# Science with paper folding – I

Friends, I am back. It’s almost two years break and I hope I’ll become as regular as I used to be here. Today I’ll talk about mathematics and its DIY kind of application. This is related to something that I demonstrated in this year’s Science Day Program in IMSc, Chennai. This post is the part I of it.

Do you know how many times you can fold a paper into half? Using any of your daily sort of papers, you cannot fold beyond 7, i.e. mathematically $n_{\text{max}} <= 7$. Just give a try right now if you never have realized it. OK, once you have tried, you might have felt that it’s due to the limitation of human muscle that doesn’t allow you press harder and hence you actually need something else. What else? Maybe a powerful hydraulic press? Well, let’s check out this video.

Watched it? So you see that the paper breaks down like a biscuit after the successful 7-th fold! What could be the reason? Your physics mind may say that the paper is made out of some material which is not very much ductile, doesn’t allow it to withstand high pressure of a hydraulic press, and hence breaks down.

Surprisingly the answer contains more maths than physics. Think it this way : Suppose you actually succeeded in folding a piece of paper into half 42 times. Now for each folding the size of the paper reduces to half and the thickness of the paper gets doubled. So the thickness vs number of folds plot  should look something like this :

You can see that the thickness ($t$) grows exponentially with number of folds ($n$) on the paper : $t=2^n$ and for an unfolded paper with 1 mm thickness 42 folds will give rise to finally a thickness of $2^{42} \times 0.1$ mm $= 2^{42} \times 10^{-7}$ km which is nearly equal to 4,39,804 km. Now the moon’s average distance from earth is 3,85,000 km. This means, if you could indeed cross the number of 7 limit and fold 42, you would have built a paper made road to reach the moon! Whoa!

Now you’re close to the actual catch, i.e. you know that such thing is not possible. Every time you fold a paper, you double its thickness definitely, but you lose its area by half at the same time, i.e. an A4 size paper of length 297 mm will end up $2^{-42} \times 300$ mm = 6.75 \times 10^{-14}\$ m,  which is far smaller than size of any atom (typically $10^{-10}$ m). So practically you’ll no longer have a trace of the paper you began with!

However, folding for more than 7 times (say 8) may not lead to that much disaster if you manage to do it alternatively. For simplicity say you have a paper of 300 mm $\times$ 200 mm size paper. Now you set up a rule: Fold always along the longest side of the paper. So the paper size will change in the following way :

Fold 1 : 150 mm $\times$ 200 mm $\rightarrow$ Fold 2 : 150 mm $\times$ 100 mm $\rightarrow$ Fold 3: 75 mm $\times$ 100 mm $\rightarrow$ Fold 4: 75 mm $\times$ 50 mm $\rightarrow$ Fold 5: 37.5 mm $\times$ 50 mm $\rightarrow$ Fold 6: 37.5 mm $\times$ 25 mm $\rightarrow$ Fold 7: 18.75 mm $\times$ 25 mm $\rightarrow$ Fold 8: 18.75 mm $\times$ 12.5 mm $\rightarrow$. However, this doesn’t work as you’ve already seen in the video above.

In 2001 Britney Galivan, who was doing her junior high school at that time, took up this challenge and made a record of folding a long toilet paper 11 times. In the next year she made it to 12. And 12 is more or less the highest record (though there’s a claim of 13 folds with a long taped paper). However, what’s the most important contribution of her is that she found exact mathematical relations between the thickness and number of folds. For folding in the single direction the formula becomes

$L=\frac{\pi t}{ 6} (2^{n}+4) (2^{n}-1)$  (1)

and for folding in alternative direction it is

$W=\pi t 2^{3(n-1)/2}$  (2)

where $L$  or $W$ are the minimum length or width of a rectangular or square paper, $t$ is the thickness of the paper, and $n$ is the number of folds of the paper. Surprisingly, Britney didn’t earn that much fame for her formulas although no one has challenged the validity of them. In fact, Mathworld mentions her formula in its topic page on Folding and Wikipedia mentions that she was a keynote speaker in 2006’s National Council of Teachers of Mathematics convention.  It’s also said that she chased the folding craze in order to earn extra credit in maths in her school.  Most of Britney’s achievements and follow-ups have been archived in Historical Society of Pomona Valley, California’s webpage. The same society has published a booklet titled “Folding Paper in Half 12 Times: The story of an impossible challenge solved at the Historical Society office” , which I have not checked.

The derivation of Eq. (1) is available in Richard Condo’s (University of Michigan) page, in Gaurish Korpal’s article in At Right Angles, and some details in this blog post. The key thing considered in this derivation is that folding a paper of thickness $t$ into its half needs the paper to bend with arch $\pi t$ before forming a crease. This gives rise to the $\pi$-factor in the equation. $L$ in Eq. (1) is also known as the Loss function in the literature of Origami Mathematics.

