Month: January 2018

During a lunch in the summer of 1950, physicists Enrico Fermi, Edward Teller and Herbert York were chatting about a recent New Yorker cartoon depicting aliens abducting trash cans in flying saucers. Suddenly, Fermi blurted out, “Where is everybody?”

He reasoned: “Since there are likely many other technological civilisations in the Milky Way galaxy, and since in a few tens of thousands of years at most they could have explored or even colonised many distant planets, why don’t we see any evidence of even a single extraterrestrial civilisation?”

This has come to be known as Fermi’s Paradox.

Clearly the question of whether other civilisations exist is one of the most important questions of modern science. Any discovery of a distant civilisation – say by analysis of microwave data – would rank as among the most far-reaching of all scientific discoveries.

Drake equation

At a 1960 conference regarding extraterrestrial intelligence, Frank Drake (1930 —) sketched out what is now the Drake equation, estimating the number of civilisations in the Milky Way with which we could potentially communicate:

where

N = number of civilisations in our galaxy that can communicate.

R* = average rate of star formation per year in galaxy.

fp = fraction of those stars that have planets.

ne = average number of planets that can support life, per star that has planets.

fl = fraction of the above that eventually develop life.

fi = fraction of the above that eventually develop intelligent life.

fc = fraction of civilisations that develop technology that signals existence into space.

L = length of time such civilisations release detectable signals into space.

The result? Drake estimated ten such civilisations were out there somewhere in the Milky Way.

This analysis, led to the Search for Extraterrestrial Intelligence (SETI) project, looking for radio transmissions in a region of the electromagnetic spectrum thought best suited for interstellar communication.

But after 50 years of searching, using increasingly powerful equipment, nothing has been found.

So where is everybody?

Proposed solutions to Fermi’s paradox

Numerous scientists have examined Fermi’s paradox and proposed solutions. The following is a list of some of the proposed solutions, and common rejoinders:

• Such civilisations are here, or are observing us, but are under orders not to disclose their existence.

Common rejoinder: This explanation (known as the “zookeeper’s theory”) is preferred by some scientists including, for instance, the late Carl Sagan. But it falls prey to the fact that it would take just one member of an extraterrestrial society to break the pact of silence – and this would seem inevitable.

• Such civilisations have been here and planted seeds of life, or perhaps left messages in DNA.

Common rejoinder: The notion that life began on Earth from bacterial spores or the like that originated elsewhere, known as the “panspermia theory”, only pushes the origin of life problem to some other star system – scientists see no evidence in DNA sequences of anything artificial.

• Such civilisations exist but are too far away.

Common rejoinder: A sufficiently advanced civilisation could send probes to distant stars, which could scout out suitable planets, land and construct copies of themselves, using the latest software beamed from home.

So, the entire Milky Way galaxy could be explored within, at most, a few million years.

• Such civilisations exist but have lost interest in interstellar engagement.

Common rejoinder: As with the zookeeper theory, this would require each civilisation to forever lack interest in communication and transportation – and someone would most likely break the pact of silence.

• Such civilisations are calling, but we don’t recognise the signal.

Common rejoinder: This explanation doesn’t apply to signals sent with the direct purpose of communicating to nascent technological societies. Again, it is hard to see how a galactic society could enforce a global ban.

• Civilisations invariably self-destruct.

Common rejoinder: This contingency is already figured into the Drake equation (the L term, above). In any event, we have survived at least 100 years of technological adolescence, and have managed (until now) not to destroy ourselves in a nuclear or biological apocalypse.

Relatively soon we will colonise the moon and Mars, and our long-term survival will no longer rely on Earth.

• Earth is a unique planet in fostering long-lived ecosystems resulting in intelligent life.

Common rejoinder: Perhaps, but the latest studies, in particular the detections of extrasolar planets point in the opposite direction. Environments like ours appear quite common.

• We are alone in the Milky Way galaxy. Some scientists further conclude we are alone in the entire observable universe.

Common rejoinder: This conclusion flies in the face of the “principle of mediocrity,” namely the presumption, popular since the time of Copernicus, that there’s nothing special about human society or environment.

Stephen Webb’s 2002 book If the Universe Is Teeming with Aliens … Where is Everybody, provides numerous other proposed solutions and rejoinders.

