The Living Smart Grid

guest post by Todd McKissick

The last few years, in energy circles, people have begun the public promotion of what they call the smart grid.  This is touted as providing better control, prediction and utilization of our nation’s electrical grid system.  However, it doesn’t provide anyone except the utilities more benefits.  It’s expected to cost much more and to actually take away some of the convenience of having all the power you want, when you want it.

We can do better.

Let’s investigate the benefits of some changes to their so called smart grid.  If implemented, these changes will allow instant indirect control and balance of all local grid sections while automatically keeping supply in check with demand.  It can drastically cut the baseload utilization of existing transmission lines.  It can provide early benefits from running it in pseudo parallel mode with no changes at all by simply publishing customer specific real-time prices.  Once that gains some traction, a full implementation only requires adding smart meters to make it work. Both of these stages can be adopted at any rate and benefits only as much as it is adopted. Since it allows Demand Reduction (DR) and Distributed Generation (DG) from any small source to compete in price fairly with the big boys, it encourages tremendous competition between both generators and consumers.

To initiate this process, the real-time price must be determined for each customer.  This is easily done at the utility by breaking down their costs and overhead into three categories.  First, generation is monitored at its location.  Second, transmission is monitored for its contribution.  Both of these are being done already, so nothing new yet.  Third, distribution needs to be monitored at all the nodes and end points in the customer’s last leg of the chain.  Much of this is done and the rest is being done or planned through various smart meter movements.  Once all three of these prices are broken down, they can be applied to the various groups of customers and feeder segments.  This yields a total price to each customer that varies in real time with all the dynamics built in.  By simply publishing that price online, it signals the supply/demand imbalance that applies to them.

This is where the self correction aspect of the system comes into play.  If a transmission line goes down, the affected customers’ price will instantly spike, immediately causing loads to drop offline and storage systems and generation systems to boost their output.  This is purely price driven so no hard controls are sent to the customer equipment to make this happen.  Should a specific load be set to critical use, like a lifeline system for a person or business, they have less risk of losing power completely but will pay an increased amount for the duration of the event.  Even transmission rerouting decisions can be based on the price, allowing neighboring local grids to export their excess to aid a nearby shortfall.  Should an area find its price trending higher or lower over time, the economics will easily point to whatever and wherever something is needed to be added to the system.  This makes forecasting the need for new equipment easier at both the utility and the customer level.

If CO₂ or some other emission charge was created, it can quickly be added to the cost of individual generators, allowing the rest of the system to re-balance around it automatically.

Once the price is published, people will begin tracking their home and heavy loading appliances to calculate their exact electrical bill.  When they learn they can adapt usage profiles and save money, they will create systems to automatically do so.  This will lead to intelligent and power saving appliances, a new generation of smart thermostats, short cycling algorithms in HVAC and even more home automation.  The result of these operations is to balance demand to supply.

When this process begins, the financial incentive becomes real for the customer, attracting them to request live billing.  This can happen as small as one customer at a time for anyone with a smart meter installed.  Both customer and utility benefit from their switchover.

A truly intelligent system like this eliminates the necessity of full grid replacement that some people are proposing.  Instead, it focuses on making the existing one more stable.  Incrementally and in proportion to adoption, the grid stability and redundancy will naturally increase without further cost. The appliance manufacturers already have many load predictive products waiting for the market to call for them so the cost to advance this whole system is fully redundant with the cost of replacement meters which is already happening or planned soon. We need to ensure that the new meters have live rate capability.

This is the single biggest solution to our energy crisis. It will standardize grid interconnection which will entice distributed generation (DG).  As it stands now, most utilities view DG in a negative light with regards to grid stability.  Many issues such as voltage, frequency and phase regulation are often topics they cite.  In reality, however, the current inverter standards ensure that output is appropriately synchronized.  The same applies to power factor issues.  While reducing power sent via the grid directly reduces the load, it’s only half of the picture.

DG with storage and vehicle-to-grid hybrids both give the customer an opportunity to save up their excess and sell it to the grid when it earns the most.  By giving them the live prices, they will also be encouraged to grow their market.  It is an obvious outgrowth for them to buy and store power from the grid in the middle of the night and sell it back for a profit during afternoon peaks.  In fact this is already happening in some markets.

Demand reduction (DR), or load shedding, acts the same as onsite generation in that it reduces the power sent via the grid.  It also acts similar to storage in that it can time shift loads to cheaper rate periods.  To best take advantage of this, people will utilize increasingly better algorithms for price prediction.  The net effect is thousands of individuals competing on prediction techniques to flatten out the peaks into the valleys of the grid’s daily profile.  This competition will be in direct proportion to the local grid instability in a given area.

According to Peter Mark Jansson and John Schmalzel [1]:

From material utilization perspectives significant hardware is manufactured and installed for this infrastructure often to be used at less than 20-40% of its operational capacity for most of its lifetime. These inefficiencies lead engineers to require additional grid support and conventional generation capacity additions when renewable technologies (such as solar and wind) and electric vehicles are to be added to the utility demand/supply mix. Using actual data from the PJM [PJM 2009] the work shows that consumer load management, real time price signals, sensors and intelligent demand/supply control offer a compelling path forward to increase the efficient utilization and carbon footprint reduction of the world’s grids. Underutilization factors from many distribution companies indicate that distribution feeders are often operated at only 70-80% of their peak capacity for a few hours per year, and on average are loaded to less than 30-40% of their capability.

