Not-so-random post about randomness

One of the most common misconception about randomness is that it is often confused with uniformity.
Could you tell which is the most random distribution of dots between the two below?

Which is random? (click to enlarge)

The most common answer would be the one on the right, which is wrong. In fact, the pattern on the right is generated by applying a small wobble (or uncertainty) on an uniform distribution. The pattern on the left, instead, has been generated using a pseudorandom [1] number generator in BASH (details in this blog post). Randomness allows for "clumps" to form and it's because of those clumps that the unpredictable behaviour of randomness comes from (unless, of course, you know the "shape" of the random distribution a priori). This appears more clearly whenever we look at the distributions shown above in 1D rather than 2D:

A random and an uniform distribution (click to enlarge)

It is clear that in the case of a uniform distribution, we have a greater level of predictability. Imagine you have a series of events which follows the uniform distribution. You are filling the histograms with events one by one by reaching 5000 events which recreate the distribution above. Whenever you approach higher numbers of events filled if there is a lack of events in one region of the histogram, there will be a higher chance for the next even to fall in that region.

You can picture in practice a 2D uniform distribution by imagining pouring a layer of marbles in a box, one by one. At the beginning their position will look random, because they can move around and take any possible position, but the fuller it gets the less available spaces there are and soon it will be easy to predict which are the only available spots for the next marble to go in.

The key thing is that whenever a position is occupied, it cannot be occupied again. This allows for a level of predictability, and this is not true of random distributions. Randomness is not predictable, its profile is unknown by definition, and it allows for the same position to be occupied again. This caused by the assumption that each random event is independent from the previous (Poisson process). Maybe this is also the reason why we believe uniform distributions to be "random". We are more familiar with those in practice and random events are either mostly abstract or harder to visualize.

This explains tendency of people to estimate randomness wrongly. This happens every time during lotteries. If a number has not come up for a long time, we feel that it is due soon. This is because we imagine random events as uniform. In fact, the probability of picking any number in a lottery is the same for every extraction, making all number equally probable to be picked, and not anyone more probable, because every extraction is independent from the previous.

An equivalent example is found in coin flipping. We feel that repeated head or tail events in a sequence of coin flips is a rare event. In fact, the chance of obtaining heads or tails is still 50% even after any number of heads have come up already. For example, events of the kind: THHTHTTHTTH and TTTTTHHHHH are equally as likely.

I created a computer-generated set of tosses to work out the frequencies of occurrences of repeated heads or tails (details on how this is done are in here). These are the results for 100000 occurrences of 15 tosses each:

 3 repetitions:  93894/100000
 4 repetitions:  64634/100000
 5 repetitions:  34667/100000
 6 repetitions:  16723/100000
 7 repetitions:    7789/100000
 8 repetitions:    3487/100000
 9 repetitions:    1584/100000
10 repetitions:     719/100000
11 repetitions:     332/100000
12 repetitions:     125/100000
13 repetitions:       53/100000
14 repetitions:       18/100000
15 repetitions:         7/100000

15 repetitions would mean having a full set of heads or tails, which seems impossible, but it happens 0.006% of the times. In fact, over 100000 occurrences, it happened 7 times. Close enough.

The data shows that three repetitions is an event which happens almost all the times, with four repetitions more than half of the times. Up to 6 repetitions, in fact, is not that much of an uncommon event. Almost half of the sequence! I am sure it would be hard to define "random" - in the common sense - anything which has more than 4 repetitions, although this simple test shows that it's almost the norm.

Another great example of how bad we are at understanding randomness is given by this small application at Nick Berry's DataGenetics. It will distinguish a randomly generated sequences of tosses (from a real coin) from the ones you made up yourself. It is not infallible, as it is based on the Pearson Chi-squared test (so it cannot always predict which is which) and after some trials it is easy to trick it to make it believe your sequence is random. Yet, I am sure it will give you a better understanding of randomness!



[1] I have used the word pseudorandom because every computer-generated programs simulates randomness. True randomness can only be found in some not completely understood natural phenomena. Being a simulation, there are different "qualities" of generated random numbers. I am not sure about the one used in BASH, but it is usually good to stick with a higher quality random number generator for more serious business (e.g. GSL)

Connection between derivative and integral

I remember that most of my struggle in understanding geometrical properties of calculus in first year undergrad classes was spent in working out why the integral is the inverse operation of derivative.

