The tangent shaped track of a rain drop on a slanted tube

This picture made me think:

Fig. 1 - Constellation of metal tubes.

I took it on a railway station. The constellation of metal tubes keeps the first floor up in the air. A close up of the central tube makes clear what attracted my interest:

Look at the black tracks. Apparently rain drops fall down on the tube and create these dirt tracks while on their way down. All tracks converge to the gray strip.

I wondered whether it would be possible to compute the track of a drop. This is a typical physics problem. It turned out to be possible, as these pictures demonstrate:

Fig. 2 - Close up of Fig. 1.

The tube at the left is a different one than above and oriented vertically, to ease analysis. The blue track at the right resulting from theory resembles the track shape quite well. Note that it starts at the top at a position at the 'back' of the tube, visible in the plot right but invisible in the picture left.

Surprisingly, it turns out that the track resembles the plot of a tangent function. Straight lines, parabolas, circles, ellipses, sines, exponential functions and others - I know where one can literally see them in the real world. To my knowledge this drop track is the only example of a tangent function coming to life. Let's proceed to sketch the proof of this.

I used the Lagrangian formalism to solve the problem. This formalism is a generalization of Newton's laws. It allows one to use a suitable coordinate system, to handle constraints and to include frictional forces. For the latter I used a force which is linearly proportional to the speed of a drop. This is probably wrong, nevertheless it gives the reasonable result of above. Possibly the solution is not very sensitive to the type of friction one uses, as long as it leads to a constant velocity. Experiments show that drops move at a constant speed over an inclined surface.

For such a cylindrical problem it is only natural to use cylindrical coordinates (z, phi). The angle phi varies from 0 to 2xpi when traversing a circle at constant z. Surprisingly, when the friction is relatively large the problem can be solved analytically. These equations for the motion result in this case:

z= c1 * t

phi= 2 x atan(exp(-c2 * t + c3))

t is time in seconds. atan(x) is the inverse of tan(x). The  constants are:

• c1= -mgcos(theta)/k; m= drop mass [kg], g= gravitational acceleration (~9.8 m/s²), the frictional constant [kg/s], theta= the angle of the tube symmetry axis w.r.t. the direction of the gravitational force [deg].
• c2= mgsin(theta)/(kr); r= tube radius [m]
• c3= defines phi at t= 0 s.

Apparently the drop travels at constant speed c1 in the -z-direction, down the tube. No acceleration, hence no net force is excerted on the drop. This occurs because the frictional and gravitational force cancel each other.This is similar to a snow flake that falls at constant speed during quiet winter weather.

Fig. 3 - Tube pictured such that it appears to be vertical (left) and the theoretical result (right). Due to the rotation the gravitational force is directed parallel to the vector pointing from (0, 0, 0) to (0, 1, -1). I.e., the force vector points towards the reader, down right.

I guessed reasonable values for the parameters like the mass and tuned them subsequently to get reasonable resemblance between the analytic solution and the picture. The plots of z (left) and phi (right) I got are:

Fig. 4 - Plot of the cylindrical coordinates z (left) and phi (right) as a function of time t.

The blue curve of Fig. 2 (right) results from transforming these to (x, y, z) coordinates and plotting the result. phi asymptotically approaches 0 degrees. This corresponds to the dark band of Fig. 1 (left)where all drops converge to.

Note that one can recognize the tangent function in the right plot. This is due to the fact that atan(exp(t)) is approximately equal to a shifted and scaled version of atan(t) and hence the two have nearly the same shape. Because z is proportional to t, the track of drops on a tube thus resembles the tangent function. Check for yourself in the right plot of Fig. 3. As said before, this is the only case I know of where this function is visible in the real world.

A note on programs: I used Wolfram Alpha (https://www.wolframalpha.com/) to solve the differential equations and Octave (https://www.gnu.org/software/octave/) on Linux Mint (http://www.linuxmint.com/) to visualize the solution. The best things in life are free (https://www.youtube.com/watch?v=-GoYa1uNhYs)!

The road to reality - or beyond?

I held Roger Penrose's book “Road to reality” popularizing physics highly but I changed my my view a bit. This was due to discussing his short article “On the second law of thermodynamics” (1994) with some philosophy of physics students.

The article deals with the second law of thermodynamics. This law states that the entropy - the amount of disorder - of a closed system on average increases with time. It implies that the entropy of the universe goes up in the future. But what about the past? Penrose's solution is his "Weyl curvature hypothesis". This hypothesis states that the Weyl curvature is zero (or at least very small) at the Big Bang, and with it the entropy. More important than knowing what Weyl curvature is, is understanding what it explains:

1. The existence of the aforementioned second law.
2. The observation that the universe is homogeneous and isotropic at large scales.

The far more popular and rivaling inflation theory explains only 2. 

From the discussion it became clear that entropy has different definitions. The well established definition by Boltzmann is suited for the computation of the entropy of a gas. But what about that of the universe as a whole? Cosmology seems quite speculative.

Penrose is a good popularizer of mathematics and physics. His accompanying illustrations are unique. However, the danger is that the uninformed readers (as his readers will generally be) are seduced to see his speculative theories as being main stream. This occurs because he starts out in his books with accepted mathematical and physical material, slowly introducing his own speculations after that. Like the hypothesis of above and also his twistor theory. 

The question is whether Penrose leads one to reality or beyond.