At this time the utilities are limiting adoption rates to a couple percent.  A well known standardization could replace that with a call for much more.  Instead of discouraging participation, it will encourage innovation and enhance forecasting and do so without giving away control over how we wish to use our power.  Best of all, it is paid for by upgrades that are already being planned. How's that for a low cost SMART solution?

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[1] Peter Mark Jansson and John Schmalzel, Increasing utilization of US electric grids via smart technologies: integration of load management, real time pricing, renewables, EVs and smart grid sensors, The International Journal of Technology, Knowledge and Society 7, 47-60.

Energy, the Environment, and What We Can Do

A while ago I gave a talk at Google. Since I think we should cut unnecessary travel, I decided to stay here in Singapore and give the talk virtually, in the form of a robot:

I thank Mike Stay for arranging this talk at Google, and Trevor Blackwell and Suzanne Broctao at Anybots for letting me use one of their robots!

Here are the slides:

• Energy, the Environment, and What We Can Do.

To see the source of any piece of information in these slides, just click on it!

And here’s me:

This talk was more ambitious than previous ones I’ve given—and not just because I was struggling to operate a robot, read my slides on my laptop, talk, and click the pages forward all at once! I said more about solutions to our problems this time. That’s where I want to head, but of course it’s infinitely harder to describe solutions than to list problems or even to convince people that they really are problems.

The 1990 IPCC Climate Projections

Over on Google+, Daniel Lemire writes:

The IPCC predictions from 1990 went for a probable rise of the global mean temperature of 0.3 °C per decade and at least 0.2 °C per decade. See the IPPC assessment overview document, Section 1.0.3. (It is available online here.) We have had two decades. Fact: they predicted more warming than what actually materialized. Quite a bit more. See:

• Instrumental temperature record, Wikipedia.

What’s the full story here? Here are some basic facts. The policymaker summary of the 1990 report estimates:

under the IPCC Business-as-Usual (Scenario A) emissions of greenhouse gases, a rate of increase of global-mean temperature during the next century of about 0.3 °C per decade (with an uncertainty range of 0.2 °C to 0.5 °C per decade); this is greater than that seen over the past 10,000 years. This will result in a likely increase in global-mean temperature of about 1°C above the present value by 2025 and 3°C before the end of the next century. The rise will not be steady because of the influence of other factors…

I believe we are going along with the ‘business-as-usual’ emissions of greenhouse gases. On the other hand, Wikipedia shows a figure from Global Warming Art:

based on NASA GISS data here. In 1990, the 5-year mean Global Land-Ocean Temperature Index was .27, meaning .27 °C above the mean temperature from 1961 to 1990. By 2006, the 5-year mean was .54.

So, that’s a .27 °C increase in 16 years, or about .17 °C per decade. This is slightly less than the bottom-end estimate of 0.2 °C per decade, and about half the expected rise of 0.3 °C per decade.

Is there any official story on why the 1990 IPCC report overestimated the temperature rise during this time? In 2007 the IPCC estimated a temperature rise of about 0.2 °C per decade for the next two decades for all the scenarios they considered. So, it seems somehow 0.3 °C went down to 0.2 °C. What was wrong with the original estimate?

Dysnomia

This morning I was looking at a nice website that lets you zoom in or out and see objects of different sizes, ranging from the Planck length to the entire observable Universe:

• Cary Huang, The Scale of the Universe.

When I zoomed out to a size somewhat larger than Rhode Island, I saw something strange:

Huh? I recently saw a movie called Melancholia, about a pair of sisters struggling with fear and depression, and a blue planet of that name that’s headed for Earth. And indeed, dysnomia is another mental disorder: a difficulty in remembering words, names or numbers. Are astronomical bodies named for mental disorders catching on?

I don’t know—but unlike Melancholia, Dysnomia is real! It’s the speck at left here:

The larger blob of light is the dwarf planet Eris. Dysnomia is its moon. Both were discovered in 2005 by a team at Palomar led by Mike Brown.

Why the funny name? In Greek mythology, Dysnomia (Δυσνομία), meaning ‘lawlessness’, was the daughter of Eris, the goddess of strife and discord. You may remember the dwarf planet Eris under its tentative early name ‘Xena’, from the TV character. That was deemed too silly to stand.

Eris is 30% more massive than Pluto, and thus it helped lead to a redefinition of ‘planet’: both are now called dwarf planets, because they aren’t big enough to clear their neighborhood of debris.

Eris has a highly eccentric orbit, and it takes 580 years to go around the Sun. Right now it’s at its farthest from the Sun: three times as far as Pluto. So it’s very cold, about 30 kelvin (-243 °C), and its surface is covered with methane ice. But in about 290 years its temperature will nearly double, soaring to a balmy 56 kelvin (-217 °C). Its methane ice will melt and then evaporate away, giving this world a new atmosphere! That’s pretty amazing: an annual atmosphere.

The surface of Eris is light gray. So, it’s quite different than Pluto and Neptune’s moon Triton, which are reddish due to deposits of tholins—complex compounds formed by bombarding hydrocarbons and nitrogen with sunlight.

Remember the smoggy orange surface of Titan?

