A gripping and short account of all the human knowledge about our world. Should be of interest to both, the initiated and the novice. Some discussions can get a little involved but overall, a delightful read.

Language: Enjoyably rich and not overly pompous like some;

Style: Very nicely written and illustrated as well.

The book has some seriously interesting insights, even for someone who has already studied the topics formally. Here’s the table of contents.

• Playing with numbers
  • What’s the largest number you can think of?
  • Natural and Artificial Numbers
• Space, Time and Einstein

  • Unusual properties of space

    Topology

  • World of four dimensions

    Relativity prep

  • Relativity of space and time
• Microcosmos

  • Descending Staircase

    Till inside the atom

  • Modern Alchemy

    Particle Physics

  • “Law of disorder” [skipped]
• “The riddle of life” [skipped]
• Macrocosmos

  • Expanding Horizons

    Size of the universe

  • The Days of Creation

    Formation of planets

Playing with numbers

What’s the largest number you can think of?

This is related to the question I already discussed on c’est ça. The basic idea is to show that there are three kinds of infinities. Two of these infinities we can still wrap our heads around but the third is larger than the other two and the most elusive so far. It is this that can be thought of as our current understanding of the “largest number” which we can associate to something physical.

Natural and Artificial Numbers

Pure: It discusses open problems in pure mathematics such as

• a formula that could produce only and all primes
• Goldbach’s conjecture: each even number can be expressed as a sum of two primes
• Fermat’s Last Theorem: there are no integer solutions to $x^k+y^k=z^k$ for $k\in {3,4,5\dots}$, something which was only recently proven by Andrew Wiles.

Imaginary: Discusses the notion of complex numbers.

Space, time and Einstein

Unusual Properties of Space | Topology

Different coordinate systems are introduced. Then “Euler’s theorem” is stated and proved. $F+V=E+2$ where $F$ is the number of faces, $V$ is the number of vertices and $E$ the number of edges of any polyhedron. The proof uses triangulation to proceed. Surprisingly, this statement entails that there can be only five regular polyhedrons!

The book then makes the user imagine a polyhedron with a hole where this rule fails. This has to do with the topology of the space being discussed.

• Colouring problem: The observation that four colours suffice to colour any map is conjectured to be true but hasn’t been proven (it has been proven that five are enough using an Euler’s theorem like reasoning).

It turns out that the colouring problem can be solved for doughnut like shapes.

The book then asks the colouring problem for three dimensions. This immediately yields the question, what be an object which is to a sphere, that a doughnut is to a plane? The book answers this question by discussing double eaten apples, a three dimensional object that has no boundaries but still has a finite area. In this respect it is like the surface of a sphere (or that of a doughnut). The book then shows that this double eaten apple is topologically equivalent to a doughnut.

The last thing discussed is the notion of parity, the difference between the right and left hand. It turns out that making such an object traverse a Möbius strip flips its orientation!

World of four dimensions

One can describe three dimensional objects as projections of four dimensional objects. The distinction between space and time is emphasised—one can use the same yard stick to measure all three directions of space but it can’t be used to measure time. A standard velocity could be used to meaningfully discuss time. Ways of measuring the velocity of time are discussed. Then the distinction between space and time is said to be captured by defining the distance in four dimensional space with opposite signs for space and time.

Relativity of Space and Time

The consequence of using the aforementioned distance to a given event and two different observers is discussed. The fact that simultaneity of two events is a relative concept is motivated. How ether and its associated winds were ruled by Michelson’s experiment is discussed. Finally, it’s clarified why none of these effects can be observed without approaching the speed of light.

General relativity is discussed next. The notion of a geodesic is introduced. The hypothesis that mass warps space-time is made and how one can observe it using astronomical observations is discussed. The chapter ends with discussing the possibilities of having a finite, closed universe or an infinite one.

Formation of planets

It starts with discussing the earlier philosophical ideas. One was about there being an envelope around the sun which ended up forming the planetary system. The other was about a comet or some such colliding with the Sun resulting in the planetary system. None of these were mathematically studied. Apparently Immanuel Kant was involved in one of these.