There are many mathematical proofs of this, which make perfect sense, but their connection is counter-intuitive as they perform seemingly different operations on curves. The derivative finds the slope at any point and the integral sums infinitesimal slabs of area under the curve. All derivatives can be solved analytically, while that is not true for integrals. Integration depends to an additive constant, while derivatives do not. Dissimilarities surely are numerous.

I have recently found this paper I wrote during my undergraduates in the quest to understand the geometrical origin of the connection between. Hope it is of any help to you, if you are having trouble in this like I had. I remember it made much more sense when I - very unrigorously - put it in this simple way:

P.S. for "Derivate" I mean "Derivative"

Research

P.S. Beware of local minima

Of Einstein's genius

This man oozes brainpower. Maybe not in this picture, though.

The real genius in Einstein's work is in the fact that he discovered a fundamental property of the universe from only a few assumptions.
He set the speed of light as constant in different frames of reference and developed the mathematics of Lorentz transforms further to come up with one of the most beautiful theories in physics.

While usually in physics direct evidence and data is studied to deduce a phenomenon, he inductively derived his theory having no evidence at all. He made his own axioms and derived everything again from them. And the best part is that it worked, at least in theory. Despite a few physicist immediately recognized his genius, his theories were not completely recognized by the scientific community at the beginning, especially when the paper for his theory of Special Relativity presented no references. Einstein's reasoning was outside the lines, but sharp, nonetheless, and it proved no mistakes.

Evidences for his special theory of relativity started to come up only much later, around 1932, with the Kennedy-Thorndike experiment testing the dependence of the speed of light on the velocity of the measuring device.
Direct evidences of the general theory of relativity are still feeble, even though it is now a firmly accepted theory.

No, it is not photoshopped

The difficulty in testing this theory (linked to the difficulty to discover it) lies in the fact that we need astronomically strong gravitational fields in order to detect the space-time fabric and its tiny ripples: the gravitational waves. I did not use the word astronomical by chance, as the best empirical evidences come from deep space. Usually quasars (active galactic nuclei), which are very far, but bright x-ray sources, are used as indirect test for the theory of general relativity, as their ginormous gravitational field can bend light. Pictures of galaxies behind quasars could be worked out from their light being bent right into our eyes (or telescopes), forming "rings" of light around the quasars.

A more direct evidence of general relativity would be detecting its evident result: gravitational waves. Unfortunately, these waves are far less energetic than anything we can imagine, making it easy to let them disappear into the much higher and chaotic background of radiation. Experiments exist trying to achieve the impossible through interferometers, but a much more concrete evidence, at least for now, is presented by binary pulsars.

Not the most scientifically
accurate gif I could find,
but surely the coolest.

Binary pulsars move into space and their gravitational field emits gravitational waves, which is a leak of energy from the system. Everything, in fact, emits gravitational waves, but for small (less heavy, to be precise) objects their emission is undetectable. As energy is leaking from the binary system of pulsars, their period of revolution will get slightly slower than usual and with enough time passed, this accumulated time will get significantly big to be measured. This is not a direct evidence, as the energetic leak that slows the period down is not necessary given by gravitational waves, but that is the only phenomenon we know that could cause it, for now, and Einstein's model works really well when applied to these bodies, so it is still evidence, even if indirect.

When I started writing this post, it was meant to be an introduction to one big recent news in science in one of the FSND series. I was so excited writing about how marvelous Einstein's theories are that it got too long and I decided that it would easily be a blog post on its own.

The news regards the recent first visible-light evidence of gravitational waves from a pair of dwarf stars, and I'll now leave you in a pointless cliff-hanger (pointless as the news is already out elsewhere) as I will talk about it in the next Fortnightly Science News Digest on 15 September.

How to balance two forks on a toothpick

I posted a video a while ago about balancing forks on the rim of a cup:

This trick can be easily made using just 2 forks, 1 stick and a glass.
Your eyes won't believe it, but physics laws have not been violated in the making of this video.

The physics behind it

The center of mass of the system (forks + stick) falls around the middle of the stick, which lies exactly on the pivot.

That is the most stable position giving then stable equilibrium. Even for small displacements the system is balanced by a restoring torque.

Moreover, the system remain balanced even if half of the stick is burned, because the missing weight of the burned stick is negligible with respect to the weight of the whole system, then the center of mass approximately stays in the same position as before.

