .

The above is a definition of a tensor for a discontinous function .

 

Riemann's Zeta function.

This can describe any delineated system .

If Zeta = 0 , conservation laws apply .

 

Zeta = Sigma(1/j )^z for j=1,2,3,…,infinity and z=a+ib , where z is complex and i =(-1)^0.5

The z bit is in two dimensions as discussed above .

 

This function has a deep underlying meaning for infinite systems .

If you unpack the Right-Hand side on a x-yi plane you get a graph that looks like a random walk .

 

If every point is visited that a random walk would visit over infinity (ie all) , without clumping , then Zeta can only be non-trivially zero if a=(-1/2) .

 

Why (x – yi) plane ? See "Why 2 " above . The system is fractal . Two dimensions are necessary in any delineated system .

 

Remember , randomwalk distance from origin = step*sqrt(number of steps) .

So if the steps = 1/ ( sqrt(number of steps) ) , then the Origin might be reached if and only if a= -1/2

This is easily proven .

 

If a= - 1/2 , then b can be any real function . This would include Beth(0) and Beth(1) , but not higher orders of beth .

 

If a= -1/2 and b is an unreal number , then a cannot be equal to -1/2 anymore . Conservation cannot hold at any level .

 

Consequences:

Conservation Laws can only hold for Beth(0) and Beth(1) systems .

 

This is forced by the two dimensions of delineation .

 

Mathematically , this means that Beth(2+) systems of feedbacks can only be described in terms of attractors or/and fractal systems (ie not in isolation)

 

Physically , conservation of energy and momentum need not hold for Beth(2+) systems .

 

This has an interesting corollary in decryption (unpacking) . A Beth(2) mind unpacking Beth(0) or Beth(1) encryption is functionally equivalent to Non-Conservation of Energy

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Appendix B

For Blast

Blast from the Past

Andre Willers

13 Jan 2012

 

Synopsis:

High intensities of gamma- and X-ray radiation on Earth from the center of our galaxy can be expected during 2013 AD , due to an infalling gas cloud into the central Milky Way black hole .

 

Discussion :

The discovery was made by Stefan Gillesson of the Max Planck Institute for Extraterrestrial Physics

in Garching , Germany .

References :

1. "Nature" , DOI 10.1038/nature10652
   
2. "New Scientist" 17 Dec 2011 p16 "Cloud suicide could transform black hole"
   

 

A gas cloud of an estimated three earth masses is expected to impact the supermassive , rotating black hole at the center of our galaxy ("Sagittarius A*") in 2013 AD . Note that radiation from this event can be expected shortly afterward in local time .

 

A large part of this mass will be converted to electro-magnetic energy .

 

Now , three earth masses is not a lot of energy in the galactic scheme of things , but the way it is distributed does .

 

Briefly , the energy release is characterized by very short wavelenghts (gamma or x-ray) and lobe-shaped distributions of the pulse-wavefronts in the galactic ecliptic .

 

There is a combination of relativistic and quantum effects in the last few cm's before the event horizon .

 

1.There is a slingshot effect , because the gas cloud must be lumpy , even if only at atomic scale . The gas cloud vanishes in a few Planck -time units , emitting very short-wave radiation due to differential tidal friction .

 

2.But the black hole is large and rotating , which forms time-bands of slower time , which gives enough time for feedback . Depending on the size and rotation speed of the black hole , this will lead to 2n lobes of wavefronts on the rotation equator , where n=1,2,3,...

 

3.Hawking Burps .

This process in para 2 above "foams" the event horizon , leading to a drastically increased Hawking radiation . (The "foam" would be something like Sierpinsky chaotic triangles)

The density and time-band concentrations of energy separates particle-anti-particle pairs to give Hawking radiation .

 

This burp of energy sweeps matter from the neighborhood of the black hole and reduces the mass of the black hole , preventing black holes from swallowing everything . Elegant .

 

4.Turning a Hawking Burp into a White Hole .

There is a narrow window where the Burp becomes self-sustaining . A large black hole then evaporates in an awesome release of energy . Even the tiniest random fluctuation (eg matter-antimatter) will get magnified and perpetuated .

Not the place to take your mother for a picnic .

 

5.Foaming Space-time .

Well , space-time is already foamed , but only to a simple level . (Quantum foam)

Call it Foam(0)

We can churn this to higher orders of foam (Foam(1) , Foam(2) , Foam(3) ,...,Foam(b) )

where b can be derived from Kantor's Aleph classes combined with chaos theory .

 

Taken to the logical extreme , this leaves only Foam ordered into Branes .

 

The smile on the Cheshire cat just after he brushed his teeth .

 

So what does this mean to us ?

A full frontal impact from a lobe will give a definitive answer to Fermi's Paradox : "Where are they ?" Why , extinct .

 

More likely is an immersion in the higher radiation levels alongside the lobes for a long time .

Can be survived , given enough warning .

Apply to the Max Planck Institute for Extraterrestrial Physics for more .

 

Friday the thirteenth can be quite a gas .

 

Andre

 

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Appendix C

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

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