That’s due to tholins! It’s possible that on Eris the tholins are currently covered by methane ice.

I wish I knew more about what tholins actually are—their actual chemical structures. But they’re a complicated mess of stuff, and they vary from place to place: people talk about Titan tholins, Triton tholins and ice tholins.

Indeed, the term “tholin” comes from the Greek word tholós (θολός), meaning “not clear”. It was coined by Carl Sagan to describe the mysterious mix of substances he created in experiments on the gas mixtures found in Titan’s atmosphere… experiments a bit like the old Urey-Miller experiment which created amino acids from stuff that was present on the early Earth—water, methane, ammonia, hydrogen—together with lots of electrical discharges.

Might tholins become complicated enough to lead to life in the outer Solar System, at least on relatively peppy worlds like Titan? There’s a complex cycle going on there:

Here ‘Da’ means daltons: a dalton is a unit of mass equal to about one hydrogen atom, so a molecule that’s 2000 Da could be made of 2000 hydrogens or, more reasonably, 2000/12 ≈ 166 carbons, or various other things. The point is: that’s a pretty big complicated molecule—big enough to be very interesting!

On Earth, many soil bacteria are able to use tholins as their sole source of carbon. Some think that tholins may have been the first food for microbes! In fact, some scientists have speculated that Earth may have been seeded by tholins from comets early in its development, providing the raw material necessary for life. But ever since the Great Oxygenation Event, the Earth’s surface has been too oxidizing for tholins to exist here.

Tholins have also been found outside our Solar System:

• Red dust in disk may harbor precursors to life, Carnegie Institute news release, 5 January 2008.

I bet tholins often go hand-in-hand with PAHs, or polycyclic aromatic hydrocarbons. PAHs are also common in outer space. In Earth you can find them in soot, or the tarry stuff that forms in a barbecue grill. Wherever carbon-containing materials suffer incomplete combustion, you’ll find PAHs.

PAHs are made of hexagonal rings of carbon atoms, with some hydrogens along the edges:

Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens. You’ve probably heard about naphthalene, which is used for mothballs: this consists of two hexagonal rings stuck together. True PAHs have more. With three rings you can make anthracene:

and phenanthrene:

With four, you can make napthacene:

pyrene:

triphenylene:

and chrysene:

And so on! The game just gets more complicated as you get to use more puzzle pieces.

In 2004, a team of scientists discovered anthracene and pyrene in an amazing structure called the Red Rectangle!

Here two stars 2300 light years from us are spinning around each other while pumping out a huge torus of icy dust grains and hydrocarbon molecules. It’s not really shaped like a rectangle or X—it just looks that way from here. The whole scene is about 1/3 of a light year across.

This was first time such complex molecules had been
found in space:

• Uma P. Vijh, Adolf N. Witt, and Karl D. Gordon, Small polycyclic aromatic hydrocarbons in the Red Rectangle, The Astrophysical Journal 619 (2005), 368-378.

By now, lots of organic molecules have been found in interstellar or circumstellar space. There’s a whole "ecology" of organic chemicals out there, engaged in complex reactions. Life on planets might someday be seen as just an aspect of this larger ecology. PAHs and tholins are probably among the dominant players in this ecology, at least at this stage.

Indeed, I’ve read that about 10% of the interstellar carbon is in the form of PAHs—big ones, with about 50 carbons per molecule. They’re common because they’re incredibly stable. They’ve even been found riding the shock wave of a supernova explosion!

PAHs are also found in meteorites called "carbonaceous chondrites". These space rocks contain just a little carbon: about 3% by weight. But 80% of this carbon is in the form of PAHs.

Here’s an interview with a scientist who thinks PAHs were important precursors of life on Earth:

• Aromatic world, interview with Pascale Ehrenfreund, Astrobiology Magazine.

Also try this:

• PAH world hypothesis, Wikipedia.

This speculative hypothesis says that PAHs were abundant in the primordial soup of the early Earth and played a major role in the origin of life by mediating the synthesis of RNA molecules, leading to the (also speculative) RNA world.

Another radical theory has been proposed by Prof. Sun Kwok, author of Organic Matter in the Universe. He claims that instead of PAHs, complex molecules like this would do better at explaining the spectra of interstellar clouds:

Would this molecule count as a tholin? Maybe so: I don’t know. He says:

Our work has shown that stars have no problem making complex organic compounds under near-vacuum conditions. Theoretically, this is impossible, but observationally we can see it happening.

For more see:

• Sun Kwok and Yong Zhang, Mixed aromatic–aliphatic organic nanoparticles as carriers of unidentified infrared emission features, Nature 479 (2011), 80–83.

This paper isn’t free, but here’s a summary that is:

• Astronomers discover complex organic matter exists throughout the Universe, ScienceDaily, 26 October 2011.

However, I’d take this with a grain of salt until more confirmation comes along! They’re matching very complicated spectra to hypothetical chemicals, without yet any understanding of how these chemicals could be formed in space. It would be very cool if true.

Regardless of how the details play out, I think we’ll eventually see that organic life across the universe is a natural outgrowth of the organic chemistry of PAHs, tholins and related chemicals. It will be great to see the whole story: how much in common life has in different locations, and how much variation there is. It may be rare, but the universe is very large, so there must be statistical patterns in how life works.

