{"loaderData":{"0-0-56":{"bookData":{"_id":"66ced12dc3d3f3f05eac023f","category_id":"66ceced8c3d3f3f05eac01a2","author_id":["66d4e75cb6378709e9cd4f27"],"book_name":"Puzzles to Unravel the Universe","book_image":"/books/PuzzlesCoverpng-1724829997641.png","book_description":"

Puzzles to Unravel the Universe is a wonderful survey of concepts central to modern physics and mathematics explored through the medium of puzzles. This is one of the most unusual and captivating approaches I have ever encountered, allowing the reader—whether novice or expert—to learn through the enjoyment of solving brain-teasers. What a fun and effective way to actively absorb essential and cutting-edge ideas from one of the world's greatest physicists. —Brain Greene

This book offers a fascinating and unusual tour of advanced ideas in physics and mathematics, illustrated with elementary and entertaining puzzles. Readers will find much to enjoy and learn from. —Edward Witten

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Many fundamental aspects of physics have simple mathematical underpinnings that may be concealed by the complexity of the formalism–both the unfamiliar language and the sometimes daunting equations. The same is true of many abstract mathematical ideas, which often involve simple concepts that can, nevertheless, get obscured due to the setting in which they are presented. Deep ideas in physics and mathematics often share a common core, which is perhaps not surprising given the closeness of these two disciplines. More surprising is the fact that some of those same ideas can emerge in the course of solving mathematical puzzles.

This book is about puzzles and their relations to mathematics and physics. While puzzles can be fascinating and entertaining in and of themselves, we are going to see how they can serve as a bridge between the two fields and reveal some of the common bonds they share. It does not take an advanced knowledge of mathematics and physics to solve the puzzles presented in this book, nor is it assumed that the reader has a deep background in either field. But a keen interest, as well as some training, in those subjects would certainly be helpful in appreciating this book, with college students or advanced high school students being among the intended audience.

Although physics and mathematics are tightly intertwined, they have very different cultures and philosophies. Mathematics starts from fundamental axioms, drawing upon logical inferences to build from there. Physical laws, rather than being logically deduced in a hierarchical fashion, have been developed to explain how different parts of nature work and how the laws of nature fit together. Physics stresses the relationships between these laws, rather than their logical dependencies. Of course, logical cohesion of these ideas is still a necessary ingredient of physical laws. In mathematics, it is important to be clear as to what your underlying axioms and assumptions are. On the other hand, as we will see later, the axioms, or fundamental principles, of physics can change as new evidence or theoretical ideas come to light.

History shows that important progress in the field can occur when what was first regarded as a consequence of a physical law is subsequently determined to be a guiding principle in its own right. A good physicist, therefore, should always be open to revisions or “reshufflings” of this sort because the newly recognized principle often ends up being more fundamental, and having a wider range of applicability, than the original principle from which it was initially thought to have stemmed. The principle of conservation of momentum offers a good example. Although it was first viewed as a consequence of Newton’s laws of motion, it was subsequently determined–more than 225 years after Newton’s laws were presented in Principia Mathematica –that conservation laws were more fundamental than laws of motion because they stem from underlying symmetries of nature.

It is for this reason that physicists try to maintain a flexible attitude as to what the fundamental principles are, which is constantly evolving. Rather than attaching too much value to the hierarchical nature of ideas, physicists are willing to reorder the structure at any moment, which is contrary to the way mathematicians usually view mathematics. A mathematical theorem, if proven correct, is regarded as eternally true–unlike physical principles that are subject to change as new empirical findings come to the fore.

There are other differences, as well. Explaining complicated phenomena in physics, for example, often involves the kinds of approximations that mathematicians may be loath to make. The question, for instance, of whether a space is “continuous,” containing no gaps whatsoever, or is instead made up of points that are close together, may not matter to physicists focusing on the outcome of experiments carried out over much larger distance scales. For mathematicians, on the other hand, the smoothness of a given space or lack thereof, is a key feature, rather than being an irrelevant distraction.

The goal for this chapter is to provide a brief overview of the landscape of physics. This will be a quick review that is light on exposition; the aim here is not to be comprehensive, as that is essentially impossible in the course of a single chapter. Instead, we intend to touch on a few examples from the history of physics that can provide a sense of where we are today in the longstanding quest to understand the fundamental laws of nature.

