The nose job

A duel to resolve scientific arguments? Well those were the days my friend..

"While studying at the University of Rostock[12] in Germany, on 29 December 1566 Tycho Brahe, a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations, lost part of his nose in a sword duel against fellow Danish nobleman (and his third cousin), Manderup Parsberg.Tycho had earlier quarrelled with Parsbjerg over the legitimacy of a mathematical formula, at a wedding dance at professor Lucas Bachmeister's house on the 10th, and again on the 27th. Since neither had the resources to prove the other wrong, they ended up resolving the issue with a duel. Though the two later reconciled, the duel two days later (in the dark) resulted in Tycho losing the bridge of his nose.

In his De nova stella (On the new star) of 1573, Tycho Brahe refuted the Aristotelian belief in an unchanging celestial realm. His precise measurements indicated that "new stars," (stellae novae, now known as supernovae) in particular that of 1572, lacked the parallax expected in sub-lunar phenomena, and were therefore not "atmospheric" tailless comets as previously believed, but were above the atmosphere and moon."

Resource: http://en.wikipedia.org/wiki/Tycho_Brahe

Why do we love reading some books?

Why do we love reading some books, but not the others?

Inspired by mathematician Roger Penrose's depiction of Platonic Mathematical World, Mental World, and Physical World, I would like to present my theory.

From the book: Roger Penrose, The Road to Reality, Vintage Books, 2005.

I begin by breaking the problem into several worlds.

The world the author perceives, the world he describes, and the world we perceive.

The discrepancies between these worlds are inevitable; they would have cues on their own, as well as hindrances for our view.

Regardless of the author’s intentions, we begin to build our own world from page one. We continue to fit our perception on the world we have been constructing, not on the world that the author saw or described. Similar to divergence in mathematics we diverge from intended world-view.

So the secret to good authorship must be to relieve both the author and the reader from a burden, the burden of mapping author's world.

Rather than pushing an answer down the throat of the reader, the author should pose a question, such that the world he perceives disappears, and the world his reader perceives gradually unfolds.

---

Links:

http://en.wikipedia.org/wiki/The_Road_to_Reality

http://aminotes.tumblr.com/post/1025534467/the-concept-of-laws-the-special-status-of-the

References:

1. Roger Penrose, The Road to Reality, Vintage Books, 2005

A toroidal lamp

I woke up in Istanbul around 6am staring at a toroid shaped lamp hung from the ceiling.

Toroid’s surface has peculiar properties compared to a much simpler shape such as a sphere.

On a spherical surface you may start travelling in any direction and end up where you started. This property is valid for any point on the surface.

A spherical surface is a finite two-dimensional surface with no boundaries. It is a finite surface hence you end up reaching the same point. It has no borders or sharp edges that you may fall off.

As a result a spherical surface is a useful model to study a universe with no boundaries but a finite shape like ours.

A spherical surface has also constant curvature. Nowhere on it there is a point that sits on an area that is more or less curved than any other point on the sphere.

On a toroid on the other hand you may find points across its surface sitting on different curvatures. A point on its outer rim is on a surface with positive curvature, where a point on the inner rim is on a surface with negative curvature.

A simplified modelling can explain this “curvature business” that is the essence of Einstein’s general relativity.

You may construct a toroid by bending a cylinder along axis and joining its ends.

Imagine you make a toroid from a rubber cylinder. You will see that the outer surface of the toroid will stretch, and the inner surface will squash forming wrinkles.

Take a rubble rectangular slab, and on its surface paint little circles with identical diameter placed in equal distance from each other. Form a cylinder from the slab by rolling it so that the circles we drew can be viewed from outside. We should observe that despite introducing a positive curvature, the sizes of circles and the distance between them remain the same. There is no stretching or squashing; the curvature is positive but constant.

In contrast when we form a toroid using a rubber cylinder by bending it along its axis and joining both ends, we should observe that the circles on toroid’s outer surface grow in size and the distance between them increases too. The circles on toroid’s inner surface should shrink in size and the distance between them should become shorter. This is because we have introduced variable curvature changing from positive (stretching) to negative (squashing) from outer to inner rim respectively.

Imagine the toroid represents a certain space-time configuration.

It is possible a light beam to travel from circle A to its neighbour B on the inner surface in shorter time than between neighbouring circles on the outer surface. Space-time is clearly stretched on the outer rim, shrunk on the inner one.