It goes to show how everything is connected. Starting from a chance encounter with Dysnomia, we’ve been led to ponder another planet whose atmosphere liquifies and then freezes every year… and then wonder about why so many objects in the outer solar system are red… and why the same chemicals you find in the tarry buildup on a barbecue grill are also seen in outer space… and whether life on Earth could have been started by complex compounds from comets… and whether life on planets is just part of a larger interstellar chemical ‘ecology’. Not bad for a Saturday morning!

Tidbits of Geometry

Since I grew up up reading Martin Gardner, I’ve often imagined it would be fun to write about math and physics in a way that nonexperts might enjoy. Right now I’m trying my hand at this on Google+. You can read that stuff here.

Google+ encourages brevity—not as much as Twitter, but more than a blog. So I’m posting things that feature a single catchy image, a brief explanation, and URL’s to visit for more details.

Lately I’ve been talking about geometry. I realized that these posts could be cobbled together into a kind of loose ‘story’, so here it is. I couldn’t resist expanding the posts a bit, but the only really new stuff is more about Leonardo Da Vinci and the golden ratio, and five puzzles—only one of which I know the answer to!

The golden ratio

Sure, the golden ratio, Φ = (√5+1)/2, is cool… but if you think ancient Greeks ran around in togas talking about the “golden ratio” and writing it as “Φ”, you’re wrong. This number was named Φ after the Greek sculptor Phidias only in 1914, in a book called The Curves of Life by the artist Theodore Cook. And it was Cook who first started calling 1.618… the golden ratio. Before him, 1/Φ = 0.618… was called the golden ratio! Cook dubbed this number “φ”, the lower-case baby brother of Φ.

In fact, the whole “golden” terminology can only be traced back to 1826, when it showed up in a footnote to a book by one Martin Ohm, brother of Georg Ohm—the guy with the law about resistors. Before then, a lot of people called 1/Φ the “Divine Proportion”. And the guy who started that was Luca Pacioli, a pal of Leonardo da Vinci who translated Euclid’s Elements. In 1509, Pacioli published a 3-volume text entitled De Divina Proportione, advertising the virtues of this number.

Greek texts seem remarkably quiet about this number. The first recorded hint of it is Proposition 11 in Book II of Euclid’s Elements. It also shows up elsewhere in Euclid, especially Proposition 30 of Book VI, where the task is “to cut a given finite straight line in extreme and mean ratio”, meaning a ratio A:B such that A is to B as B is to A+B. This is later used in Proposition 17 of Book XIII to construct the pentagonal face of a regular dodecahedron.

The regular pentagon, and the pentagram inside it, is deeply connected to the golden ratio. If you look carefully, you’ll see no fewer than twenty long skinny isosceles triangles, in three different sizes but all the same shape!

They’re all ‘golden triangles’: the short side is φ times the length of the long sides.

And the picture here lets us see that φ is to 1 as 1 is to 1+φ. A little algebra then gives

which you can solve to get

and thus

For more, see:

• John Baez, Tales of the Dodecahedron.

Da Vinci and the golden ratio

Did Leonardo da Vinci use the golden ratio in his art? It would cool if he did. Unfortunately, attempts to prove it by drawing rectangles on his sketches and paintings are unconvincing. Here are three attempts you can see on the web; click for details if you want:

The first two make me less inclined to believe Da Vinci was using the golden ratio, not more. The last one, the so-called
Vitruvian Man, looks the most convincing, but only if you take on faith that the ratio being depicted is really the golden ratio!

Puzzle 1. Carefully measure the ratio here and tell us what you get, with error bars on your result.

It would be infinitely more convincing if Da Vinci had written about the golden ratio in his famous notebooks. But I don’t think he did. If he didn’t, that actually weighs against the whole notion.

Indeed, I thought the whole notion was completely hopeless until I discovered that Da Vinci did the woodcuttings for Pacioli’s book De Divina Proportione. And even lived with Pacioli while this project was going on! So, we can safely assume Da Vinci knew what was in this book.

It consists of 3 separate volumes. First a volume about the golden ratio, polygons, and perspective. Then one about the ideas of Vitruvius on math in architecture. (Apparently Vitruvius did not discuss the golden ratio.) Then one that’s mainly an Italian translation of Piero della Francesca’s Latin writings on polyhedra.

De Divina Proportione was popular in its day, but only two copies of the original edition survive. Luckily, it’s been scanned in!

• Luca Pacioli, De Divina Proportione.

The only picture I see that might be about using the golden ratio to draw the human figure is this:

The rectangles don’t look very ‘golden’! But the really important thing is to read the text around this picture, or for that matter the whole book. Unfortunately my Renaissance Italian is… ahem… a bit rusty. The text has been translated into German but apparently not English.

Puzzle 2. What does Luca Pacioli say on this page?

The picture above is on page 70 of the scanned-in file. Of course some scholar should have written a paper about this already… I just haven’t gotten around to searching the literature.

By the way, here’s something annoying. This picture on the Wikipedia article about De Divina Proportione purports to come from that book:

Again most of the rectangles don’t look very golden, even though it says “Divina Proportio” right on top. But here’s the big problem: I can’t find it in the online version of the book! Luca Luve, who spotted the online version for me in the first place, concurs.

Puzzle 3. Where is it really from?

