Category Archives: Books

The Golden Ratio and Understanding the Universe

When you stop and think about it, it is truly astonishing how well we can describe the universe around us using mathematics.  That equations as simple as F=ma and E=mc^2 can describe so much of what we observe is really amazing.

The Golden Ratio by Mario Livio is essentially an examination of one of the most remarkable numbers to be discovered as a pretext for ultimately exploring the question why do mathematics work so well.  The golden ratio, known from the time of the ancient Greeks, is a pretty simple concept: take a line (defined by points A and B) and divide it (at point C) such that the length of the entire line over the length of the longer portion of the division is the same as the length of this longer portion over the shorter portion.  That is, if the longer portion is CB and the shorter is AC, then AB/CB=CB/AC.  A pretty simple concept and definition.  It turns out, this has many profound consequences.  This golden ratio, normally dubbed phi by mathematicians, is one of the first irrational numbers discovered and is, in some sense, the most irrational of all.  It shows up in many branches of math, especially geometry, and has been observed in nature in the patterns formed by petals on flowers, the seeds in sunflowers, the shape of certain seashells, and the shape of galaxies.  It is ubiquitous in nature.  Why should that be so?  That is the real point of Livio’s book.

Livio spends a lot of time on the history of phi, how it was discovered, how it was understood, and what it means to math and science.  When he is focusing on the role of phi in science, the book is wonderful.  There were many very interesting insights that I was unaware of that were real gems to discover.  Livio also spends a lot of time on the supposed role the golden ratio played in art, poetry, music, and architecture, including the constrution of the pyramids of Egypt.  His goal is to debunk those who claim that the golden ratio was an instrumental part in many such works, and he does so convincingly.  Unfortunately, I found this a huge distraction and very uninteresting.  I would much rather that he had filled those pages with more discussion of how the golden ratio is found in nature and science.  I understand that he felt a need to determine where the golden ratio is really to be found, but I felt it was over done.

Ultimately, the fact that the golden ratio, and by extension math as a whole, figures in so much of what we see around us leads Livio to examine why that is so.  He describes two alternative views.  First, that the universe is objectively Platonic; that humans are discovering the laws of the universe and those are written in the language of math.  Any civilization in the universe would uncover the same mathematical laws.  The second view is that math is a human language, a human construct, and we are using it to interpret our observations of the universe.  Civilizations with different formulations of mathematics would have a different view of how the universe works.  Livio falls somewhere in the middle, and, to be honest, I did not overly understand his reasoning for his position.

It is an interesting question.  I guess I would tend a bit more towards the Platonic view.  I think that it is just too striking that our math and physics can not only describe but predict what is happening around us so well.  I once had a chat with a fellow grad student at the UW physics department that was about this.  His point, as far as I could understand, is that maybe we create the reality around us via our investigations and our interpretations of our observations.  Essentially, that there is no objective reality, that reality is created by the observer.  Thus, as we develop our math, we view the universe through that math and thus shape it to conform to our math.  This “the observer shapes reality” perspective seems like an extreme view of the Copenhagen interpretation of quantum mechanics.  I definitely wouldn’t go so far as this.  But, it is an interesting question.

Wicked by Gregory Maguire

The idea of taking a familiar story — a fairy tale, a common legend, or a children’s story that all of us know — and turning it on its head appeals to me.  I like the idea of taking the familiar and presenting it from a different point of view.  One of the best examples of this that I’ve encountered is Neil Gaiman’s retelling of Snow White in his Smoke and Mirrors.  Gregory Maguire does it in his own way with Wicked: The Life and Times of the Wicked Witch of the West.

Wicked tells the story of the Wicked Witch of the West, known and feared from the Wizard of Oz stories, form her point of view, giving her a name — Elphaba — and a history.  Is Elphaba really wicked, as described in the original Wizard of Oz, or is she on the losing end of history, her story being told by those who hated her most?

Maguire does a great job of creating a background for Elphaba and her sister, Nessarose, the eventual Wicked Witch of the East.  Especially as a young girl and a young woman, Maguire gives Elphaba a depth and richness that is really captivating.  He creates a setting in which the land of Oz becomes a complex place, full of history and politics.  Elphaba grows up in this land and becomes the woman known as the Wicked Witch of the West as a result of the injustices around her.  That is, her becoming “wicked” is a result of the events happening around her.

Overall, I greatly enjoyed the novel and had a hard time putting it down.  I really felt that Maguire did a good job of creating a character in Elphaba that was both interesting and one that I cared for.  The novel centers on her — other characters come and go as time passes.  There are often big jumps in her life as we move from one major period to another.  All eventually lead to the preordained outcome — Elphaba is killed when Dorothy douses her with water.  But, the events that lead to that eventuallity are still compeling, maybe even more so, because the outcome is known.