How to make it

The making of follows three simple steps:

1.   Put a toothpick (or any stick that can stand the weight of two forks) between the teeth of a fork.

2.   Take another fork (of the same kind) and push its teeth between the ones of the first fork. This is the most difficult bit as most of the times, the two forks and the toothpick will not stick together (you could use glue at this point without making the trick pointless, but I managed to do it without glue).

3.   Put the structure on any edge, trying to find the point on which it will balance (no glue allowed at this point). You will find it by the pressure on your fingers while you try to do it. The point depends on the shape of the forks and in my case the centre of mass fell on the middle of the toothpick.

3b.   Impress your friends!

Higgsteria - How to interpret the mass spectrum graph

The seminar on the 4th of July held at CERN in Geneva - despite the use of Comic Sans font - gave very interesting news to the Physics community.

ATLAS and CMS, two experiments from LHC, both discovered a new particle (5-sigma level is a requirement for a discovery) at 126 GeV in the mass spectrum.

Jargon apart, if you are not a physicist, you can read info about the Higgs boson (such as what is the Higgs, what is a boson, how does it give mass to other particles...) pretty much everywhere nowadays.

What many people may wonder is: what is this 5σ? And what about this 95% confidence level (CL)? But, most importantly, what is 126 GeV?
In this post I will give you an idea, with simple language, about these concepts and you will be finally able to understand this (not-so) mysterious graph.

5 "must-see at least once in a lifetime" Celestial events

Most of the travellers choose their destinations based on the wonders of the world. Some of them, choose them for the wonders of the universe.

There are some beautiful celestial events that are visible only on certain locations on Earth at certain times. Their rarity and poor availability, combined with their beauty, drives people to travel to be able to see them.

Here is a list of 5 events worth seeing at least once in a lifetime. For each one of them I wrote a blog post on their nature and how and when to find them:

1. Aurorae
2. Solar Eclipse
3. Midnight Sunset
4. Milky Way
5. Meteor Shower

Enjoy.

1. Aurora

Aurorae are one of the most impressive views of a night sky and they are very famous to be an event that not all the skies can host.
They happen around the polar regions, both in the northern and southern hemisphere taking respectively the more common name of northern and southern lights.
The curvy movements and the lightness with which they fly across the sky is a really hypnotizing and fascinating sight that cannot leave even the most apathetic man without a "wow" out of his lips.

[This is a post which is part of the series: 5 unmissable celestial events]

2. Solar Eclipse

Solar eclipses are natural phenomena that occur when the Moon passes between the Earth and the Sun, obscuring the latter.
This event is so unnatural that it astonishes and amazes not just humans, but many others living creatures. Studies have shown that animals react strangely to solar eclipses. Their behaviour is driven by the absence of sunlight where there should be and, in fact, depending on the animal, they usually prepare to sleep.

 [This is a post which is part of the series: 5 unmissable celestial events]

3. Midnight Sunset

The midnight sun is a quite surreal phenomenon happening in very northern or southern latitudes, nearby the polar regions. It is nothing more than having the sun out in the sky, only, at midnight!
What is stunning, though, is that (it depends on the latitude and the season) the sun does not set, but remains still on the horizon before rising again and giving sleepless "nights" to visitors.

 [This is a post which is part of the series: 5 unmissable celestial events]

4. Milky Way

A starry sky is always a very relaxing and touching view, and often, if we spend some time to contemplate it, when our eyes are well adapted to darkness, we could spot some steady "white clouds" between the stars.
Fortunately that is no premonition of rain, because those clouds are well beyond the Earth's atmosphere. That is the Milky Way, no less than the very galaxy we live in!

[This is a post which is part of the series: 5 unmissable celestial events]

5. Meteor Shower

Seeing a falling star is a peculiar event. Its rarity permitted the birth of the popular conception of expressing a desire when seeing one. But maybe this habit founds more solid roots in the fact that people nowadays spend a very little time looking at a night sky, than on the rarity of the event itself.
Millions of meteoroids are in orbital collision with the Earth every day and we should thank our atmosphere that only a tiny percentage reach the ground with much smaller sizes than the original object. On certain seasons the frequency of falling meteors is so high that the event is called meteor shower. But why the Earth is tormented by these intruders?

 [This is a post which is part of the series: 5 unmissable celestial events]

Shower

Having a shower is a relaxing moment of the day. Some people sing, others just imagine, but whatever you do, the mind is surely free to think about anything and fly (freely) over a lot of disparate topics, often non-related between each other.

