Preview of a Homemade Chaotic (Magnetic) Pendulum

About a year ago while I was at a "curiosity" shop in the Singapore Science Centre, I came across an interesting table-top display sold under a hefty price tag. It was marketed with a catchy name "Random Oscillating Magnetic Pendulum" or ROMP and so I thought, hey, since the mechanism is essentially a perturbed pendulum, so why not make one myself that is worthy of display?

Sometime during mid April, I went to my father's workshop for its carpentry. I covered the base of the pendulum with white Formica laminated under a sheet of stainless steel which enable permanent magnets to stick on (picture above). The magnets provide perturbative force which disturbs the oscillation of a rigid pendulum (with ferrite magnetic tip) hanging from a static boom (arm). The designated poles of the magnet does not matter as we are only interested in its interaction between the magnetic tip and the magnets on the base - either repulsive or attractive. So, three bobs of different length (305, 311 and 314 mm respectively) was cut from a single piece of square wooden rod and after gluing a magnet to its tip, I aesthetically finished it with stained lacquer. 

The picture above (I digitally "stitched" two photos together) shows free-oscillations of two geometrical axis relative to the base of the pendulum using long-exposure photography. I gummy tacked a blue L.E.D to the tip of the bob and connected its electrodes to a battery with a fine copper wire from a stripped cable to minimize erroneous dampening and it is essentially a photographic technique called "light painting". 

The pendulum motion from this apparatus can never be a harmonic oscillator. This is due to natural dampening sources which eventually causes the pendulum to stop: in addition to friction between the hooks at the top of the bob (check video at the end of this article), air resistance and torsional force, we also need to note some residual "attractive" forces exist between the tip magnet and the stainless steel plate below the base affects the periodic motion. 

When a single magnet with same pole as the one on the pendulum tip was introduced at the center of its (where the pendulum tip will be when rest) oscillating path, the resultant trajectories can be drastically modified. This demonstration is included in the video while photographic example exhibits similar motion in 30 second exposure as the deflected path painted a chrysanthemum flower pattern around the central base magnet. 

These examples shows obvious changes in the pendulum's motion when natural oscillation under the influence of gravity was disturbed by the opposing magnetic force as it approaches the base magnet. However, the interesting part was that the geometry (square cut) of the magnets actually provide "sites of lowest energy" or metastable spots as the tip of the bob shows repeated, predictable, "dying out" oscillation when the system's energy is dissipating away.

Even with one base magnet, it begin to show this system exhibits chaos as the trajectory evolution of the pendulum depends highly on its initial conditions. We will soon see why it is almost impossible to get the exactly-same "chrysanthemum pattern" even though we tried our best to start the oscillation at the same spot. 

See, a chaotic system is characterized by having exponential growth (Lyapunov exponent) in error or more popularly known as the butterfly effect. Lets say if we have an instrument which allow us to measure initial velocity and position of the pendulum with extreme precision. Therefore, we should in theory be able to predict the trajectory outcome of the pendulum based on known physical laws like Newtonian mechanics and electromagnetism even though the analysis is extremely tedious. 

The catch is, there is a limit on how accurate our measurements can be done (for example, we cannot measure 0.00001 mm with a ruler). In fact, quantum mechanics dictate it is fundamentally impossible to know exactly a particle's position and momentum at a same time. Thus, our measurements are bound to have a slight uncertainty no matter how good the instrument is. As we mentioned earlier about the dynamics of the butterfly effect, these initial uncertainty from our measurements, although small, will grow exponentially as the we let the pendulum swing under influence of the magnet. Every single moving moment of the pendulum constitute a larger uncertainty to its motion and eventually it grows so large we are not able to predict which direction the next move will be. 

In the picture above, I used a red laser pointer to trace the path of oscillation (long-exposure photography) when 8 more magnets are introduced, all under "repulsion" with the magnetic tip. Tiny changes of either the position of the base magnet(s) or starting pendulum position will always produce a completely different movement. This is the essence of a chaotic system: given the knowledge of all initial positions, it is still difficult to predict the outcome even though the natural laws governing this system are fully deterministic.