There were a couple of things that didn’t quite ring true to me.  For a character that is supposed to be so menacing in the original story, there just didn’t seem to be any real reason for Elphaba to be viewed as overly menacing.  She doesn’t do anything that really threatens the Wizard, for example.  She maybe has an outpost in a backwards part of Oz, but how that matters to the Wizard is never really demonstrated, not in a concrete way.  And the last few acts in which Elphaba engages — creating the flying monkeys and obsessing with the shoes Dorothy got from her dead sister — seem forced, as they had to happen to mesh with the original story but they seem either trivial or of no real importance here.  For example, regarding the monkeys, they play a role in helping get Dorothy to the Witch’s castle, but that is the only time they leap into action.  I felt that these types of details, necessary to make Maguire’s Witch become the Witch of the Wizard of Oz, could have been done in a more meaningful way.

Similarly, the appearance of Glinda, the Good Witch, a character that Elphaba knows well from school days, is brief and anticlimactic.  I expected much more from the encounter, more sparks to fly, as it were.  However, again, Glinda’s appearance late in the novel serves to link this story to the original, and not much more.  I felt the relationship between the two could have been developed more, their mutual antagonisum developed more.

But, maybe this is the point of the novel: why was Elphaba viewed as wicked?  What did she do to deserve this reputation?  In the end, maybe not much.  She certainly wasn’t evil by any definition of the word.  Her sister, who ruled with religious conviction, did more harm in the world and was hated by more than Elphaba ever was.  And Glinda, the “Good” Witch, is a socialite who throws around her money and her status and has no real conviction of doing anything worth while.  Only Elphaba, who fails miserably at most of her undertakings, does what she does out of a sense of justice and right and wrong.  And, as a result, is the one who is perceived as the most evil.

Overall, while the ending maybe wasn’t what I expected, it was still a very engaging novel.  I’m not sure I will immediately delve into the other books Maguire has written set in Oz, but they will be on the bookshelf, waiting for a future day.

Maguire has a website dedicated to Wicked.

The Graveyard Book by Neil Gaiman

Neil Gaiman has become quite a well known author.  With the recent movie Coraline, based on the book of the same name by Mr. Gaiman, he is fast becoming a household name.  I’ve read a few of his previous efforts, including American Gods and his short story collection Smoke and Mirrors, both of which I greatly enjoyed.  His newest book, a children’s book like Coraline, is The Graveyard BookThe Graveyard Book recently won the The John Newbery Medal for “the most outstanding contribution to children’s literature,” an honor which I think it richly deserved.

I don’t want to give away much of the plot, but I’m sure that there might be some spoilers in what follows.  I would rather describe my general thoughts about the story.  The story centers around the childhood of a young boy, Nobody Owens, as he grows up after a tragedy in his family.  I imagine it won’t be much of a spoiler, since the title of the book essentially gives this point away, to say that Bod, as Nobody is nicknamed, grows up in a graveyard.  I’ve read that Gaiman was inspired on this point by The Jungle Book, putting a young boy in a very odd environment in which to grow up.  The plot revolves around Bod growing up and learning about the graveyard and the world around him, as well as the mystery surrounding the events that led him to the graveyard in the first place.

The story is fast paced, with several adventures as Bod discovers new corners of the graveyard.  The reader essentially grows up with Bod, learning about both the world in which Bod lives as well as the greater world beyond the physical world in which most people live.  We learn that Ghouls, Werewolves, and, while never explicitly stated, Vampires exist in this world.  Bod has to learn to navigate both the everyday world as well as this supernatural world in order to survive.

There are three main aspects of the story that I particularly enjoyed.  First, there is a diverse cast of characters and, while we don’t get to know most of them very well, they all add a lot of color to the universe of The Graveyard Book.  Second, the plot is definitely suspenseful, and at the peak I definitely didn’t want to put it down.  It is a real page turner.  Finally, the book is meant for children.  Maybe not the youngest, but maybe preteens or so.  As such, I like that it doesn’t offer a world-view that is all roses.  That is, bad things happen to Bod and, even when he does the right thing, it doesn’t always work out for him.  And the ending is bitter-sweet.  I’ll leave it at that.

I highly recommend this book.  It is full of imagination and I expect that most kids would love the world that Gaiman has created.  I am torn in hoping that Gaiman further explores the world of the graveyard, but, at the same time, it is maybe better to leave those corners too to the imagination.

There is a website dedicated to the book.

