Journal of Physics Special Topics
P4 3 Hubble Space Origami: The Cosmological Fold-Limit in
Concordance Cosmology
J. S. Baker, R. R. Norisam, K. E. Wright, T. W. Buggey
Department of Physics and Astronomy, University of Leicester. Leicester, LE1 7RH.
October 22, 2015
Abstract
In this article, we attempt to answer the age-old question: ”How many times can you fold a piece of
paper?”. We make use of Gallivan’s equations and λ-CDM Cosmology to evaluate the maximum fold
count at the present age of the universe and beyond. We find the present maximum number of folds to
be 67. We also find that our universe has only just entered an epoch in which the maximum number of
folds can be completed. We discuss the philosophical implications of this. Is the universe fine-tuned for
origami?
Introduction
”How many times can you fold a piece of paper?”,
thought Britney Gallivan to herself in her high-school
mathematics class. At the time, it was thought that
the maximum number of times a piece of paper could
be folded was seven or eight times. Gallivan went
about proving this false by deriving formulas for both
uni and bi-directional folding; imposing the theoretical
maximum limit for the number of folds possible given
the dimensions of a sheet of paper. Gallivan then went
on to use the uni-directional method of folding to beat
the standing record for fold count, clocking in at eleven
folds [1]. Furthermore, it is found that Gallivan’s equations can actually be extended to the number of times
any flat surface can be folded.
We now pose the question: ”how many times can you
fold a piece of paper in our observable universe?” Using
Gallivan’s equations and concordance cosmology, we
hope to answer the question once and for all.
Fig. 1: Square paper confined to the Hubble Sphere.
Gallivan’s Equations
Gallivan’s first equation for bi-directional folding [1] is
as follows
3
W0 (t0 , n) = πt0 2 2 (n−1)
(1)
expanding universe. The rate of expansion is proportional to distance from your own frame of reference [2].
This is the Hubble law
where W0 is the initial width of the surface (before
v = H(t)d
(3)
folding), t0 is the initial thickness of the surface and n
is the maximum number of possible folds. Gallivan’s Where v is recessional velocity, H(t) is the time evolvsecond equation for uni-directional folding [1] is then ing Hubble parameter and d is the distance from our
own frame of reference. At some finite distance from
πt0 n
n
L0 (t0 , n) =
(2 + 1)(2 − 1)
(2) our frame of reference, the rate of universal expansion
6
exceeds the speed of light: becoming ”super-luminal”
Where L0 is the original length (in the direction of fold- [4]. This distance is known as the Hubble distance dh
ing) of the surface. A consequence of both equations is and is approximately
c
that for a fold to be completed, a surface’s width must
dh ≈
(4)
be at least π times larger than it’s thickness.
H(t)
The Hubble Sphere
Where c is the speed of light in vacuum. We can then
In 1929, Edwin Hubble discovered that we live in an impose a sphere of radius dh with our frame of reference
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Hubble Space Origami: The Cosmological Fold-Limit in Concordance Cosmology, October 22, 2015
at its centre. Outside this sphere, light and therefore
information (also deemed to propagate at the speed of
light) can never reach our frame.
Suppose now we fit the largest square piece of paper
possible within this sphere (Fig.1). This is the largest
piece of paper that can be folded in the Universe. If
the paper were any larger, the ’information’ about the
incoming fold could never reach our frame. It is for
this reason also that we have to use the bi-directional
folding method as information would have to travel 2dh
(an impossibility) in the uni-directional regime.
√ Using the Pythagorean theorem on fig 1 yields W0 =
2dh . We can now solve (1) for n
!
√
2
2c
n(H) =
+1
(5)
ln
3 ln 2
πH(t)t0
Equation (5) describes the evolution of the maximum
number of folds possible by a square piece of paper with
the Hubble Parameter. We can chose to evaluate n at
our present epoch with a suitable value of the Hubble
Constant or we can examine the evolution of n with
time using λ -CDM cosmology [3]
p
3p
ΩΛ,0 H0 t
(6)
H(t) = ΩΛ,0 H0 coth
2
Fig. 2: Maximum number of folds versus time since the big
bang.
fold due to it’s tiny velocity relative to the expansion
of space.
Where ΩΛ,0 ≈ 0.7 and t is the cosmic time. Eq (6)
is obtained by using the scale factor, a(t), found in [3]
and the fact that the Hubble parameter is defined by
1 da(t)
H(t) = a(t)
dt .
Conclusion
We have found that there is a limit on the maximum
number of times paper can be folded in a universe obeying concordance cosmology. That number is 67 and
we have arrived at this maximum point in the recent
history of the universe. Although we have asserted
the theoretical maximum value, practical verification
of this cosmic fold-limit could prove a very difficult task
indeed. We therefore recommend further research into
a method of verification by experiment. For now, the
scientific method determines this limit to be a mathematical curiosity.
Results
We calculate the maximum number of folds possible at
the present epoch to 67. We used an estimate of the
thickness of A4 paper (0.1mm) and a Hubble Parameter of 70kms−1 M pc−1 [3]. A more interesting result is
the plot of n against t (Fig. 2). It appears the we live
in a time where dn
dt has just become equal to zero.
Discussion
We appear to live in a universe which has just matured
to the state where we can fold paper of the thickness of
A4 the maximum amount of times. Origamists (across
the cosmos, potentially) may argue that the universe is
indeed fine-tuned for origami but as philosopher David
Hume once said regarding a drop of water in a puddle: ”This is an interesting world I find myself in, an
interesting hole I find myself in, fits me rather neatly,
doesnt it? In fact it fits me staggeringly well, must
have been made to have me in it”. This exposes the
possible fallacies with the fine-tuning argument.
There are obvious shortcomings with the method in
which, practically, paper of this size can be folded. Especially the time it could take for the first fold to be
completed. Even if the paper closest to the edge of the
Hubble Sphere moved near the speed of light, it could
still take a staggering amount of time to complete the
References
[1] Wikipedia, ’Britney Gallivan’, 2015. [Online]. Available:
https://en.wikipedia.org/wiki/Britney_
Gallivan. [Accessed: 16- Oct- 2015].].
[2] L. Jardine-Wright, Hubble’s Law, 1st ed. Cambridge:
Cavendish Laboratory, 2007.
[3] ’Dark Energy and the Accelerating Universe’, Annual
Review of Astronomy and Astrophysics, vol. 46, pp.
385-432, 2008.
[4] Universe Adventure, ’Hubble Distance’, 2015. [Online].
Available:
http://www.universeadventure.org/
fundamentals/media/model-hubbledistance.swf.
[Accessed: 16- Oct- 2015].
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