A black hole is like a toroid that its inner hole is infinitely small hence the circles you drew on its inner surface collapse onto each other. There is no way to identify or differentiate those circles from one another, the information about them are sucked by black hole’s event horizon.

It is fascinating to think that there may be oddly shaped universes with multitude of curvatures. A universe that its space-time properties are shaped like a toroid is possible. But other weirder shapes too. Some of them may even have collapsed regions where curvature is transformed in odd directions that we may not easily imagine their shape in our Euclidian minds.

The reason we could visualise and construct a toroid is because we may construct it from Euclidian shapes that we are familiar with such as a cylinder. However we should note that we might not do so if we could not stretch and squash the outer and inner surfaces respectively. You cannot construct a non-Euclidian shape without introducing variable curvature.

A toroid is a non-Euclidian shape whereas a cylinder is a Euclidian one. Our education system has given emphasis on shapes with Euclidian geometry perhaps because there are economical benefits of realising them. Euclidian geometry has given us ability to make useful approximations. We can build pipes by modelling them as cylinders for instance.

However studying non-Euclidian geometry such as toroids, can be crucial in modelling and understanding the cosmos.

Evocom

I have been quiet for a while totally immersed myself into evolutionary computing.

I have developed a computer program named evocom to solve The Eight Queens Problem. I have also written a paper quantitively analysing and demonstrating fundamental aspects of biological evolution.

You can download evocom program and related paper at:

http://members.iinet.net.au/~coruh/evocom/

I have used great open source tools and free products in my work. Here they are:

• TextWrangler : A brilliant text editor for Mac. Simple, fast, reliable, user friendly, got syntax-coloring.
• Python: Mac comes with Python, free to download for Windows at http://www.python.org/
• LaTeX: LaTeX is a high-quality typesetting system; it includes features designed for the production of technical and scientific documentation. LaTeX is the de facto standard for the communication and publication of scientific documents.
• TeXShop: Award winning LaTeX editor for Mac, robust, simple, easy to use, reliable.

You are a null set

The "theory of probability" was one of my favorite subjects during university education.

Our lecturer was a man with great sense of humor, then Assoc. Prof. Fatih Canatan.

In probability theory there were abundant of interesting real life examples that would pull you out of the realm of photons (electromagnetic fields’ constituent particles) and put you on the ground standing firm on your feet, well almost firm, as we are talking about probabilities here. The theory of probability gave us the students a rare sense of belonging in our otherwise weird electrical engineering curriculum.

I remember we also had a classmate, an annoying type, you know the guy, a definitive 'nerd', who constantly asks stupid questions without giving a fig about how disruptive he is being for the class. 

Then one day following a chain of dumb and tiring interruptions he asked “what is a ‘null set’ professor?”. Mr Canatan turned, calmly puffed his long pipe twice, deeply thinking, but his face now turning slightly pinkish, answered:

“Mmm. A null set is an empty set as I already mentioned number of times. For example you, you are a null set!”.

The whole class suddenly cracked up, tears in eyes.


A classical problem in introductory probability lectures is:

"What is the probability of finding at least two people with the same birthday in a group of people?"

Most people underestimates the value of such probability. To give you an idea in a group of 20 people the probability is 41%.

In my hand writing the solution is given as follows :

If you also want to plot how probability varies with the group size first of all you need a good calculator which won’t give you overflow with large factorials. Most likely your computer’s calculator application would fail. You would need a tool like Wolfram Alpha.

Wolfram Alpha is a magnificent online math tool. Go to http://www.wolframalpha.com then in the search box type;

1 - 365!/((365-k)!*365^k)

After a few seconds Wolfram will give you a set of interesting plots and numbers in various forms:

If you want to zoom into the probability distribution for a group size between 20 and say 40, simply follow your intuition:

1 - 365!/((365-k)!*365^k) from 20 to 40

And you would get this;

Or if you want to calculate the probability for a group of 40 people simply type:

1 - 365!/((365-k)!*365^k) where k=40

And you would get the result as 89.1232%:

Isn’t this amazing! I mean the whole thing. It simply feels great NOT to be a null set.

Forgetful Loops

Consider a software algorithm that builds a tree in a computer program. Typically when you build a tree you would have a recursive loop in which tree nodes are created and connected. The loop will have an entry and an exit point.

You might also recycle the same tree builder algorithm to build a parallel tree, an almost identical tree with only its target context being different. For example tree1 is supposed to be an observer of explorer style representation, whereas tree2 observes a column based representation. So the nodes of each tree might have a different type, their properties are determined and their behavior are dictated by the context they live in.