Microcosmos

Descending Staircase

This starts with asking if taking a smaller sample of water indefinitely would ever yield something different. We know it does. It discusses how Greek ideas and alchemy paved the path for the current understanding. What is the size of a molecule? An experiment is proposed where one keeps thinning a layer of oil on water until it is only one-molecule thin and starts breaking thereby allowing an estimation of the size of a single molecule. It goes on to describing how the speed of molecules could be determined by constructing molecular beams. These speeds could then be used to correlate with the temperature of the gas to get a direct proof of the kinetic theory of heat! X-rays are then discussed as a method to seeing molecules (or crystals really). There were attempts of describing physical and chemical properties of various atoms using geometrical shapes which failed. It then discusses Thomson’s model which was way off. Then came Rutherford’s model with the nucleus at the centre and the electrons moving around. Apparently the nucleus contained 99.97 percent of the atomic weight and the sun contains 99.87 percent of the total weight of the solar system. There was an analogous relation between the interplanetary distance with the diameters. Then there was the analogy between the electric force of attraction and gravity—the inverse square law. Meanwhile the periodic table was discussed, the electrons seemed to govern the properties of material. The trouble was that no consistent model of the atom could be conceived. Enter quantum mechanics. Disturbance due to observation can not be avoided. An anolgy to placate the failure of the classical theory is given by considering the situations where classical optics fails—diffraction. It ends with describing de Broglie’s and Schrödinger’s wave mechanics.

Modern Alchemy | Particle Physics

Prout’s hypothesis that the weight of all atoms must be integer units of a hydrogen atom’s weight. The book then builds on to how neutrons, protons and electrons together are the building blocks for all material substances. Why don’t we have free positive charges? What about negative protons? These are used as pretexts for discussing “annihilation” and pair production. Cosmic rays are discussed. Then the story of the discovery of neutrinos is told. At some point, protons, neutrons, electrons and neutrionos are claimed to be fundamental (which we now know they’re not) and the focus is shifted to nuclear physics. The notion of surface tension is discussed and then the stability of different nuclei are discussed using it. Experiments for artificial synthesis of elements are discussed. The basic scheme of a Wilson cloud-chamber is discussed. To get good images using the said device, one needs particle accelerators. To this end, the cyclotron is described and so is a linear accelerator. Various reactions are discussed, including their tracks on the cloud-chamber image.

Nucleonics

Here various conditions are discussed which make the development of progressive branching chain reactions possible. The conclusion is that only a specific isotope fits the bill, U-235 (Uranium 235). Naturally, however, the U-238 and U-235 are mixed. Separating them is the main challenge. Two classes of techniques for doing this are discussed. The first uses the difference in their mass to separate (because all other properties are the essentially same). The trouble is that because the difference in mass is so small, they must run the procedure repeatedly. The other class uses what are called mediators. The problem with using a mixture is that U-238 captures the much needed neutrons. Mediators slow down the neutrons which makes the absorption by U-238 very inefficient while U-235 continues to work well.

Finally, estimates are made on the kind of energy one can get from nuclear fission. The estimate is that 16 g of uranium and thorium (from say 1 ton of rock) would be equivalent to 320 tons of ordinary fuel.

Macrocosmos

Expanding Horizons | Size of the universe

“Above” and “below” were absolute directions in space at some point so falling off of the earth was a legitimate fear. Only after sailing around the world was the spherical shape of the earth finally established. The question then was:

Size of the earth: Smart Greeks figured it by measuring shadow sizes and distance between cities. The notion of parallax is then discussed, a method which was used for estimating the distance of enemy warships before radars were invented.

Distance to the moon: Using surrounding stars as reference, the expected parallax for the moon was found by comparing its photographs taken from two points on the earth.