I myself tested how Eq. (1) will tell the maximum number of pages for an A4 sized paper. The equation is actually a quadratic equation and this can be easily solved for a given length $L$ and thickness $t$. Let me just show below.

Eq. (1) implies ${(2^n-1)}^2+5(2^n-1) =6 L/({\pi t})$
$\Rightarrow x^2+5x-c=0$
where $x \equiv 2^n-1$ and $c\equiv 6 L/({\pi t})$.

Then we get
$x=(-5+\sqrt{25+4c})/{2}\,.$

Typical paper size (A4) : 210 $\times$ 297 mm$^2$. We take $L=297$ mm. Typical thickness $t=$ 0.1 mm. Then $c=\frac{6}{\pi}\times 2970 \simeq 5672$. Then $x=2^n-1 \simeq 38$. Then $n=\log 38/(\log 2)=5.24$. Thus if we keep on folding A4 paper in the direction of longest side only, we cannot cross more than 6 folds ! So more or less validates the 6-7 maximum number of possible folds theory, isn’t it.

Readers, you can check it for the second formula as well and let me know. I’ll end up by putting the Discovery  channel’s Mythbusters video, in case you doubt folding is not a serious business.

Upcoming (part II) : So far I have talked about folding a paper. What about cutting a paper? Could there be math involved too? That I’m going to talk about in the next episode. So just stay tuned. 🙂

# Can an insect teach architecture?

Can you guess what the above pic is about?

These giant sky-lantern shaped architecture on the rural landscape of Ethiopia are meant for harvesting water from the air, known as the Warkawater Project Architecture and Vision group. Warka is the name of a common Ethiopian fig tree.

The next question could be: Well, how does it work?

The detail may not be given by the key designer of the group, Arturo Vittori, but he discusses how such beautiful structures are built out of simple stuff and his vision towards changing the deprived areas where women and children spend a long journey just to collect water from the available resources.

Not only the journey is long, the water resources are not clean not and there are high risks of rape and abduction.

Now let’s get back to the question: what could be the engineering mechanism behind the architecture? Don’t laugh, an insect might be the best buddy to answer you.

Namib Beetle

Namib Beetles or the Namibian Desert Beetles (Stenocara gracilipes) are found in the Namib Desert of the southwest coast of Africa. It is one of the driest desert of the world. Despite the arid climate, Namib Beetles know how to survive. Their wind-covering shells have a perfect combination of uneven hydrophilic (water attracting) and hydrophobic surfaces (water repelling) which finally helps them to store water from a moist surrounding.

The idea of fog harvesting is not new. In fact, one should know this Indian and ex-IItian chap, named

Shreerang Chhatre, who came into the news during his PhD+MBA program in MIT for developing a mesh that can harvest water from the moist. The Namib Beetle purely inspired him (a nice presentation here) and if such an inspiration comes into a practical implementation, it is called biomimicry or biomimetics (e.g. Velcro tapes mimic burrs).

# Origami+optics=Foldscope, a poor man’s microscope

When I typed microscope in google, I found the following advertisement from eBay India.

From that we get the idea that microscope cost in India varies in the range: ₹1,500-7,000, which may not sound too cheap to be provided all health centers for diagnostic requirements in India or many other countries. Now Manu Prakash,  who is presently an assistant professor in the Bioengineering Dept. of Stanford University (an IIT Kanpur alumni as well), came up with an innovative invention: the Foldscope. It’s a microscope obtained just by folding papers in a way that a 2.4mm ball lens can be mounted by using the capillary-encapsulation  technology. Manu claimed that the whole fabrication costs around 50 cents which will be below ₹50 in Indian market. Watch out this brilliant TED show.

Could you ever imagine it? He called it a use-and-through microscope! The invention sounds like a revolution!

# “It doesn’t matter, does it anti-matter ?”

What could be your worst enemy whom you will always feel afraid to touch? He/she looks like you, but he/she is exactly opposite of you, your anti-matter. He/she can exactly perform the trick that you always can do. Because you both preserve a symmetry called the CPT symmetry. C stands for Charge conjugation, P stands for Parity transformation, and T means Time reversal. So even if your anti-matter looks like your twin, once you two come into touch, you both get annihilated.

Anti-matter is made of fundamental particles and we call them anti-particles. Like positron is electron’s anti-particle. Existence of positron was predicted by Paul Dirac in 1928 when found two solutions, one with positive and other negative energy, of the relativistic version of the Schroedinger’s equation (Now we call it Dirac equation). Particles obeying Dirac equation having positive energies and $-e$ charges are electrons, and particles with negative energies and $+e$ charges (the opposite sign in the charge is followed by the C symmetry) are positrons. Four years later positrons were discovered by C. D. Anderson in 1932 and this discovery awarded him Nobel prize in 1936. Similar many other anti-particles, e.g. antiprotons (by Emilio Segrè and Owen Chamberlain in 1955), antineutron (by Bruce Cork in 1956) and probably many others later on. When a particle collides its antparticle they both disappear by emitting energy as a photon (particle version of light). A photon is its own antiparticle and hence it automatically satisfies this collision rule.