Two of Drake’s key terms – fp (the fraction of stars that have planets) and ne (the average number of planets that can support life, per star that has planets) are subject to measurement.

Scientists once thought stable planetary systems and Earth-like planets were a rarity. But recent evidence suggests otherwise.

Thanks to Kepler and other projects, these two terms have been found to have reasonable values, although not quite as optimistic as Drake and his colleagues first estimated.

With every new research finding in the area of extrasolar planets and possible extraterrestrial living organisms, the mystery of Fermi’s paradox deepens.

“Where is everybody?” is a question that now carries even greater resonance.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Jonathan Borwein (Jon), University of Newcastle and David H. Bailey, University of California, Davis.

We are a product of evolution, and are not surprised that our bodies seem to be well-suited to the environment.

Our leg bones are strong enough to allow for Earth’s gravitational pull – not too weak to shatter, not so massively over-engineered as to be wasteful.

But it could also be claimed we are special and the environment was formed and shaped for us.

This, as we know, is the basis of many religious ideas.

In recent years, such ideas have been expanded beyond Earth to look at the entire universe and our place within it.

The so-called Fine-Tuning Argument – that the laws of physics have been specially-tuned, potentially by some Supreme Being, to allow human life to arise – is the focus of Victor J. Stenger’s book.

Stenger presents the mathematics underpinning cosmic evolution, the lifetime of stars, the quantum nature of atoms and so on. His central is that “fine-tuning” claims are fatally flawed.

He points out that some key areas of physics – such as the equality of the charges on the electron and proton – are set by conservation laws determined by symmetries in the universe, and so are not free to play with.

Some flaws in the theory, he argues, run deeper.

A key component of the fine-tuning argument is that there are many parameters governing our universe, and that changing any one of these would likely produce a sterile universe unlike our own.

But think of baking a cake. Arbitrarily doubling only the flour, or sugar or vanilla essence may end in a cooking disaster, but doubling all the ingredients results in a perfectly tasty cake.

The interrelationships between the laws of physics are somewhat more complicated, but the idea is the same.

A hypothetical universe in which gravity was stronger, the masses of the fundamental particles smaller and electomagnetic force weaker may well result in the following: a universe that appears a little different to our own, but is still capable of producing long-lived stars and heavy chemical elements, the basic requirements for complex life.

Stenger backs up such points with his own research, and provides access to a web-based program he wrote called MonkeyGod.

The program allows you to conjure up universes with differing underlying physics. And, as Stenger shows, randomly plucking universe parameters from thin air can still produce universes quite capable of harbouring life.

This book is a good read for those wanting to understand the fine-tuning issues in cosmology, and it’s clear Stenger really understands the science.

But while many of the discussions are robust, I felt that in places some elements of the fine-tuning argument were brushed aside with little real justification.

As a case in point, Stenger falls back on multiverse theory and the anthropic principle, whereby we occupy but one of an almost infinite sea of different universes, each with a different law of physics.

In multiverse theory, most universes would be sterile (though we should not be surprised to find ourselves in a habitable universe).

While such a multiverse – the staple of superstring and brane ideas of the cosmos – is often sold as science fact, it actually lies much closer to the world of science speculation (or, to many, fiction).

We are not out of the fine-tuning waters yet, but Stenger’s book is a good place to start getting to grips with the issues.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Geraint Lewis*

I work on the mathematics of sharing resources, which has led me to consider emotions such as envy, behaviour such as risk-taking and the best way to cut a cake.

Like, I suspect, many women, my wife enjoys eating dessert but not ordering it. I therefore dutifully order what I think she’ll like, cut it in half and invite her to choose a piece.

This is a sure-fire recipe for marital accord. Indeed, many mathematicians, economists, political scientists and others have studied this protocol and would agree. The protocol is known as the “cut-and-choose” procedure. I cut. You choose.

Cut-and-choose

Cut-and-choose is not limited to the dining table – it dates back to antiquity. It appears nearly 3,000 years ago in Hesiod’s poem Theogeny where Prometheus divides a cow and Zeus selects the part he prefers.

In more recent times, cut-and-choose has been enshrined in the UN’s 1982 Convention of the Law of the Sea where it was proposed as a mechanism to resolve disputes when dividing the seabed for mining.

To study the division of cake, cows and the seabed in a more formal way, various mathematical models have been developed. As with all models, these need to make a number of simplifying assumptions.