Distance to the stars: From planets to stars, can’t we find the distances to stars using this method? Bessel measured the position of a star, 61 Cygni, by using the revolution of the earth around the sun as two points of observation. Then William Herschel figured, using a self-made telescope, that the faintly luminous belt cutting across the night sky is actually composed of a multitude of stars which appear so faint because they are so distant. The Milky Way was finally recognised. Gradually it was realised that it is a lens shaped object with our sun being close to its outer edge.

Galaxies: A characteristic property of the stars in a galaxy is that they move around what’s called the galactic centre, much like the planets rotate around the sun. How might one prove such a statement? It turns out that just like planets appear to follow a peculiar trajectory (which seems to forward then backward and then forward again) as seen from earth as an artefact of the planetary rotation around the sun, so do stars. Oort was able to show that the corresponding phenomenon must exist by observing the Doppler shifts of the associated stars. Some interesting numbers: The stellar system is about 5 billion years old and (from the aforesaid observations we know that) the sun has since its birth (along with the planetary system) has made about 20 complete rotations. The sun is only 20 “solar years old”!

Distance to other galaxies: There’s andromeda but then it is impossible to estimate how far they are using parallactic measurements. We would’ve been stuck at this if it weren’t for a curious observation: There are giant stars that pulsate as regularly as the heart beat and larger the star, longer is the period of pulsation. How does it help? Well, bigger stars are known to also be brighter. So there’s a relation between the brightness of a star and its period of pulsation. Simply observe the period of such a star and measure its apparent brightness. From the period, find out what is the real brightness and comparing with the apparent brightness, one can determine how far the star is!

Shapes of Galaxies: Galaxies come in various shapes, spherical, elliptic, transitional, closed spiral, open spiral. These seem like stages of evolution of galaxies. It is apparently well known that a slowly rotating sphere of gas gets flattened and then at a 7/10 ratio of the polar to the equatorial radius, the body takes a lenticular shape (like a lens) and finally the gas along the sharp edge starts flowing away creating a thin veil. These were apparently proved by James Jeans. While this explains a great deal of what we see, it is not the complete answer.

• We don’t have a satisfactory explanation of why and how the spiral forms are formed and what causes the difference between the simple and barred spirals. (It was found that the “spiral-arm” usually contains very hot and bright stars which are absent from the centre)

Distance to far-away galaxies: Pulsating stars can work only to a limit. Beyond this, the observation that all galaxies of a given type have about the same size, is used to find distances.

Fundamental question—Size of the universe: Does the universe extend into infinity (meaning better and bigger telescopes will always reveal something new and unexplored) or must we believe that the universe is finite (where recall that finite does not necessarily mean there’s a boundary, it could “close on itself”; a hypothetical explorer might come back to the point from which he/she started). Recall there were two kinds of curvatures, positive—corresponding to the closed space of finite volume—and negative—corresponding to the saddle-like opened infinite space. It turns out that that for closed space (positive curvature) the number of uniformly scattered objects increases more slowly than the cube of the distance (from an observer). For open spaces (negative curvature) the opposite is true. Hubble did such a counting and found that the universe has positive curvature (and thus should be finite). But this is very slight and assumes that galaxies are all equally bright (which might not be the case; their brightness might drop over time). Thus the fundamental question remains open for the moment (as per the book).

The Days of Creation

Thin crust: How do we know that the “solid ground”, the surface of the earth, is just a thin crust? The temperature increases at the rate of about 30° C as we go deeper from the surface. So it started off as a hot object which cooled but how did it form? Two hypotheses were proposed: the collision hypothesis (1749)—the planetary system was a result of a collision between the sun and a comet that came from the depth of interstellar space—and the ring hypothesis (decades later)—no collision, the sun cooled, increased its rotation speed, a series of rings were ejected and condensed into the various planets. Clerk Maxwell later showed that Kant’s view (which was adopted by Laplace as well) when looked at mathematically yields thin rings which should remain as rings just like those of Saturn. The collision theory was then studied (after appropriate modernisation) and it also ran into trouble: one couldn’t explain why the trajectories would be almost circular instead of being elongated ellipses. Finally Weizsäcker (1943) showed that the ring hypothesis is in fact the right one using newer observations about the chemicals that constitute our planet. Here’s the final version.