I don't know how, but when I was showering, today, I happened to think about an awesome quotation from Tanenbaum (besides, a Physics doctorate and one of the man that "triggered" Linus Torvalds to write his famous UNIX operative system):

Never underestimate the bandwidth of a station wagon full of tapes hurtling down the highway.

What's so interesting about this expression?
Even if it has been used in 1981 on a different context, its irony could still be up to date, since Sneakernet is a still very common practice and couldn't (still) be easily substituted by virtual data transfers (p2p, ftp...).
I've heard new versions of this quote with "hard disks" instead of "tapes", indeed.

The next step on my train of thoughts - during the shampoo - was the realization that I can calculate the actual bandwidth, a kind of average in kb/s, of a car bringing an hard disk.
How? Dimension analysis, obviously!

The dimensions of velocity are: [Length] / [Time] , so to obtain the dimensions of bandwidth [Data] / [Time] it is just necessary to multiply by the data carried and divide by the distance covered.
The result is a formula for the bandwidth depending on velocity of the car v, the distance travelled l, and the data carried d.
${b = \frac{v \cdot d}{l}}$
(if you see incomprehensible signs enveloped by dollar signs, Latex script is not working properly)

If we pick a car travelling a distance of 1000 km at 130 km/h carrying a Terabyte:

${b = \frac{130}{1000}\,TB/h \sim 133\,GB/h \sim 38\,MB/s}$

Quite satisfying.

This is useful (maybe not) but it's not finished.
If you can calculate the bandwidth of a bunch of data travelling in a car, why don't we calculate the velocity of a bunch of data travelling through the wires knowing the bandwidth?
In this case data should be considered like a solid packet, and d is the distance between the host and the server. After some easy math:

${v = \frac{b \cdot l}{d}}$

Wow, you're still reading... crazy. Anyway, that's not all, not yet:
if you pick a value for d (data) smaller than b (bandwidth), it almost doesn't make any sense and the result could be a velocity higher than the speed of light, indeed.
Since the speed of light is an upper limit:

${v = \frac{b \cdot l}{d} \leq c}$

and so:

${b \leq \frac{d \cdot c}{l}}$

that is the absolute upper limit in bandwidth.
If we put numbers, the result still doesn't make any sense.
For example, since the net works in small packets, for a 64 byte packet from a server 7000 km far, the upper limit speed is about 2.7 kb/s.
This result could be useful if we make a small modification to find the time required to transfer that packet at c (dimension analysis again):

${t = \frac{d}{b} = \frac{64\,byte}{2.7\,kb/s} \sim 0.02\,s}$

That is 20 milliseconds: the lowest time physically possible needed to receive something from a distance of 7000 km.
Real timings are about 150ms for good servers, to send and receive, so about 75ms one way, that is in the limits...

Now I ended the shower and then also this small trip around physics and the internet. The result of this brainstorming thoughts? I really don't know, but I still like it.

100 points for the first who analyse relativistic effects on the packets and find how much kilobytes the file loses travelling at that speeds.

Moonset and Jupiter again

Same telescope as in the previous post, different situation: waxing moon.
After some time wasted (the temptation to point the telescope to the ground and spy innocent people from the top of my house is unimaginable) I convinced myself to watch the orange setting moon a little nearer (20mm eyepiece):

 

And after I proved that Jupiter's moons (obviously) move and then (indirectly) that gravitational laws are true. As you can see in the pictures below, with the same telescope (20mm eyepiece) the moons have a different configurations and it agrees with Stellarium. (mouse rollover to circle the moons):

Stellarium (mouse rollover to name the moons):

I've seen what Galileo saw

Clear sky, Tramontana (a Northern wind known to be very dry in Southern Italy), new moon, neighbourhood lights off: perfect occasion to dust off my brother's telescope and do some night-sky observations.

A refracting telescope, 10cm objective lens and two 34mm and 20mm eyepieces; not so bad for the Moon and planets.