So theories aside, I have not really quantified all the parameters which allows me to calculate the Lyapunov exponent. After all, the aim of this project was simply to prove "display-worthy" demonstrations can cost much lesser compared to similar products on the shelf. However, given free time in the future, with decent computer programming skills, it might just be possible to consider a mathematical modelling of this system to calculate its exponent value. 

On a side-note, I find it amusing that even chaos can be classified into subtle or complex types. Mathematical physicists are still working out if there is a fractal pattern called strange attractors emerging from the chaos generated by noise of certain systems. Strange isn't it?   

The occurrence of chaos in nature is not rare by all means. The weather is a good example. It is one of the reason why scientist find it tremendously difficult to accurately predict due to chaotic nature of air-flows, wind currents, temperature variations et cetera. Dynamic phenomena such as the pattern of Saturn's rings, SARS outbreak and the trigger event of heart attack are other examples of chaotic systems.

So, after discussing about the occurrence of chaos, it is not surprising to get one thinking (especially geophysicists) about the so called butterfly effect - "If a kupu-kupu (butterfly) flaps its wing in Taman Tasik Perdana, will it dramatically change the weather pattern in San Francisco?"

Moiré and Carbon Atoms

Here is one example of an optical phenomena coming from a common kitchen object, when briefly examined, I am surprised at the remarkable resemblance with something I've come across while attending microscopy lectures not too long ago. 

Figure 1

Anyway, this is a demonstration of Moiré pattern using a metallic teapot filter. If we pay attention to our surrounding, Moiré patterns are actually quite common! It is an intereference pattern caused by two grids overlaid at different grid size or orientation angle. This effect is the easiest to find when we cross two thin fabric over a bright light source. Try doing it! 

What I have, as mentioned, is a metallic filter that comes in cheap teapots. It has an arranged structure (like wire gauze or weaving of a thin linen) forming a mesh of individual holes with diameter not more than 0.5 mm. I used a white L.E.D as light source, trying to illuminate this object from various angle and capture the result Moiré pattern with my digital camera. 

Figure 2

The pattern itself is obvious even without specific lighting. But when we started to manipulate the light source, we can see how the contrast changes which are essentially reciprocal to each other. i.e. white change to black, vice-versa. 

Comparing figure 1 and 2, the patterns are reciprocal, albeit at a slightly different superperiod (superperiod is defined as the distance from one end of a repeating Moiré unit to another). Apparently the superperiod are affected by the camera's focal ratio and the distance of the teapot filter to the camera. I haven't yet explore this variable but I believe I would come back to that some day when I have a flat Moiré pattern generating object (like two pieces of transparency film with parallel lines) instead of a cylindrical (more accurately, a circular conical section) holey filter I'm using now. 

Anyway, you can say this Moiré pattern was generated by two superimposed mesh of pinholes. The pinholes are uniform in size so it is the tilt angle that creates this pattern.

Hence, from the superperiod, D, of the imposed pattern, we can estimate the angle of tilt, Z, following a few quantities such as the diameter of the pinhole, d, by this expression:

D = d / [2 sin(Z/2)]

From figure 1, we note that the superperiod is about 7 times the diameter of the pinholes. i.e. D ≈ 7d. So putting this into the equation above, should give us Z ≈ 8 degrees. Now, how do we verify this? 

Because the Moiré pattern was formed by superimposing symmetrical pinholes around a conic section,  which means the pattern is formed by overlaying the front and back part of a "cylinder", the angle of the cone can represent the angle of tilt of the pinhole mesh. 

Figure 3

Figure 3 shows the image of my teapot filter. By measuring the diameter of the bottom and the top of the conical section and the height of the filter, we are able to use simple geometry to calculate the angle of the cone, which roughly correspond to the angle of orientation tilt of the pinholes.

What I got, was 6.1 ± 0.9 degrees, taking account into the uncertainty of measuring all the parameters. So, comparing this result to the estimated tilt angle, well, it doesn't coincide perfectly but we see comparable results under forgivable error. After all, the superperiod was based on estimation of pinhole diameter without actual physical measurement. 