However, to see a real matter made out of antiparticles we first need to build a matter, whose smallest unit is the atom. That means to create anti-partner of hydrogen, we need to bring an antiproton and a positron (antielectron) together in a similar way such that the positron orbits around the antiproton. Now this situation is very tricky to produce in the lab since anything in our world is made of matters, so any antiparticle once gets created will interact with its particle counter part present in the real world stuff (say, air, table, wall, etc.) and get annihilated.

The ASACUSA experiment at CERN (Courtesy: CERN)

This is a remarkable achievement since now probably can attempt to answer a few fundamental question:This week people in CERN’s ASACUSA (Atomic Spectroscopy And Collisions Using Slow Antiprotons) have claimed in their Nature Communication paper the success of creating 80 antihydrogen atoms by applying magnetic fields, detected 2.7 meter downstream after their creation.

1. Do antimatters obey CPT symmetry?

2. Will antihydrogen show the same spectral lines as normal hydrogen does?

3. Why galaxies and stars made out of antimatters have not been observed yet, whereas the Big Bang Theory supports equal amount of matter and antimatter at the beginning of the universe?

3. Do antimatters feel antigravity (instead of getting attracted like matters, two antimatters can feel repulsion from each other) ?

4. Can we use matter and antimatter annihilation energy to fuel up high-speed spaceships?

Antihydrogen project (Max-Planck institute)

The matter-antimatter asymmetry problem (article from CERN)

The other side (my previous blog post)

# Beer prank: a lesson to fluid-dynamics

I might be a bit late, already this has come in the news: A common beer bottle prank can teach a lot about fluid dynamics. The prank is the following. When you are in a party and you find your long-term enemy at the same party, you just approach him/her, say “Hello!” and hit the mouth of his/her beer bottle by the bottom of your own bottle. Voila! Your enemy will watch all of his/her beer erupting out as foam and leaving almost nothing to drink inside his/her bottle.

Javier Rodríguez-Rodríguez, Almudena Casado, and Daniel Fuster explained this weird effect in the last Annual Meeting of the APS Division of Fluid Dynamics by using a well-known phenomenon in fluid-dynamics called cavitation. Cavitation is the formation of vapor cavity/void/bubble due to the act of reduced pressure in a liquid.

So probably the above video says every thing. Nevertheless, let me try to put this in my own language. A sudden hit on the mouth of the bottle generates a shock wave, causing a sudden rise in pressure and creates a compression wave. When the compression wave hits bottom wall of the beer bottle, it gets reflected and forms an expansion (compression in the opposite direction) wave. During the expansion, pressure is reduced, which causes cavitation in the liquid. Thus throughout the beer liquid a train of compression and expansion waves form, which break the larger bubbles into daughter bubbles of smaller radii. These daughter bubbles move rapidly towards the neck of the bottle in the form of bubble-plumes. Since the surface of the neck is open and the daughter bubble’s motion is so fast, it finally erupts out by exhausting almost all the liquid inside the bottle.

Rodríguez said that understanding this kind of phenomenon may help to understand many other natural caviatation events, e.g. volcanic carbon dioxide in 1986 Lake Nyon disaster in Cameroon. His group submitted the explanation in the arXiv.org.

Now if you haven’t tried this prank yet, go and try it. But be aware of accidents, as the foam always doesn’t not come out instantly, it does certainly.

Now do you know (click if you don’t)

Why do bubbles in Guinness sink?

How To Make A Beer Freeze Instantly?

What happens when you do the same “hit beer bottle prank” with water inside instead of beer?

# Physics and Entertainment

Do you think physics are fun only for curious people and the real entertainments are matters of art and skill (e.g. movies, games, and sports)? If you think so, then you are absolutely wrong!

Tennis for Two ( Photo credit: Brookhaven National Laboratory)

The first video game, which was shooting with a light flash from a cathode ray tube (CRT), was invented by Thomas T. Goldsmith Jr. and Estle Ray Mann in 1947. The shooting was controlled by the knobs that are used to control the trajectory of CRT’s light beam. The inventors named the device Cathode-Ray Tube Amusement Device. Later in 1958, William Higinbotham of Brookhaven National Laboratory extended this idea to attract the visitors and get a break from the monotony of research. Higinbotham was a physicist, so he used cathode ray oscilloscope (CRO), which I guess any physics student uses at least once in life. His game was called Tennis for Two.

Now another excellent entertainment stuff came from IBM, which is a movie made out atoms only. The movie name is A Boy And His Atom . The atoms were imaged from the scanning tunneling microscopy (STM) measurement (thanks to quantum mechanics!).

How was the movie made?

I am grateful to Prof. Shobhana Narsimhan of JNCASR, Bangalore, who sent us the youtube links.