Let us now assume that we are living in a universe U_R.

One way of comprehending our universe is thinking of it as a gigantic Hilbert Space. A vector-space that represents the state-spaces of quantum mechanical systems is also a Hilbert Space.

Think about every atomic particle that makes you, they have a certain position and momentum at any given time, i.e. they can be represented as vectors. And when I say 'at any given time' I mean the time of our vector space. Together with other quantum mechanical systems, such as the coffee mug, the earth, the sun, the milky way galaxy and so on, we are all part of one big vector space.

Another way of visualizing our universe is thinking it as a space-time sheet or fabric, on which every vector stays on the same fabric, moving in random directions and changing their magnitude all the time.

Returning back to our tree building problem, there is no reason why you would not build parallel trees by running the tree-builder algorithm sequentially, so you build the tree1 first followed by the tree2 and so forth.

After all our program runs in the realm of our universe, in the same space-time fabric. So the loops building these trees would occupy different localities on our universe's space-time since they are built at different times, nevertheless they lie on the same sheet of space-time.

Suppose now that there is a glitch in your program so that while executing the first loop to build tree1, the program suddenly switches its context, enters into the second loop and starts building tree2.

You may be surprised to find that when the program finishes building the second tree, and returns control to the loop of tree1, instead of the iteration continues where it left, it might leave the loop prematurely with tree1 left incomplete.

Suppose also that the glitch causing your program to suddenly leave the first loop at the first place is not avoidable. This also implies that your trees are NOT built sequentially contrary to your intention (design).

The only way to fix this software problem is to SAVE the loop's iterator prior to leaving the first loop, and RESTORE it back when the control is returned prior to the loop iterator is incremented.

The loop needs memory of where it left prior to making the jerky switch, otherwise it fails to restore itself.

This is a remarkably common pattern in mathematics, theoretical physics and literature no matter how speculative its representation might turn out to be.  Think about wormholes, multiverses, Alice's Adventures in Wonderland, The Magician's Nephew, and The Lion the Witch and the Wardrobe.

The loop is like consciousness. Suddenly leaving the realm of loop1 is like entering into a shortcut, a wormhole and finding yourself in the realm of loop2, a different consciousness in a different universe.

However there is one crucial difference. In literature we are biased to think that the heroes maintain their consciousness. They remember other realms when they return. This gives the audience a comforting cozy feeling.

I tend to think though there seems no need to remember your previous conscious presence in one realm or the other. All you need is a functioning anchor when you return. You might pretty much succeed as a forgetful loop provided that your anchor works.

Poincaré's Great Topological Papers

Henri Poincaré had learned of non-Euclidean geometry from Beltrami who first noticed that hyperbolic geometry was a constant curvature geometry in which the curvature is negative. Beltrami pointed out that a surface called the pseudosphere carried such geometry. He also discovered the disk model and realized that Riemann's conception of geometry provided the link that united the two conceptions. Poincaré worked out the details and discovered more models. Moreover, Poincaré did not stop at two dimensions. He extended this work to higher dimensional geometries. Here he is, in his own words, describing his model of three-dimensional hyperbolic space:

"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows: If R be the radius of the sphere, and r is the distance of the point considered from the centre, the absolute temperature will be proportional to R²-r². Further, I shall suppose that in this world all bodies have the same coefficient of dilatation so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall suppose that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment. There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the first place, that although from the point of view of our ordinary geometry this world is finite, to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of laws according to which invariable solids move, to these imaginary beings it will be the study of laws of motion deformed by the differences of temperature alluded to..

Let us make another hypothesis: suppose that light passes through media of different refractive indices, such as the index of refraction is in R²-r². Under these conditions it is clear that the rays of light will no longer be rectilinear, but circular.... If they [the beings in such a world] construct a geometry, it will not be like ours, which is the study of movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be 'non-Euclidean displacements', and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours."

from The Poincaré Conjecture Donal O'Shea

The Poincaré Conjecture

I've been reading this fascinating book "The Poincaré Conjecture - in search of the shape of the Universe" by Donal O'Shea.

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds.

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold.

Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane (an idealized flat surface) and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry. This is the type of geometry we learn in the high school.

The Poincaré conjecture concerns a space that locally looks like ordinary three dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is just a three-dimensional sphere.

A sphere can be represented by a collection of two dimensional maps; therefore a sphere is a two dimensional manifold.