Luca Pacioli

Luca Pacioli had many talents: besides books on art, geometry and mathematics, he also wrote the first textbook on double-entry bookkeeping! This portrait of him multitasking gives some clue as to how he accomplished so much. He seems to be somberly staring at a hollow glass cuboctahedron half-filled with water while simultaneously drawing something completely different and reading a book:

Note the compass and the regular dodecahedron. The identity of the other figure in the painting is uncertain, and so is that of the painter, though people tend to say it’s Jacopo de’ Barbari.

Piero della Francesca

This creepy painting shows three people calmly discussing something while Jesus is getting whipped in the background. It’s one of the first paintings to use mathematically defined rules of perspective, and it’s by Piero della Francesca, the guy whose pictures of polyhedra fill the third part of Pacioli’s De Divina Proportione.

Piero della Francesca seems like an interesting guy: a major artist who actually quit painting in the 1470’s to focus on the mathematics of perspective and polyhedra. If you want to know how to draw a perfect regular pentagon in perpective using straightedge and compass, he’s your guy.

Constructing the pentagon

I won’t tell you how to do it in perspective, but here’s how to construct a regular pentagon with straightedge and compass:

Just pay attention to how it starts. Say the radius of the circle is 1. We bisect it and get a segment of length 1/2, then consider a segment at right angles of length 1. But

so these are the sides of a right triangle whose hypotenuse has length √5/2, by the Pythagorean theorem!

Yes, I know I didn’t explain the whole construction… just the start. But the golden ratio is √5/2 + 1/2, so we’ve clearly on the right track. If you’re ever stuck on a desert island with nothing to do but lots of sand and some branches, you can figure out the rest yourself.

Or if you’ve got the internet on your desert island, read this:

• Pentagon, Wikipedia.

But here’s the easy way to make a regular pentagon: just tie a simple overhand knot in a strip of paper!

The pentagon-decagon-hexagon identity

The most bizarre fact in Euclid’s Elements is Proposition XIII.10. Take a circle and inscribe a regular pentagon, a regular hexagon, and a regular decagon. Take the edges of these shapes, and use them as the sides of a triangle. Then this is a right triangle!

How did anyone notice this??? It’s long been suspected that this fact first came up in studying the icosahedron. But nobody gave a proof using the icosahedron until I posed this as a challenge and Greg Egan took it up. The hard part is showing that the two right triangles here are congruent:

Then AB is the side of the pentagon, BC is the side of the decagon and AC’ is the radius of the circle itself, which is the side of the hexagon!

For details, see:

• John Baez, This Week’s Finds in Mathematical Physics (Week 283).

and

• Pentagon-decagon-hexagon identity, nLab.

The octahedron and icosahedron

Platonic solids are cool. A regular octahedron has 12 edges. A regular icosahedron has 12 vertices. Irrelevant coincidence? No! If you cleverly put a dot on each edge of the regular octahedron, you get the vertices of a regular icosahedron! But it doesn’t work if you put the dot right in the middle of the edge—you have to subdivide the edge in the exactly correct ratio. Which ratio? The golden ratio!

This picture comes from R. W. Gray.

According to Coxeter’s marvelous book Regular Polytopes, this fact goes back at least to an 1873 paper by a fellow named Schönemann.

Puzzle 4. What do you get if you put each dot precisely in the center of the edge?

The heptagon

The golden ratio Φ is great, but maybe it’s time to move on? The regular pentagon’s diagonal is Φ times its edge, and a little geometry shows the ratio of 1 to Φ equals the ratio of Φ to Φ+1. What about the regular heptagon? Here we get two numbers, ρ and σ, which satisfy four equations, written as ratios below! So, for example, the ratio of 1 to ρ equals the ratio of ρ to 1+σ, and so on.

For more see:

• Peter Steinbach, Golden fields: a case for the heptagon, Mathematics Magazine 70 (Feb., 1997), 22-31.

He works out the theory for every regular polygon. So, it’s not that the fun stops after the pentagon: it just gets more sophisticated!

Constructing the heptagon

You can’t use a straightedge and compass to construct a regular heptagon. But here’s a construction that seems to do just that!

If you watch carefully, the seeming paradox is explained. For more, see:

• Heptagon, Wikipedia.

Trisecting the angle

When I was a kid, my uncle wowed me by trisecting an angle. He wasn’t a crackpot: he was bending the usual rules! He marked two dots on the ruler, A and B below, whose distance equaled the radius of the circle, namely OB. Then the trick below makes φ one third of θ.

Drawing dots on your ruler is called neusis, and the ancient Greeks knew about it. You can also use it to double the cube and construct a regular heptagon—impossible with a compass and straightedge if you’re don’t draw dots on it. Oddly, it fell out of fashion. Maybe purity of method mattered more than solving lots of problems?

Nowadays we realize that if you only have a straightedge, you can only solve linear equations. Adding a compass to your toolkit lets you also take square roots, so you can solve quadratic equations. Adding neusis on top of that lets you take cube roots, which—together with the rest—lets you solve cubic equations. A fourth root is a square root of a square root, so you get those for free, and in fact you can even solve all quartic equations. But you can’t take fifth roots.

Puzzle 5. Did anyone ever build a mechanical gadget that lets you take fifth roots, or maybe even solve general quintics?