One typical assumption is that the people employing the cut-and-choose method are risk-averse. They won’t adopt a risky strategy that may give them less cake than a more conservative strategy.

With such assumptions in place, we can then prove what properties cake cutting procedures have and don’t have. For instance, cut-and-choose is envy free.

You won’t envy the cake I have, otherwise you would have taken this piece. And I won’t envy the piece you have, as the only risk-averse strategy is for me to cut the cake into two parts that I value equally.

On the other hand, the cutting of the cake is not totally equitable since the player who chooses can get cake that has more than half the total value for them.

With two players, it’s hard to do better than cut-and-choose. But I should record that my wife argues with me about this.

She believes it favours the second player since the first player inevitably can’t divide the cake perfectly and the second player can capitalise on this. This is the sort of assumption ignored in our mathematical models.

My wife might prefer the moving-knife procedure which doesn’t favour either player. A knife is moved over the cake, and either player calls “cut” when they are happy with the slice.

Again, this will divide the cake in such a way that neither player will envy the other (else they would have called “cut” themselves).

Three’s a crowd

Unfortunately, moving beyond two players increases the complexity of cutting cake significantly.

With two players, we needed just one cut to get to an envy free state. With three players, a complex series of five cuts of the cake might be needed. Of course, only two cuts are needed to get three slices.

The other three cuts are needed to remove any envy. And with four players, the problem explodes in our face.

An infinite number of cuts may be required to get to a situation where no one envies another’s cake. I’m sure there’s some moral here about too many cake cutters spoiling the dessert.

There are many interesting extensions of the problem. One such extension is to indivisible goods.

Suppose you have a bag of toys to divide between two children. How do you divide them fairly? As a twin myself, I know that the best solution is to ensure you buy two of everything.

It’s much more difficult when your great aunt gives you one Zhu Zhu pet, one Bratz doll and three Silly Bandz bracelets to share.

Online

More recently, I have been studying a version of the problem applicable to online settings. In such problems, not all players may be available all of the time. Consider, for instance, allocating time on a large telescope.

Astronomers will have different preferences for when to use the telescope depending on what objects are visible, the position of the sun, etcetera. How do we design a web-based reservation system so that astronomers can choose observation times that is fair to all?

We don’t want to insist all astronomers log in at the same time to decide an allocation. And we might have to start allocating time on the telescope now, before everyone has expressed their preferences. We can view this as a cake-cutting problem where the cake is made up of the time slots for observations.

The online nature of such cake-cutting problems poses some interesting new challenges.

How can we ensure that late-arriving players don’t envy cake already given to earlier players? The bad news is that we cannot now achieve even a simple property like envy freeness.

No procedure can guarantee situations where players don’t envy one another. But more relaxed properties are possible, such as not envying cake allocated whilst you are participating in the cutting of the cake.

Ham sandwich

There’s a brilliantly named piece of mathematics due to Arthur H. Stone and [John Tukey](http://www.morris.umn.edu/~sungurea/introstat/history/w98/Tukey.html, the Ham Sandwich Theorem which proves we can always cut a three-layered cake perfectly with a single cut.

Suppose we have three objects. Let’s call them “the top slice of bread”, “the ham filling” and “the bottom slice of bread”. Or if you prefer “the top layer” of the cake, “the middle layer” and “the bottom layer”.

The ham sandwich theorem proves a single slice can always perfectly bisect the three objects. Actually, the ham sandwich theorem works in any number of dimensions: any n objects in n-dimensional space can be simultaneously bisected by a single (n − 1) dimensional hyperplane.

So, in the case of the three-layered cake, n = 3, and the three-layered cake can be bisected (or cut) using a single, two-dimensional “hyperplane”. Such as, say, a knife.

Who would have thought that cutting cake would lead to higher dimensions of mathematics by way of a ham sandwich?

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Toby Walsh*

I work on the mathematics of sharing resources, which has led me to consider emotions such as envy, behaviour such as risk-taking and the best way to cut a cake.

Like, I suspect, many women, my wife enjoys eating dessert but not ordering it. I therefore dutifully order what I think she’ll like, cut it in half and invite her to choose a piece.

This is a sure-fire recipe for marital accord. Indeed, many mathematicians, economists, political scientists and others have studied this protocol and would agree. The protocol is known as the “cut-and-choose” procedure. I cut. You choose.