How did the planets come about?: Our sun was formed by the condensation of interstellar matter. A large part of it this remained outside as a giant rotating envelope. It had about a hundred times the present mass of the sun plus the planets. This rapidly rotating envelope had gasses (hydrogen, helium primarily) and dust-particles (such as iron oxides, silicon compounds, water droplets and ice crystals). The formation of planets was due to the collisions between the dust particles and their eventual aggregation. This is just a high level picture which is justified in the text rather convincingly (there is a description of how to view the collisions, why the collisions contributed to aggregation as opposed to destroying lumps; there is also an explanation of the periods of rotation of planets by looking at collisions as traffic; and the time needed for the appearance of planets). Weizsäcker’s contribution was to show mathematically that this conclusion is valid.

Consequence: Planets must have formed around essentially all the stars. Contrast this with the collision theory where these would be rare events. Since there are billions of stars, what about life on their planets? This is still an area of investigation, clearly. Interestingly, the book ends with predicting that “nuclear-power propelled space ships” would allow us to visit nearby planets which we have.

How did stars form?: Let us start with our sun. For a long time, we couldn’t account for how the sun shines. Studying nuclear physics gave us the answer. A series of reactions, together termed the carbon cycle, explained how one could convert hydrogen into helium at high temperatures using carbon and nitrogen as catalysts, releasing energy. Bethe showed that the temperature associated with this reaction would be 20 million degrees which exactly matches the total amount of energy radiated by our sun. In fact this can be fitted exactly to determine the exact chemical composition of the sun.

All stars are the same?: For the “normal” stars, their mass determines all their properties: the radius, the luminosity and the density. Then there are the insubordinate “red giants” which although have the same total matter for a given luminosity as the “normal” stars, their radii are much larger.

• The forces which caused the “red giants” to swell up can still not be explained

There are also “white dwarfs” which are smaller then “normal” stars for a given luminosity. These almost certainly represent among the last stages of stellar evolution.

Life of a star: Stars consume hydrogen to shine. Once the supply of hydrogen is exhausted, they start shrinking (“white dwarfs”). They still shine bright because of high surface temperature. Their “last mile” is often explosive: novae and supernovae. The book explains this further but I will skip it here.

Final remark about the universe: From present observations, it seems that the kinetic energy of the galaxies is significantly higher than their gravitational potential meaning that our universe is expanding into infinity without any chance of ever being pulled more closely together again by the forces of gravity. The universe seems to be bouncing back from a very dense configuration. The same high density must have “completely obliterated all records of the earlier compressive stages” allowing nature to maintain her mystery around her origin.

Trajectorial Serendipity

H. Bethe: I was reading the star formation section when the book mentioned H. Bethe who figured the Carbon cycle which is used to fuse Hydrogen and it became the sole candidate for explaining the luminosity and life of the sun; thus it answered the question: Why does the sun shine?

I was also watching the Nobel Lecture by Haldane (Kishor had mentioned his name to me earlier). And, wouldn’t you be surprised to note that, H. Bethe had apparently made some interesting contribution to a problem in condensed matter before tackling the astrophysical problem of the sun. He had solved this in some specific dimension, I think two, and then had stated he’d generalise it later. Haldane then explained that while the solution was fine, why it worked took some twenty years of further development. Now it is clear that it works only in that specific dimension and would fail otherwise indicating that Bethe also did not know about this even though he found a solution.

Andrew Wiles: The book mentioned Fermat’s Last Theorem as an open problem. I then read the book “Fermat’s Last Theorem” where Simon Singh describes how Andrew Wiles proved Fermat’s Last Theorem.

Venky: The book, Gene Machine, which is about the structure of the RNA motivates the interest in the subject by trying to address the question of origin of life. Here the same question appears in the context of planet formation.