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Symmetry is alluring to the eye and perhaps reassuring to the senses, as if reinforcing the notion that there is some underlying (or overarching) structure and order to our world. People have long marveled at the intricacy of hexagonally shaped snowflakes, an often-cited example of nature’s beauty. On a similar note, human faces that appear symmetrical are generally regarded as more attractive. Symmetry has also had a profound and rather obvious influence on architecture, which can be dramatically seen, for instance, from a straight-on view of the Taj Mahal, one of the Seven Wonders of the modern world.

Symmetry also plays a surprisingly powerful role in the laws and workings of physics, a role that goes far beyond matters of aesthetic appeal. To a physicist, symmetry is not merely a property of an object–such as a regular hexagon or octagon, that reflects a perfectly balanced construction and inner constitution. Symmetry often means an operation performed on an object or system that leaves it completely unchanged. Operations of this sort include rotations of an equilateral triangle about its center by 120 degrees or rotations of a square by 90 degrees. Rotations of a circle or sphere about its center are examples of continuous symmetries, valid for any and all degrees. A regular pentagon, on the other hand, has discrete symmetry: rotations of 72 degrees, or multiples thereof, leave it unchanged.

Yet the sweep of symmetry extends much further still, a fact illustrated by a theorem proven more than a century ago by Emmy Noether, a German mathematician. Noether showed that for every continuous symmetry of nature there is a corresponding conservation law. Using this theorem, one can derive important principles of physics–such as the conservation of energy, linear momentum, and angular momentum–strictly from mathematical arguments stemming from symmetry, as will be discussed later in this chapter.

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In the last chapter, we illustrated some of the power of symmetries in solving puzzles, in studying physics and in the world and universe around us. We noted that symmetries are equivalent to conservation laws in physics and conservation laws, as you may have noticed, can be very useful. One basic application, as we saw in the puzzle with the cards, is that if things do not add up, we then know something is missing, and we can get information about the missing thing by counting what is, and is not, there. In this chapter we discuss the opposite concept: situations where symmetries are broken. And you may be surprised to find out that, in some cases, these broken symmetries can be more interesting and consequential in nature than unbroken ones.

One example is the asymmetry between matter and antimatter. The Big Bang, in principle, should have created equal amounts of matter and antimatter. If that situation had persisted, matter and antimatter particles would have eventually come into contact and annihilated each other in bursts of pure energy. But somehow the symmetry between matter and antimatter was broken by a tiny bit with one in a billion more of matter than antimatter, leaving behind a preponderance of matter after annihilation, to which we owe our existence!

Another example, less vital to our presence in the cosmos, concerns a pencil balanced perfectly on its tip. It is an unstable configuration because the pencil will eventually fall. But while the pencil is upright, it’s in a symmetric configuration, because it could fall any which way–no one direction is preferred or preordained. When the pencil eventually topples over, that symmetry–beautiful while it lasted–is then spontaneously broken. While a moment before, it might have fallen in any direction from 0 to 360 degrees, it has now singled out just one.

Here is a more mathematical example: Suppose we have a smooth real function of one variable, \\(f(x)\\) , and let us further suppose that it is an even function. In other words, \\(f\\) enjoys a reflection symmetry: \\(f(x)=f(-x)\\) . Our initial task is to find the critical points of \\(f\\) , i.e., points where \\(df/dx=0\\) . Using symmetry we can immediately find one of the solutions \\[\\left.\\frac{df}{dx}\\right|_{x}=\\left.\\frac{df(-x)}{dx}\\right|_{-x}=-\\left.\\frac{df}{dx}\\right|_{-x}\\] Therefore at \\(x=0\\) , we have \\(\\left.\\frac{df}{dx}\\right|_{0}=-\\left.\\frac{df}{dx}\\right|_{0}\\) . The only way for this to be true is \\(\\left.\\frac{df}{dx}\\right|_{0}=0\\) .

Suppose we instead asked: Find a local minimum of \\(f(x)\\) (assuming there is one). One’s immediate guess may, on symmetry grounds, be \\(x=0\\) . However, this is not necessarily so: One of two situations is possible as shown in Fig. 22 and, depending on which of those it is, the local minimum may or may not be at \\(x=0\\) .

Figure 22. The minimum of an even function can preserve (as in the top figure) or break (as in the bottom one) the reflection symmetry.

If it is not at \\(x=0\\) , we say the reflection symmetry is “ spontaneously broken .” In other words, symmetry may mislead us in our thinking as to where the actual minimum should be. In the event that the symmetry is broken, there must be at least two minima.

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In this chapter, we talk about:

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In this chapter, we talk about:

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In this chapter, we talk about:

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By now it is probably apparent that not all aspects of physics are intuitive. In fact, some of the most exciting things about physics involve cases where our intuition positively fails us. Sometimes our intuition tells us that a situation cannot possibly be true, and yet it is. Let us start our exploration of counter-intuitive physics by going back to the subject of buoyancy.