Cut-and-choose

Cut-and-choose is not limited to the dining table – it dates back to antiquity. It appears nearly 3,000 years ago in Hesiod’s poem Theogeny where Prometheus divides a cow and Zeus selects the part he prefers.

In more recent times, cut-and-choose has been enshrined in the UN’s 1982 Convention of the Law of the Sea where it was proposed as a mechanism to resolve disputes when dividing the seabed for mining.

To study the division of cake, cows and the seabed in a more formal way, various mathematical models have been developed. As with all models, these need to make a number of simplifying assumptions.

One typical assumption is that the people employing the cut-and-choose method are risk-averse. They won’t adopt a risky strategy that may give them less cake than a more conservative strategy.

With such assumptions in place, we can then prove what properties cake cutting procedures have and don’t have. For instance, cut-and-choose is envy free.

You won’t envy the cake I have, otherwise you would have taken this piece. And I won’t envy the piece you have, as the only risk-averse strategy is for me to cut the cake into two parts that I value equally.

On the other hand, the cutting of the cake is not totally equitable since the player who chooses can get cake that has more than half the total value for them.

With two players, it’s hard to do better than cut-and-choose. But I should record that my wife argues with me about this.

She believes it favours the second player since the first player inevitably can’t divide the cake perfectly and the second player can capitalise on this. This is the sort of assumption ignored in our mathematical models.

My wife might prefer the moving-knife procedure which doesn’t favour either player. A knife is moved over the cake, and either player calls “cut” when they are happy with the slice.

Again, this will divide the cake in such a way that neither player will envy the other (else they would have called “cut” themselves).

Three’s a crowd

Unfortunately, moving beyond two players increases the complexity of cutting cake significantly.

With two players, we needed just one cut to get to an envy free state. With three players, a complex series of five cuts of the cake might be needed. Of course, only two cuts are needed to get three slices.

The other three cuts are needed to remove any envy. And with four players, the problem explodes in our face.

An infinite number of cuts may be required to get to a situation where no one envies another’s cake. I’m sure there’s some moral here about too many cake cutters spoiling the dessert.

There are many interesting extensions of the problem. One such extension is to indivisible goods.

Suppose you have a bag of toys to divide between two children. How do you divide them fairly? As a twin myself, I know that the best solution is to ensure you buy two of everything.

It’s much more difficult when your great aunt gives you one Zhu Zhu pet, one Bratz doll and three Silly Bandz bracelets to share.

Online

More recently, I have been studying a version of the problem applicable to online settings. In such problems, not all players may be available all of the time. Consider, for instance, allocating time on a large telescope.

Astronomers will have different preferences for when to use the telescope depending on what objects are visible, the position of the sun, etcetera. How do we design a web-based reservation system so that astronomers can choose observation times that is fair to all?

We don’t want to insist all astronomers log in at the same time to decide an allocation. And we might have to start allocating time on the telescope now, before everyone has expressed their preferences. We can view this as a cake-cutting problem where the cake is made up of the time slots for observations.

The online nature of such cake-cutting problems poses some interesting new challenges.

How can we ensure that late-arriving players don’t envy cake already given to earlier players? The bad news is that we cannot now achieve even a simple property like envy freeness.

No procedure can guarantee situations where players don’t envy one another. But more relaxed properties are possible, such as not envying cake allocated whilst you are participating in the cutting of the cake.

Ham sandwich

There’s a brilliantly named piece of mathematics due to Arthur H. Stone and John Tukey. The Ham Sandwich Theorem which proves we can always cut a three-layered cake perfectly with a single cut.

Suppose we have three objects. Let’s call them “the top slice of bread”, “the ham filling” and “the bottom slice of bread”. Or if you prefer “the top layer” of the cake, “the middle layer” and “the bottom layer”.

The ham sandwich theorem proves a single slice can always perfectly bisect the three objects. Actually, the ham sandwich theorem works in any number of dimensions: any n objects in n-dimensional space can be simultaneously bisected by a single (n − 1) dimensional hyperplane.

So, in the case of the three-layered cake, n = 3, and the three-layered cake can be bisected (or cut) using a single, two-dimensional “hyperplane”. Such as, say, a knife.

Who would have thought that cutting cake would lead to higher dimensions of mathematics by way of a ham sandwich?

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Toby Walsh