Anyway, what was important, is because this Moiré pattern was formed by circular pinholes, any physical phenomena that is caused by superimposing two grids of circles with identical diameter should yield the same Moiré pattern.

And that is precisely what I found in an article from the Cambridge Nanoscience Centre. According to the article, graphite (the thing that made pencil write on papers) which are made of stacks of one-atom-thick sheets of carbon (graphene) weakly "stick" to each other, can dislocate and slide from one another fairly easily. Because atoms are spherical (simply speaking), so when we see these superimposing sheets of atoms under special microscope (Scanning Tunneling Microscope) it will show interesting Moiré pattern, formed by the carbon atoms themselves! 

Figure 4

I mean, look at figure 4! When I photograph the demonstration, I used a L.E.D to cast a shadow of my teapot filter on the wall and the Moiré pattern emerged (left) matches so well with the Moiré pattern coming from overlaying two sheets of graphene that was seen using specialized microscopes (right). 

Now I'll never see my my teapot filter the same again - if at all, after I have broke the glass teapot itself which render its filter useless other than the purpose of novel photography. 

Hell Money: Cosmic Sum

I couldn't help but always used to think: "wouldn't the underworld suffer hyper-inflation i.e. buying a small piece of plain mantou might cost 1000000000 "hell currency" (lets name the currency HC for simplicity) when everyone give their dead so huge a sum, every time?" The thought is always synonymous with seeing the amount of paper-stuffs we burn as offerings to the deceased each year. 

To figure that out, lets indulge ourselves into the concept of a world which exist beyond our sensory realm. A place the decease will drift among themselves, living their lives with materialistic viewpoints which mirror our lives.

So let's start to picture this materialistic life after death: Initially, life might seems to be perfect, given the most filial child (or relatives) will burn incense, HCs, luxurious materials and offer fresh food every year during Ching Ming, or let us consider the things they might additionally offer in Chinese New Year. We will realize, each human year, they will have two cycle of "income" in which food and immense wealth will be supplied. 

But as we well know in our realm, food is a primary need. Which means, it would be ranked the most important matter to keep a person alive other than oxygen and water. (Ironically, I rarely see people offer  drinking water) They would have shelters and possibly new mode of transport every year, but food only comes twice a human year, so here's a logic block. 

Unless they eat twice a year, you may argue that the perception of time could be different between realms. So lets say, we condense their time such that each year in our world is equivalent to their day such that they can enjoy at least, a meal per "underworld-day".

Now, think about this. At some point of after their death, we will stop offering food and goods - not because the most filial son failed to adhere to his self-promise (or culture if you wish), but we as mortals will come to pass eventually. So if the concept of "delivering" them goods while we are alive to hold true, we better give them more than they could use such that when someday, the person who provide the offering passes-away, the beneficiary spirit will have sufficient supplies to go on. 

As mentioned, other than perishable food, we tend to provide a form of legal tender (in the form of hell-notes), which supposedly enables them to barter if they need to purchase, perhaps food, from other spirit who were wandering in the same realm which somehow they have enough supplies to sell.

To see such economy possibly lead to hyperinflation due to rapid increase in demand for food for coming years compared to the relatively large sum of money they receive, we have to make a few (crude but reasonable) assumptions:

1. An average human lifetime is 75 years old.

2. The eldest child is born at their parents age of 20. 

3. Once the parent pass away, the children will burn offerings until their own death.

Assumption 1. and 2. implies that the children's parents will pass away when the child is 55, and starting from age 55, he/she will have 20 years to provide the offerings until their death. 

To calculate the sum of things he/she can offer to the spirit, we need another few assumptions:

1. Each year they are two events which the child will provide the offerings. (CNY and Ching Ming)

2. Each event, the children will burn offering materials worth 10 million billion HCs, equivalent to Chinese 一億億.

In the course of 20 years, the child would have provided the deceased (4e17) HCs. In numbers, that is 400,000,000,000,000,000 HRs to last for.. um.. eternity. 