Collapse by Jared Diamond

Guns, Germs, and Steel was one of the best books I’ve read, so I was very interested in reading Jared Diamond’s latest book, Collapse.  Browsing the reviews at Amazon, they were very mixed, with some finding the book boring, a rush job, or saying nothing new.  I guess I can see the last point, if I’d read more about the condition of the world’s environment.  But I haven’t, so, for me, it was a real thought-provoking, eye-opening read.  And I thought it was far from boring.  I don’t know enough about the facts behind Diamond’s claims, so I can’t judge at all the veracity or the bias behind the statistics or claims Diamond makes.  Even so, if even half of what he writes represents the real situation, then still the book is of great importance.

The basic theme of the book is that there are many examples of societies, both in the past and in modern times, that have failed.  Diamond’s task is to try to understand why, and he has arrived at a five-point framework to consider a given society’s collapse:

  • environmental damage by the society
  • climate change
  • hostile neighbors
  • friendly trade partners
  • the society’s response to its environmental problems

Not all of these factors contribute to any given society’s collapse, but, according to Diamond, at least one of these is a major contributing factor and for nearly all societies, the first one is often the most important.  Diamond tries to demonstrate this by looking at various past and present societies that did fail, including Easter Island, the island of Henderson, the Anasazi, the Maya, Rawanda and Burundi, and the Greenland Norse, and some that overcame their problems and developed a sustainable society, such as New Guinea, Iceland, the Greenland Inuit, and Japan.  As Diamond points out, some of the problems that faced some of these societies was essentially random luck, such as the quality of the land they settled.  For example, the Greenland that the Norse encountered looked lush, like their native Norway, but the soil was not anywhere near as productive and that led to some of their struggles.

The point of all of this is to understand what led to the failure or eventual success of each society so that we can apply the underlying lessons to our modern world.  Diamond illustrates those dangers by describing the current state of China, Australia and Montana, showing how ecological damage has affected the environment and, more important, the people and society of each.  He concludes that failure is not a given, that societies at some point essentially choose to either fail or succeed.

One might wonder why they would ever “choose” to fail.  To say that they choose to fail is a bit misleading.  Rather, Diamond gives 4 reasons that they essentially do not end up fixing their problems:

  • they fail to anticipate a problem before it arrives
  • they fail to perceive a problem that has arisen
  • they fail to try to solve a problem they do perceive
  • they may try to solve the problem, but fail

The third point, that they don’t even try to solve a problem that they do know about, is the hardest to understand, but in truth it seems that societies do indeed just fail to act.  Whether the choices involved in acting are too difficult, maybe involving abondoning core values or beliefs, or there are conflicting values, such as a profit motive.  We are at a point where we will have to make these hard choices to confront problems facing us, choices that many of us will be reluctant to make.

Finally, Diamond describes 12 problems that are currently facing the world:

  • the destruction of natural habitats, such as forests and wetlands
    – Diamond claims that deforestation was one of or the primary reason for the collapse of each previous society he analyzes
    – half of the world’s original forests have been converted to other uses and a quarter of what remains will be converted within the next 50 years
  • wild foods, a large fraction of protein for many of the world’s people, are disappearing, with many fisheries already having collapsed
  • many species have gone extinct, decreasing the world’s biodiversity, upsetting the balance of many ecosystems
  • farmland soil is being eroded at a much greater rate than it is being reformed, leading to the eventual ruination of that land; much other farmland is being destroyed by salinization
  • the primary energy sources are fossil fuels, which are a limited, non-renewable resource
  • most of the world’s freshwater is already being used, for irrigation, domestic and industrial use, or recreation, leaving very little for future expansion
  • we are near the photosynthetic capacity of the planet; that is, the way that sunlight can be used for plant growth is finite and we are already using about half of that, even assuming plants are 100% efficient at capturing photons
  • chemicals, either synthetic ones made by humans or natural ones that are made in extreme quantities by humans, are entering the environment; they have reached the furthest corners of the planet — the level of PCBs in the milk of Inuit mothers is at hazardous levels
  • alien species, introduced either intentionally or unintentionally, are upsetting ecosystems around the world, destroying native species and making farming extremely difficult in some areas
  • greenhouse gases and global warming
  • the growth of the global human population
  • finally, even more importantly, the impact per person on the environment is increasing

Upon reading his arguments, one realizes that the most alarming aspect of all of this is that these are problems today, in a world where the First World uses 32 times more resources per capita than the rest of the world, and the rest of the world is trying to catch up.  The rest of the world sees how the First World lives and wants that standard of living.  Getting there will mean that they too have a much higher per capita impact on the world, exacerbating all of the problems listed above.  For example, if China alone, which is pushing hard to achieve First World standards of living, reaches the same level as the First World, the per capita environmental impact of the world will increase by a factor of 2.  This is just if China reaches that level, and many of the other very populus countries are currently poor and working to get to First World standards.