Buoyancy Revisited

Many people think that modern physics is unintuitive, and there is certainly some truth to that sentiment. Viewed from outside, the field is getting stranger and stranger. Nevertheless, unintuitive physics is by no means a contemporary phenomenon; it began much, much earlier. Buoyancy, to take one example, has been known for more than 2,000 years, and even after all that time the subject remains largely unintuitive. For how can a very heavy ship be held up simply by the buoyant force of water? Of course, we saw in the last chapter that this concept can be made intuitive if one thinks about it correctly. We might call that a matter of correcting our intuition to make the unintuitive part of physics feel more intuitive. Yet it still seems very strange, especially when contemplating supertankers nearly a half-kilometer long that can carry cargo weighing more than a half-million metric tons!

We all know that a helium balloon will rise due to buoyancy. But what happens if you have a helium balloon in your car and suddenly stop? While you (the driver) will lean forward–hopefully restrained by a seatbelt–the balloon will move backwards . What if you turn? It will move inwards, unlike what you may think. This is because, in both cases, as viewed from the perspective of passengers in the car, there is an effective acceleration, and the balloon moves in the direction opposite to that acceleration relative to the rest of the air, which is more dense, as would be expected based on buoyancy.

There is a more extreme version of this, the so-called “Archimedes’ Paradox.” Suppose you have a huge ship and only a few buckets of water. Can you float the ship using only that much water? The answer is astonishingly yes! Since buoyant forces are local, it suffices to cover the bottom of the ship with just a very thin layer of water, as in Fig. 69 . Indeed, the argument for the buoyancy force we reviewed in the last chapter did not demand that there be a large reservoir of available water.

Figure 69. Archimedes argued that you could float a ship with a bucket of water if you had a container that fits the shape of the ship.

While there were already many unintuitive things in classical physics, the number of unintuitive things has increased rapidly since we have entered the era of contemporary physics, with much bafflement caused by relativity, quantum mechanics, string theory, etc.

Airplanes

We do not question it too much anymore, but the fact that airplanes can fly is quite astounding. We are talking about a huge and massive metal contraption flying in midair! How is this possible?

The airfoil is designed so that the air moving over the wing moves faster than the air below the wing. Bernoulli’s principle, which was formulated in the early 1700s, states that \\(P+\\rho v^2/2\\) is constant along the flow, where \\(P\\) is the pressure, \\(v\\) the speed and \\(\\rho\\) the density of the fluid. Therefore, the higher the \\(v\\) , the lower the \\(P\\) . The pressure over the wing is lower than the pressure under the wing, because the wing is designed so that the air moves faster over the wing. This creates a net upwards force, giving the airplane a lift.

The Bernoulli principle, despite its unintuitive consequences, has a simple origin: It is essentially the conservation of energy as applied to laminar flows: Increases in flow velocity (and hence kinetic energy) are caused by the work done due to the net change in pressure and therefore the change in energy is accompanied by decreases in pressure.

Puzzle

You are given a (small) powerful air blower and a (big) light beach ball. How can you suspend (as in Fig. 70 ) the beach ball in midair using only the air blower?

Solution

One may initially think that aiming the blower upwards is the way to go, but that creates an unstable equilibrium, and the ball will quickly fall.

Figure 70. Can you suspend the beach ball with an air blower?

Surprisingly, aiming the airstream just above the ball (as in Fig. 71 ), rather than below, will create lift that pushes the ball up. As we have just discussed in the context of airplanes, pressure and velocity are related by the aforementioned Bernoulli principle. So when you increase the velocity of the air above the ball, the pressure decreases and the air pushes the ball upwards countering its weight and leading to a stable situation.

Figure 71. To suspend the ball in air, you need to aim the blower just above the ball rather than below it!

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In this chapter, we talk about:

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In the last chapter, we explored the power of naturalness. Dimensionless numbers that appear in physics, we said, are expected to be of order 1 numbers. That makes everything nicer and neater–and generally more tasteful. However, this prejudice does, at times, run contrary to the facts. There are some very big, as well as some very small, dimensionless numbers that crop up in physics. That issue, as we will see in this chapter, turns out to be a rather important aspect of modern physics. For explaining the presence, and persistence, of very large and very small numbers in nature constitutes one of the biggest challenges facing physicists today.