An important scientific application of 3-manifolds is in physical cosmology, as models for the Shape of the Universe – the surface of the earth is locally approximately flat – it is roughly a 2-manifold, and globally the surface of the earth is a sphere. The universe, likewise, looks locally approximately like 3-dimensional Euclidean space, so the universe may be modeled as a 3-manifold, and one may ask which 3-manifold it is. More precisely, the universe is better described as 4-dimensional space-time, so this description is of a 3-dimensional spatial section.

Our genetic code allows us to see in 3 spatial dimensions. Most animals possess perspective vision evolved to enable them estimating distance, a vital tool for survival to avoid predators and approach prays in great efficiency. Simply said we have no provisions to see more than 3 spatial dimensions, therefore we have difficulty comprehending concepts such as 4-dimensional space-time. We have trouble picturing the shape of the universe as a whole.

It is not possible to get outside the universe. This is an important difference between the Earth and the universe. A rocket can leave the surface of the Earth, and we can look at the Earth from outside it. Since we see in three dimensions and since the surface of the Earth is two-dimensional, we can see our planet is bending in a third dimension and visualize its whole shape easily. However even if we could get outside the universe in an attempt to see what shape it had, since the universe is three-dimensional, we would need to be able to see in at least four dimensions to visualize the universe as a whole.

The conjecture in question is a claim made by the mathematician Henri Poincaré over 100 years ago. He said that all four dimensional spaces in which a loop in that space can be narrowed to a point are four-dimensional spheres -- there are no other possible shapes for the universe.

Until 2002, no one had been able to prove or disprove this claim. The framework for this book was the announcement by Russian mathematician Grigori Perelman that he had proven it.

If I travel on a 2-dimensional manifold or surface like Earth, I will always come back to the same point. There are no edges that I would fall. The proof of Poincaré Conjecture supported the idea of Universe is finite (a closed 3-manifold) and has no boundaries. So if I traverse the universe in one direction I will always come back to where I started.

I will have more posts on the shape of the universe, so stay tuned.

Ref. Wikipedia

The Pythagorean theorem

The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theory predates him. (There is much evidence that Babylonian mathematicians understood the principle, if not the mathematical significance.)

Pythagorean theorem - Wikipedia

An elegant proof of Pythagorean theorem can be found in Roger Penrose's popular science classic "The Road to Reality - A Complete Guide to the Laws of the Universe". His proof suits perfectly to the central theme of this blog; "Negative Matter", in other words the ability to see and transform negative spaces in order to convey further "meaning" into positive spaces.


First consider the pattern illustrated below. It is composed entirely of squares of two different sizes. It is 'obvious' that this pattern can be continued indefinitely, and the entire Euclidean plane can be covered in this repeating way with no gaps or overlaps by squares of these two sizes.



If we mark the centres of the larger squares, they form the vertices of another system of squares, at somewhat greater size than either, but tilted at an angle to the original ones.

Instead of taking the centres of the two large squares of the original pattern, we may choose any other point, together with its set of corresponding points throughout the pattern. The new pattern of tilted squares is just the same as before but moved along across either vertices without rotation (i.e. by means of motion referred as translation). For simplicity lets choose our starting point to be one of the corners in the original pattern.


It should be clear that the area of the tilted square must be equal to the sum of the areas of the two smaller squares. The small square at the top-left has a small triangular portion outside the tilted square, which is identical to the one inside, if we move along the vertex of tilted square. A similar observation can be made about larger triangles, proving our assertion.

Moreover it is evident that the edge-length of the large tilted square is the hypotenuse of a right-angled triangle whose two other sides have lengths equal to those of the smaller squares. We have thus established the Pythagorean theorem:: the square on the hypotenuse is equal to the sum of the squares on the other two sides.

from "The Road to Reality - A Complete Guide to the Laws of the Universe - Roger Penrose"

Ockham's Razor

Pluralitas non est ponenda sine necessitate

Plurality is not to be posited without necessity

William of Ockham (c. 1288 - c. 1348)

I like simple designs. Simplicity makes us humans happy, often unknowingly. But what does 'simple' mean anyway?

Simple is less, and less is more. If you can do with less why waste your energy and resources. Simple is elegant, and elegant is desirable.

Less is more. Why so? A bird's egg is simply round, smooth and slippery, no awkward angles. Difficult to capture intact and transport by predators. Optimised for smooth delivery, comfortable resting, and even temperature distribution. Space efficient for internal layout and external storage at the same time. i.e. 'do more with less'. A good example of natural selection favoring minimalism.