Fluid Flows and Infinite-Dimensional Manifolds I

Or: waves that take the shortest path through infinity

guest post by Tim van Beek

Water waves can do a lot of things that light waves cannot, like “breaking”:

In mathematical models this difference shows up through the kind of partial differential equation (PDE) that models the waves:

• light waves are modelled by linear equations while

• water waves are modelled by nonlinear equations.

Physicists like to point out that linear equations model things that do not interact, while nonlinear equations model things that interact with each other. In quantum field theory, people speak of “free fields” versus “interacting fields”.

Some nonlinear PDE that describe fluid flows turn out to also describe geodesics on infinite-dimensional Riemannian manifolds. This fascinating observation is due to the Russian mathematician Vladimir Arnold. In this blog post I would like to talk a little bit about the concepts involved and show you a little toy example.

Fluid Flow modelled by Diffeomorphisms

The Euler viewpoint on fluids is that a fluid is made of tiny “packages” or “particles”. The fluid flow is described by specifying where each package or particle is at a given time . When we start at some time on a given manifold , the flow of every fluid package is described by a path on parametrized by time, and for every time there is a diffeomorphism defined by the requirement that it maps the initial position of each fluid package to its position at time :

This picture is taken from the book

• V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics, Springer, Berlin, 1998. (Review at Zentralblatt Mathematik.)

We will take as a model of the domain of the fluid flow a compact Riemannian manifold . A fluid flow, as pictured above, is then a path in the diffeomorphism group . In order to apply geometric concepts in this situation, we will have to turn or some closed subgroup of it into a manifold, which will be infinite dimensional.

The curvature of such a manifold can provide a great deal about the stability of fluid flows: On a manifold with negative curvature geodesics will diverge from each other. If we can model fluid flows as geodesics in a Riemannian manifold and calculate the curvature, we could try to infer a bound on weather forecasts (in fact, that is what Vladimir Arnold did!): The solution that you calculate is one geodesic. But if you take into account errors with determining your starting point (involving the measurement of the state of the flow at the given start time), what you are actually looking at is a bunch of geodesics starting in a neighborhood of your starting point. If they diverge fast, that means that measurement errors make your result unreliable fast.

If you never thought about manifolds in infinite dimensions, you may feel a little bit insecure as to how the concepts that you know from differential geometry can be generalized from finite dimensions. At least I felt this way when I first read about it. But it turns out that the part of the theory one needs to know in order to understand Arnold’s insight is not that scary, so I will talk a little bit about it next.

What you should know about infinite-dimensional manifolds

The basic strategy when handling finite-dimensional, smooth, real manifolds is that you have a complicated manifold , but also locally for every point a neighborhood and an isomorphism (a “chart”) of to an open subset of the vastly simpler space , the “model space”. These isomorphisms can be used to transport concepts from to . In infinite dimensions it is however not that clear what kind of model space should be taken in place of . What structure should have?

Since we would like to differentiate, we should for example be able to define the derivative of a curve in :

If we write down the usual formula for a derivative

we see that to make sense of this we need to be able to add elements, have a scalar multiplication, and a topology such that the algebraic operations are continuous. Sets with this structure are called topological vector spaces.

A curve that has a first derivative, second derivative, third derivative… and so on at every point is called a smooth curve, just as in the finite dimensional case.

So should at least be a topological vector space. We can, of course, put more structure on to make it “more similar” to , and choose as model space in ascending order of generality:

1) A Hilbert space, which has an inner product,

2) a Banach space that does not have a inner product, but a norm,

3) a Fréchet space that does not have a norm, but a metric,

4) a general topological vector space that need not be metrizable.

People talk accordingly of Hilbert, Banach and Fréchet manifolds. Since the space consisting of smooth maps from to is not a Banach space but a Fréchet space, we should not expect that we can model diffeomorphism groups on Banach spaces, but on Fréchet spaces. So we will use the concept of Fréchet manifolds.

But if you are interested in a more general theory using locally convex topological vector spaces as model spaces, you can look it up here:

• Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, American Mathematical Society, Providence Rhode Island, 1999.

Note that Kriegl and Michor use a different definition of “smooth function of Fréchet spaces” than we will below.

If you learn functional analysis, you will most likely start with operators on Hilbert spaces. One could say that the theory of topological vector spaces is about abstracting away as much structure from a Hilbert space and look what structure you need for important theorems to still hold true, like the open mapping/closed graph theorem. If you would like to learn more about this, my favorite book is this one:

• Francois Treves, Topological vector Spaces, Distributions and Kernels, Dover Publications, 2006.

Since we replace the model space with a Fréchet space , there will be certain things that won’t work out as easily as for the finite dimensional , or not at all.

It is nevertheless possible to define both integrals and differentials that behave much in the expected way. You can find a nice exposition of how this can be done in this paper:

• Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bulletin of the American Mathematical Society 7 (1982), pages 65-222.

The story starts with the definition of the directional derivative that can be done just as in finite dimensions:

Let and be Fréchet spaces, open and a continuous map. The derivative of at the point in the direction is the map

given by:

A simple but nontrivial example is the operator

given by:

with the derivative

It is possible to define higher derivatives and also prove that the chain rule holds, so that we can define that a function between Fréchet spaces is smooth if it has derivatives at every point of all orders. The definition of a smooth Fréchet manifold is then straightforward: you can copy the usual definition of a smooth manifold word for word, replacing by some Fréchet space.