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The interplay between science and religion has a long history, and it would be understating things to say that the relationship has been strained at times. In 1600, for example, the Italian philosopher, cosmologist, and former Catholic priest Giordano Bruno was burned at the stake for espousing heretical views, including the notion that the universe is infinite and contains an infinite number of worlds. In 1633, Galileo Galilei was condemned by the Roman Catholic Church for his blasphemous claims that the Earth moved around the sun. Galileo, who was 69 when this judgment was passed, might have met a fate similar to Bruno had he not renounced his scientific findings. In lieu of torture, imprisonment, and possibly even execution, Galileo was placed under house arrest for eight years until his death in 1642.

It is clear that interactions between science and religion have not always been harmonious, and signs of tension and conflict are still manifest today. Nevertheless, one might say that religion was the first form of science in its effort to make sense of the world in which we live and to explain why things are the way they are. Most, if not all, religions attempt to say something about the physical universe. Furthermore, both science and religion are based on observation, even though the “methodologies”–to the extent that word applies–are completely different. What happens when the views clash? My aim in this discussion is not to come up with any conclusive statement about the relation between the two but rather to review what a sample of scientists have thought about this relation in the past without offering my personal views on the subject. And, as always, I will try to frame the discussion in the context of puzzles!

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The topic of duality, which is becoming increasingly important in both physics and mathematics, offers a nice way to tie together many of the topics we have covered so far. Nowadays, it is a common occurrence in mathematics and physics that one might be trying to answer a complicated question that is somehow equivalent, or “dual,” to a much easier question. And the answer to the easier question can almost trivialize the complicated one, showing that it is not nearly so difficult as was initially thought. Good puzzles are like that too. All it takes to solve them, sometimes, is a shift in perspective. The key lies in knowing how, and in which way, to shift. So, in this sense, the notion of seeking dualities seems to be ingrained in us, as we are naturally driven to seek the easiest way of solving a problem.

Over the years, we have learned, for instance, that five different versions of string theory are mathematically equivalent to each other; that 10-dimensional string theories are dual to 11-dimensional M-theory–a gravitational theory of membranes; and that string theories are also dual to some lower-dimensional quantum field theories. Which version one chooses depends on the problem at hand.

The AdS/CFT correspondence, ¹to pick another example, is a duality between a gravitational theory governing a region of space-time and a gravity-free quantum field theory that pertains to just the boundary of the same region. This correspondence was discovered more than 20 years ago and it continues to generate new insights, along with many surprises. “People keep finding new facets of dualities,” the physicist Edward Witten has said. “Dualities are interesting because they frequently answer questions that are otherwise out of reach. For example, you might have spent years pondering a quantum theory, and you understand what happens when the quantum effects are small. But textbooks don’t tell you what to do if the quantum effects are big; you’re generally in trouble if you want to know that. Dualities frequently answer such questions. They give you another description, and the questions you can answer in one description are different than the questions you can answer in a different description.”

Dualities represent symmetries, in a sense, as well as highly “nontrivial equivalencies,” and by the latter I mean things that are not obviously true. Dualities have recently allowed us to solve extremely complicated problems in mathematics and physics. Curiously, we do not know why many of these dualities are true. They are a puzzle solver’s dream, but they are also puzzles in and of themselves. When we solve a puzzle, we normally have some understanding of how we arrived at the answer. However, when we rely on a duality to solve a puzzle, we often do not understand how it works. That can make for a pretty awesome trick, but the fact that the explanation is beyond us can also be rather embarrassing and frustrating!

This was originally proposed by Juan Maldacena and has been developed further by many physicists. ↩︎

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Many of our discussions in previous chapters have alternated between physics and mathematics, though in the last chapter we saw them intertwined in the concept of duality. In some cases, we have discussed a given topic (A) from the mathematical as well as physical perspective. We then discussed the opposite topic (let’s call it anti-A), again from both perspectives. We also worked through some puzzles that combined ideas from physics and math into a kind of dual picture.

Figure 92. A rough diagram of the topics of the lectures.

Duality, as we have explained, has become a theme of growing importance in physics today, and it is also an important idea for summarizing what we have learned in these lectures. One lesson gleaned from an exploration of the dualities that emerge in nature is that we must be open to diverse ideas in our consideration of physical laws. We should not hold fast to a single perspective, disregarding alternative viewpoints. There are many different ways of looking at things: all may be equally legitimate and all may offer their own advantages and insights. In some cases, a better perspective can give rise to a better solution. This, in a nutshell, may be the key, overriding message to be drawn from our discussions. Let us review some of the things we have covered, while adding a few details here and there.

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Puzzles to Unravel the Universe

By Unknown Author

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