All of this made me feel very depressed and pessimistic about the future.  These are huge problems that will require huge efforts to fix, require huge changes in how we live.  It seems to me that, to reach a sustainable lifestyle, people all around the world will have to compromise.  The First World will have to realize that, even if the rest of the world stays poor, the lifestyle we have is unsustainable.  We will have to settle for a lifestyle that is less affluent.  At the same time, the rest of the world will have to realize that they cannot have the same standard of living the First World currently has, a harsh realization.  This means hard choices on both sides, choices that it is not clear to me we will all make.

Diamond does end on one cautiously optimistic note.  The problems we are facing are caused by us, meaning they can be fixed by us.  Some of them will be difficult to fix even if we decide to do everything possible today.  But, it can be done if we have the will.  Whether we choose to do so will be the big question.

There is a lot in this book that I have failed to mention.  I highly recommend this book and think it should be a topic of discussion in all classrooms in the country.  We all have to acknowledge the problems facing us for there to be any chance that we can address them.  That means we have to think beyond how we want to live and consider how we should live.

After reading this book, I am concerned about the world my daughter will live in.  Hopefully, my generation will begin to act such that her generation has a better chance for a world in which the majority of humanity can live in both a sustainable and reasonably affluent manner.

Nexus by Mark Buchanan

By now, everyone has heard about the six-degrees of separation thing, how we are all connected to Kevin Bacon by about 6 other people.  Turns out, there is nothing special about Kevin Bacon — each of us is connected to pretty much everyone in the world by about 6 or 10 other people.  In a world with 6 billion people, how can this be possible?  That is where the theories of complex networks and, in particular, small world networks come in.

In Nexus, Mark Buchanan gives an introduction to this new field (many of the seminal discoveries have occurred within the last decade).  He describes how these networks are ubiquitous in nature (e.g. the networks of streams comprising a river system), social networks (the 6-degrees thing, among others) and networks created by humans (the internet and the electricity grid, as two examples).  It turns out that there are two types of small world networks, called egalitarian and aristocratic.  Buchanan discusses how such networks arise naturally.  In particular, the aristocratic networks, characterized by special nodes that have an especially high number of links to other nodes, occur via a “rich get richer” process, in which nodes that already have a lot of links or friends or what have you are more likely to get even more.

There were a number of intriguing points in this book.  For example, when he discusses river systems, it turns out that all river systems follow the same distribution of land they drain versus the number of streams in the river system that drain that amount of land.  They follow a power law distribution.  That is, if 100 streams in a given river system each drain 50 square miles of land, then 50 streams will drain some constant times 50, and 25 streams will drain that constant times that constant times 50.  There is a power law association between the number of streams that drain a given area of land and the size of that area.  Even more interesting, just assuming the most simplest of assumptions, this distribution can be generated in a computer.  All they assume, given a random topology of land (not even a real landscape), is that water flows straight down hill.  They neglect so many seemingly important features (erosion, for example) that it seems impossible that it would represent anything about real river systems.  But, it does.

Another feature that has such a power law distribution is the amount of wealth held by a given percentage of the population.  Known as the Pareto principle, it basically is the observation that in most countries in the world, regardless of type of government or economy, about 80% of the wealth is owned by 20% of the people.  And, as you look at the number of people who own 90% of the wealth, it is a constant factor of 20%, and so on.  This, to me, is amazing.  Furthermore, again in simple computer experiments in which you allow people to exchange wealth in one of two ways — they can either (a) buy something from someone else, involving direct transfer of wealth, or (b) they can invest their money with some random rate of return — this Pareto distribution is reproduced.  This is true even if you assume that all players have the same skill in investing.  It all comes down to random luck and the rich get richer principle.  As someone gets lucky and wins on their investment, they in turn have more money to invest.  Think about the implications: the distribution of wealth in most countries might be mostly due to random luck.  Sure, government policy (e.g. taxes, etc) would change the slope of the distribution, or the power in the power law, and skill might as well, but that this distribution can be obtained without any of that suggests that the wealthy are wealthy not because they worked harder or were smarter or anything like that, it is just pure random luck.  If this is true, what does that mean about how we view the wealthy’s role in society?

This book had a number of intriguing points such as this that really make you look at the world in a different way.  These small world networks are so pervasive (they even occur in the sexual relationships of people — there are some people who have so many sexual partners that they essentially connect everyone within a few links of one another; this has implications on how you might treat, e.g., sexually transmitted diseases) that understanding how they come about and what they teach us about how the world works is absolutely essential.  Knowing that we are all connected so intimately and what that means for how we interact is fascinating.

I highly recommend this book to anyone who is interested in how people, atoms, computers, rivers — so many different things — interact with one another.

For more information about small world networks, see this Wikipedia article.