With tangent vector s, you may remember that there are several different ways to define them in the finite dimensional case, which turn out to be equivalent. Since there are situations in infinite dimensions where these definitions turn out to not be equivalent, I will be explicit and define tangent vector s in the “kinematic way”:

The (kinematic) tangent vector space of a Fréchet manifold at a point consists of all pairs where is a smooth curve

With this definition, the set of pairs forms a Fréchet manifold, the tangent bundle , just as in finite dimensions.

The first serious (more or less) problem we encounter is the definition of the cotangent bundle: is isomorphic to its dual vector space. This is still true for every Hilbert space (this is known as the Riesz representation theorem). It fails already for Banach spaces: The dual space will still be a Banach space, but a Banach space does not need to be isomorphic to its dual, or even the dual of its dual (though the latter situation happens quite often, and such Banach spaces are called reflexive).

With Fréchet spaces things are even a little bit worse, because the dual of a Fréchet space (which is not a Banach space) is not even a Fréchet space! Since I did not know that and could not find a reference, I asked about this on Mathoverflow here and promptly got an answer. Mathoverflow is a really amazing platform for this kind of question!

So, if we naively define the cotangent space as in finite dimensions by taking the dual space of every tangent space, then the cotangent bundle won’t be a Fréchet manifold.

We will therefore have to be careful with the definition of differential forms for Fréchet manifolds and define it explicitly:

A differential form (a one form) is a smooth map

where is the tangent bundle, such that restricts to a linear map on every tangent space .

Another pitfall is that theorems from multivariable calculus may fail in Fréchet spaces, like the existence and uniqueness theorem of Picard-Lindelöf for ordinary differential equations. Things are much easier in Banach spaces: If you take a closer look at multivariable calculus, you will notice that a lot of definitions and theorems actually make use of the Banach space structure of only, so that a lot generalizes straight forward to infinite dimensional Banach spaces. But that is less so for Fréchet spaces.

By now you should feel reasonably comfortable with the notion of a Fréchet manifold, so let us talk about the kind of gadget that Arnold used to describe fluid flows: diffeomorphism groups that are both infinite-dimensional Riemannian manifolds and Lie groups.

The geodesic equation for an invariant metric

If is both a Riemannian manifold and a Lie group, it is possible to define the concept of left or right invariant metric. A left or right invariant metric on is one that does not change if we multiply the arguments with a group element:

A metric is left invariant iff for all :

Similarly, is right invariant iff:

How does one get a one-sided invariant metric?

Here is one possibility: If you take a Lie group off the shelf, you get two automorphisms for free, namely the left and right multiplication by a group element :

given by:

Pictorially speaking, you can use the differentials of these to transport vector s from the Lie algebra of – which is the tangent space at the identity of the group, – to any other tangent space . If you can define an inner product on the Lie algebra, you can use this trick to transport the inner product to all the other tangent spaces by left or right multiplication, which will get you a left or right invariant metric.

To be more precise, for every tangent vectors of a tangent space there are unique vectors that are mapped to by the differential of the right multiplication , that is

and we can define the inner product of and to have the value of that of and :

This works for the left multiplication , too, of course.

For a one-sided invariant metric, the geodesic equation looks somewhat simpler than for general metrics. Let us take a look at that:

On a Riemannian manifold with tangent bundle there is a unique connection, the Levi-Civita connection, with the following properties for vector fields :

If we combine both formulas we get

If the inner products are constant along every flow, i.e. the metric is (left or right) invariant, then the first three terms on the right hand side vanish, so that we get

This latter formula can be written in a more succinct way if we introduce the coadjoint operator. Remeber the adjoint operator defined to be

With the help of the inner product we can define the adjoint of the adjoint operator:

Beware! We’re using the word ‘adjoint’ in two completely different ways here, both of which are very common in math. One way is to use ‘adjoint’ for the operation of taking a Lie bracket: . Another is to use ‘adjoint’ for the linear map coming from a linear map between inner product spaces given by Please don’t blame me for this.

Then the formula above for the covariant derivative can be written as

Since the inner product is nondegenerate, we can eliminate and get

A geodesic curve is one whose tangent vector is transported parallel to itself. That is, we have

Using the formula for the covariant derivative for an invariant metric above we get

as a reformulation of the geodesic equation.

For time dependent dynamical systems, we have the time axis as an additional dimension and every vector field has as an additional summand. So, in this case we get as the geodesic equation (again, for an invariant metric):

A simple example: the circle

As a simple example we will look at the circle and its diffeomorphism group The Lie algebra of can be identified with the space of all vector fields on If we sloppily identify with with coordinate , then we can write for vector fields and the commutator

where is short for the derivative:

And of course we have an inner product via

which we can use to define either a left or a right invariant metric on , by transporting it via left or right multiplication to every tangent space.

Let us evaluate the geodesic equation for this example. We have to calculate the effect of the coadjoint operator:

If we write for the vector fields , and , this results in

where the last step employs integration by parts and uses the periodic boundary condition for the involved functions.

So we get for the coadjoint operator

Finally, the geodesic equation

turns out to be

A similar equation,

is known as the Hopf equation or inviscid Burgers’ equation. It looks simple, but its solutions can produce behaviour that looks like turbulence, so it is interesting in its own right.