I tried to spot some stars but they were too faint because of that damn light pollution, so I decided to see one of the most luminous astral bodies in the Northern Sky: Jupiter.
After some focusing, I spotted Jupiter with its prominent lighter-hued zones.
But I also noticed what at first sight I thought were some refractions/reflections. There were smaller dots aligned near Jupiter, with different brightness. They were too strange to be some optical effect, so the second assumption was: the Jupiter's moons!
That dots were quite far from Jupiter, in my opinion, to be its moons so I wasn't so sure about that, but they were four (as the Galilean moons), aligned and with different sizes.
I took a picture with my phone (one of the most difficult things in my life, but I was determined to take it) and the result is a very fuzzy and dirty image. Unfortunately the view through the telescope was much clearer and defined, but it can get the idea across (click on the image to enlarge):

After reversing and some photoshopping (or better "gimping"):

Then I immediately checked with Stellarium what kind of bodies they could be, whether Jovian moons or stars. This is the screenshot:

Fascinating.

Why aren't there any green stars?

Sun, green wavelengths filtered

I've wondered for long before studying physics why there are no green star. There are many beautiful picture of stars around, but why green is always missing? Has Nature decided to discriminate green?

The simple answer is: no, but we don't see it because of how we perceive colours and because there are no stars that emit only green wavelengths.
Not satisfied?

Good, because we need to use some physics to understand why we don't percept green in stars.

Firstly, we see the stars because they emit light.
Light is an electromagnetic wave, also know as radiation and the different colours of light emitted depends on its wavelength. The visible range of wavelengths can be seen here.
Stars, for an empirical reason (but there is also an explanation for that), emit a spectrum, that is a range of wavelengths. Then, for example, a red star does not emit only "red wavelengths", but a specific range of wavelengths that include the red.
The star colour depends on its temperature: higher temperatures correspond to shorter wavelengths, that is "bluer" colours, and lower temperatures correspond to longer wavelengths, that is "redder" colours. Then, even if red is a warm colour and blue is a cold one, a blue star is actually much hotter than a red one.
Is this range of colours emitted with the same intensity? No, the intensity of each colour emitted follow a specific path, discovered with quantum physics, called blackbody radiation curve and you can find a cool toy (applet) to play with it here. For each temperature, there is a different path that includes a different range of colors.

Now let's see a bit how our eyes interpret spectra.
When we see a color of an object, we are actually seeing the composite color, that is the mix of the colors emitted (or reflected). Black and white are not really "colors" since black is just an absence of color, while white is formed by all the colors. A prune has that nice purple color because it basically reflects red and blue wavelengths and our eyes mix them up.
The same applies for stars, but you have to mix up a very large range of colours with different intensities.

Known that, let's use those information to answer the question.
The applet used before helps a lot: it mixes the colours under the curve and shows the composite result. You can change the temperature in Kelvin of the curve (i.e. of the star) on the bottom (our Sun has a surface temperature of 6000 Kelvin, very roughly).
When the temperature is low, the star is evidently red because there is almost no blue at all in the spectrum (or it has a very low intensity). It is clear also why very hot stars have a bluish colour, since in the spectrum the red colour has low intensity in respect to blue.
It is interesting to see what happens when we choose the peak near the green wavelength. The sum of the colours is white, not green!

You can't see green stars because of the nature of the star emission, in which the green color is included, but is not perceived by our eyes because it is mixed with all the other colors to form white.

I'm not dead

Even if I haven't written for some time, this blog is still on, but now I'm spending the whole time to study.
I've got exams in May, and I hope to pass (decently) this first year studying physics at UCL.
For those who are interested, this is the list of the exams:

13 May - Thermal Physics
18 May - Mathematical Methods II
19 May - Waves, Optics and Acoustics
22 May - Classical Mechanics
26 May - Mathematical Methods I
27 May - Physics of the Universe
28 May - Communication Skills Talk (Coursework)

See you soon.

Thermal Death

This is the second part of the sketch, made the same evening (have a try to do that holy problem sheet wasn't an option).

Skydiving or Chemtrails?

When I saw this image, I couldn't hold out against thinking that those people were sent from the NWO and that it's a new method to create chemtrails. If you don't know what the "New World Order" or the "Chemtrails" are, just google them yourself and you'll find how much ridiculous that theories are. Anyway, skydiving has always fascinated me. Seeing the world from an uncommon perspective with your own eyes (not a satellite) and feeling the adrenaline for being in free fall should be amazing. I think it's worth to do it sometime in your life. At least it's a good way to prove directly Strokes' Law.

Entropy

This is a small sketch made during a boring evening trying to do the last Thermal Problem Sheet of the term. That's very xkcd-style and I know it's not very funny but the idea is mine (at least!). Enjoy (click on the image to enlarge).
P.S. Happy Easter!