Two ways of "counting infinity"

Do these sets have the same size? :

{1, 2, 3} and {1, 4, 9}

Is your answer the same if we extend the sets indefinitely, like this? :

{1, 2, 3, 4, ...} or {1, 4, 9, 16, ...}

Probably you determined the size by counting the number of elements in answering the first question. The sets {1, 2, 3} and {1, 4, 9} have each 3 elements so the answer to the first question is: yes, the two sets have the same size.

How to count the two infinite sets of the second question? This is impossible using conventional 1, 2, 3, --- counting. It would not end. Mathematician Cantor therefore proposed a method to find out whether two infinite sets have the same size, a way of “counting infinity”:

If we can pair all the elements of two sets than they have the same size. 

Because we can pair like this: {1, 1}, {2, 4}, {3, 9}, {4, 16}, ... Cantor's answer to the second question is the same as to the first: yes, the two sets have the same size. His method also works for finite sets, by the way. As such it is a generalization of the finite case.

But wait... something strange is going on. The set {1, 4, 9, 16, ...} is a proper sub set, a part of {1, 2, 3, 4, …} (not the same, hence “proper”). Nevertheless they have the same size. How can that be?

A mathematician's response might be that we should not speak of the size of a set but of its cardinality instead. That's the word Cantor introduced. This makes clear that we are dealing with the mathematician's world which differs from the ordinary world. In the ordinary world proper sub sets of a set are always smaller than the set - and infinite sets cannot be counted in this world. In mathematics proper sub sets of an infinite set can have the same cardinality as that set. Not the same size, which is just an everyday, non mathematical word.

So in the mathematical world it seems that we gain something - cardinality as a measure for the size of sets - and we loose something: that proper sub sets of a set are always 'smaller' than that set.

This is a kind of friction I was uneasy with for decades. I realized myself all too well the difference between 'size' and 'cardinality' of above and especially the more or less arbitrary choice of Cantor's pairing principle. Would another, maybe better choice be possible?

It felt as an impasse to me but probably not to William Byers, the author of the book 'How mathematicians think - using ambiguity, contradiction, and paradox to create mathematics'. His main point is that ambiguity etc. - things that somehow do not feel right - stimulate the formation of new mathematical ideas. Difficulties advance mathematics.

This is not an idea pertaining exclusively to mathematics. For example, koans are the Zen equivalent given by zen masters to their pupils. These are paradoxical puzzles like: “"What is the sound of one hand clapping?" To solve these one has to widen one's view of the world. A poetic, musical analog is this line from Leonard Cohen's song "Anthem": “There is a crack in everything. That's how the light gets in.” Imperfections are necessary for advancement, seems to be the suggestion.

Back to the original problem regarding infinity. How was it solved? About 100 years after Cantor the theory of “numerosities” was proposed by Vieri Benci. Like cardinality the numerosity is a measure for the size of a set. An important difference is that proper sub sets of a set have always a smaller numerosity than that set (while the cardinality can be the same).

Some more information can be found here: http://www.newappsblog.com/2014/03/counting-infinities.html. From the discussion there it appears to me that numerosity is a more complicated concept than cardinality – but I can be wrong.

'Mind and cosmos' by Thomas Nagel

Many people - and especially scientists - think that our current knowledge is sufficient to explain current life. Doubting the theory of evolution is not done. Nevertheless, this is exactly what philosopher Thomas Nagel does in his small book 'Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False' (http://opinionator.blogs.nytimes.com/2013/08/18/the-core-of-mind-and-cosmos/?_r=0).

Neo-Darwinism implies that the combination of mutation and natural selection (darwinism) drives evolution. It does not explain the existence of non material aspects of life like consciousness, cognition and value. This is no surprise, because science does not include these aspects.

Nagel pleas for looking for a naturalistic theory that describes the order that governs the natural world from the inside out. That is, a theory without supernatural elements. The order should lead without surprises and maybe inevitably to consciousness, cognition and value. Non material matters might also exist in some form in the beginning to evolve from there.

After investigating several solutions Nagel concludes with a teleological theory as feasible and the most probable. Teleology dates back to Plato and Aristotle. It states that development is aimed at a goal or purpose. This in contrast to ordinary causality, where effects follow causes. Teleology could in this case work by guiding mutations such that life as we know it arises in time. This guiding is necessary because Nagel beliefs that neodarwinism is 'too slow' for the development of life as we know it. The teleological guiding law would be just another natural law.

The book surprised many people because Thomas Nagel is a respected, atheistic philosopher. Teleological views do not fit into this picture. The book is rejected by many if not most philosophers.

The book is interesting to me because I belief already quite some time that neodarwinism is too slow and could be repaired by a teleological 'fix'. It is therefore a pity that the book is difficult to read and understand - at least for me. There are many long sentences and it lacks a decent summary of the main lines of reasoning. Taking more time for reading it would increase my understanding. However, I will not do this because I seriously doubt whether philosophy can contribute much conclusive to this subject. Maybe science should come to the rescue again?

IDEOLOGIES and ideologies

Last week about twenty people were killed in Paris due to a clash of two ideologies: one religious and one of free speech.

Friday morning I boarded a rush hour train to Leiden. I sat down opposite to a young woman with a black headscarf. She telephoned softly while the Metro (a free newspaper) was lying opened in front of her on the small window table. I was curious and turned my head slightly to see if she was reading about the Parisian affair. That was not the case. Then, while continuing calling, she rearranged her newspaper and with small shocks pushed the underlying Metro in my direction.