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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