If we take a somewhat more sophisticated diffeomorphism group, we can get slightly more complicated and therefore more interesting partial differential equations like the Korteweg-de Vries equation. But since this post is quite long already, that topic will have to wait for another post!

Dolphins and Manatees of Amazonia

No, these aren’t mermaids. They’re sirenians!

Sirenians or ‘sea cows’ are aquatic mammals found in four places in the world. The three places shown here are home to three species called ‘manatees’:

For example, the sirenians shown above are West Indian manatees, Trichechus manatus, which live in the Caribbean. There’s also a big region stretching from the western Pacific Ocean to the eastern coast of Africa that’s home to the ‘dugong’.

Right now there’s one different species of sirenian in each place. But once there were many more species, and it’s just been discovered that there often used to be several species living in the same place:

• Multiple species of sea cows once coexisted, Science Daily, 8 March 2012.

The closest living relatives of the sirenians are elephants! They kind of look similar, no? More importantly, they share some unusual features. They keep growing new teeth throughout their life, molars that slowly move to the front of the mouth as the teeth in front wear out. And quite unlike cows, say, the females have two teats—located between their front limbs.

Here’s an evolutionary tree of sirenians:

You’ll see they got their start about 50 million years ago and blossomed in the late Oligocene, about 25 million years ago. Later the Earth got colder, and they gradually retreated to their present ranges.

You’ll also notice that three branches of the tree seem to reach the present day:

• Trichechus, which includes all the manatees,

• Dugong, which (surprise!) is the dugong… and

• Hypodamilis, which is another name for Steller’s sea cow.

Steller’s sea cow was discovered in the North Pacific in 1741, and hunted to extinction shortly thereafter. Ouch! It took 24 million years of evolution to refine and polish the information in that species, and it was wiped out without trace in just 27 years.

The Amazonian manatee, Trichechus inunguis, is of special interest to me today because it lives in many branches of the Amazon river:

How did it get there? Why does it live in rivers? Its nearest living neighbor, the West Indian Manatee, likes coastal waters but can also go up rivers. Another clue might be the wonderful Amazon river dolphin, Inia geoffrensis.

It’s also called a pink dolphin. Here’s why:

Their are some interesting myths about it… one of which connects it with the manatee!

In traditional Amazon River folklore, at night, an Amazon river dolphin becomes a handsome young man who seduces girls, impregnates them, and then returns to the river in the morning to become a dolphin again. This dolphin shapeshifter is called an encantado. It has been suggested that the myth arose partly because dolphin genitalia bear a resemblance to those of humans. Others believe the myth served (and still serves) as a way of hiding the incestuous relations which are quite common in some small, isolated communities along the river. In the area, there are tales that it is bad luck to kill a dolphin. Legend also states that if a person makes eye contact with an Amazon river dolphin, he or she will have lifelong nightmares. Local legends also state that the dolphin is the guardian of the Amazonian manatee, and that, should one wish to find a manatee, one must first make peace with the dolphin.”

Indeed, the range of the Amazon river dolphin, shown here, is similar to that of the Amazonian manatee:

Dolphins and other cetaceans are not closely related to sirenians. Dolphins are carnivores, but sirenians only eat plants. But they both started as land-dwelling mammals, and both took to the seas at roughly the same time. And it seems the Amazon river dolphin became a river dweller around 15 million years ago. Why? As sea levels dropped, what once was an inland ocean in South America gradually turned into what’s now the Amazon! According to the Wikipedia article:

It seems this species separated from its oceanic relatives during the Miocene epoch. Sea levels were higher at that time, says biologist Healy Hamilton of the California Academy of Sciences in San Francisco, and large parts of South America, including the Amazon Basin, may have been flooded by shallow, more or less brackish water. When this inland sea retreated, Hamilton hypothesizes, the Amazon dolphins remained in the river basin…

So maybe the manatees did the same thing. I don’t know. But I find the idea of an inland sea gradually turning into a river-filled jungle, and life adapting to this change, very intriguing and romantic!

This shows what South America may have looked like during the early-middle Miocene, when the Amazon river dolphin was just getting its start. The upper Amazon Basin drained into the Orinoco Basin at left, while the the lower Amazon Basin drained directly to the Atlantic Ocean at fight. This is from a paper on megafans, which are huge regions covered with river sediment:

• M. Justin Wilkinson, Larry G. Marshall, and John G. Lundberg, River behavior on megafans and potential influences on diversification and distribution of aquatic organisms, Journal of South American Earth Sciences 21 (2006), 151–172.

Almost needless to say, we’ll need to work a bit to protect the dolphins and manatees of Amazonia if we want them to survive. Check out this Amazon river dolphin in action:

This guy is swimming in the Rio Negro, a large tributary of the Amazon. But there are also Amazon river dolphins in the Orinoco, another huge river in South America, not connected to the Amazon! You can see it just north of the Rio Negro:

Was it ever connected to the Amazon? If not, what’s the story about how the same species of dolphins live in both river basins?

By the way, my joke about mermaids comes from the etymology of the word ‘sirenian’. There’s a legend that lonely sailors—very lonely, it seems—mistook sea cows for mermaids, also known as ‘sirens’.