So lets say for the first 20 underworld-days (remember one day of theirs is one year for us) they have no worries about their food. (rather a quick family reunion to the dead, isn't it?) And after that, they would have 400 million billion for each wandering soul to spend for as long as time lasts.. how much can they allocate themselves (average) for use each day?

Now the last paragraph seems to be a paradox. How can you divide the amount of money, with infinite amount of days? It is mathematically meaningless to divide a number by infinity. Here we need to bring up a certain aspects in cosmology to wrap this study up. 

In 1924, a Soviet mathematical physicist A. A. Friedmann developed a set of equations, that if the constants in the equations were found, are able to reveal the ultimate fate of the universe: whether time will come to an end when the entire universe collapse into a single point (like a shrinking balloon which represent our universe, in which the balloon will shrink into the size smaller than an atom) , or time will go on forever with everything in the universe eventually disappears into nothing but empty space.

Since our argument about hyper-inflation in the after-world economy would involve the limit of time, so we select, the case which the universe is collapsing. Following experimental data from astrophysics labs, physicists are able to predict when the universe will collapse into a point: roughly 17 billion (human) years from now. 

This would mean, if the underworld is a subset of our universe, it can exist for another 17 billion underworld-days, which is about 46 million underworld-years. So, dividing 400 million billion HCs into 17 billion days, we have 23.5 million (23,500,000) HCs to spend every afterworld-day. 

So? 

Inflation? Of course - given the scarcity of food over the underworld, years after everyone died and become a materialistic hungry-spirit. (human species will barely outlive half of the universe's lifetime, but I digress) I can only see the other world as a wide-spread famine, glamorously adored with jewelry and golden contemporaries that are unnecessary to sustain living (oxymoron) itself. So it is either food is necessary, and hence everyone in the underworld is probably starving in the far future, or the whole concept of food is meaningless (and hence we can cut ourselves from offering food stuffs).

How about the money? Well, turns out, the value of the hell-money they can spend a day cannot be less than the monetary value during the 1923-1924 (note the year the cosmic lifespan formulae was found) German hyperinflation of the Weimar Republic when about half a kilogram of bread costs 3 billion (3,000,000,000) Marks, half a kg of meat costs 36 billion (36,000,000,000) Marks, and a glass of beer costs 4 billion.




But it all depends how much value the printing office of hell-money decides to give them isn't it?

On Lights From Masking Tapes

About a few weeks back while I was in the dark room preparing small square pieces of x-ray films, I discovered something interesting. The x-ray films are used in plasma-focus diagnostics but that's outside of this story.

What happened was, after preparing those light-sensitive x-ray films, I have to keep them in a light tight pocket. Because x-rays can penetrate the pocket material (usually a black pvc or in my case, a home-made black paper envelope) the films contained inside the pocket will register an image depending on the intensity of the x-ray, just like ordinary films.

Now, making the paper envelope requires a type of adhesive, so I used masking tape. Since I did this in a dark environment, it led me to observe something I could not understand at first.

So this is what happened:

The moment force is applied to the peeled tape, a layer of tape is lifted from the roll. What I saw was a momentary emission of bluish-white light coming out from the contact between peeled tape and the roll. This light only emits when the tape is pulled in a sudden. Astounded by this discovery, I kept repeating it to see if there are changes when I applied force of different magnitude (strength). The colour of the light is the same, but the intensity increases with increasing applied force.

So that night I went online to search what was it and I came across this from nature.com. It appears, if I were to perform this in a vacuum condition I would get x-rays! (what's more is that the x-ray intensity would be high enough to take x-ray images of a human thumb!)

Now since I was working in a plasma technology lab, I asked my supervisor if she's able to provide me a chamber for this test (and they have x-ray detecting diodes too) but she said the set-up is too time costly and took this as a novelty phenomenon.

Well, If I have a lab myself.

It appears that visible light I saw was generated from an effect called triboluminescence, which is an effect where light is emitted when solid material is given a mechanical stress such that chemical bonds are broken/altered to produce light.

Detailed information can be obtained from Wiki.