Also available at http://math.ucr.edu/home/baez/week288.html
January 1, 2010
This Week's Finds in Mathematical Physics (Week 288)
John Baez
Happy New Decade! I hope you're doing well. This week I'll say more
about rational homotopy theory, and why the difference between
equality and isomorphism is important for understanding the weather
in space. But first: electrical circuits!
But even before that... guess what this is a picture of:
http://math.ucr.edu/home/baez/linear_dunes.jpg
Now, what about electrical circuits?
I've been thinking This Week's Finds has become a bit too far removed
from its roots in physics. This problem started when I quit working
on quantum gravity and started focusing on n-categories. Overall it's
been a big boost to my sanity. But I don't want This Week's Finds to
be comprehensible only to an elite coterie of effete mathematicians -
the sort who eat simplicial presheaves for breakfast and burp up
monoidal bicategories.
So, in an effort to prevent This Week's Finds from drifting off into
the stratosphere of abstraction, I've decided to talk a bit about
electrical circuits. Admittedly, these are less glamorous than
theories of quantum gravity. But: they actually work! And there is
a lot of nice math involved.
I rarely dare predict what *future* Week's Finds will discuss, because
I know from bitter experience that I change my mind. But lately I've
started writing a new way: long stories with lots of episodes, which I
can dole out a bit at a time. So I know that for at least a few Weeks
I'll talk about electrical circuits - and various related things.
I've been trying to understand electrical circuits using category
theory for a long time. Indeed, Peter Selinger and I are very slowly
writing a paper on this subject. The basic inspiration is that
electrical circuit diagrams look sort of like Feynman diagrams, flow
charts, and various other diagrams that have "inputs" and "outputs".
I love diagrams like this! All the kinds I've met so far can be
nicely formalized using category theory. For an explanation, try
this:
1) John Baez and Mike Stay, Physics, topology, logic and computation:
a Rosetta Stone, to appear in New Structures in Physics, ed. Bob
Coecke. Available at arXiv:0903.0340.
And after I spent a while thinking about electrical circuits using
category theory, I realized that this perspective might shed light on
analogies between circuits and other systems.
For example: mechanical systems made from masses and springs!
Indeed, whenever I teach linear differential equations, I like to
explain the basic equation describing a "damped harmonic oscillator":
for example, a rock hanging on a spring. Then I explain how the same
equation describes the simplest circuit made of a resistor, an
inductor, and a capacitor - the so-called "RLC circuit". It's a nice
easy example of how the same math applies to superficially different
but secretly isomorphic problems!
Let me explain. I hope this is a chance to help mathematicians review
their physics and ask questions about it over on the n-Category Cafe.
Let the height of a rock hanging on a spring be q(t) at time t, where
q(t) is negative when the end of the spring is down below its
equilibrium position. Then making all sort of simplifying assumptions
and approximations, we have:
m q"(t) = - c q'(t) - k q(t) + F(t)
where:
m is the mass of the rock.
c is the damping coefficient, which describes the force due to
friction: we're assuming this force is proportional to the rock's
velocity, but points the other way.
k is the spring constant, which describes the force due to the
spring: we're assuming this force is proportional to how much the
spring is stretched from its equilibrium position, but points the
other way.
F(t) is the externally applied force, e.g. if you push on the rock.
This equation is just Newton's second law, force equals mass times
acceleration. The left side of the equation is mass times acceleration;
the right side is the total force.
Now for the analogy. Everything here is analogous to something in an
RLC circuit! An RLC circuit has current flowing around a loop of wire
with 4 gizmos on it: a resistor, an inductor, a capacitor, and a
voltage source - for example, a battery.
I won't say much about these gizmos. I just want to outline the
analogy. The amount of current is analogous to the velocity of the
rock, so let's call it q'(t). The resistor acts to slow the current
down, just as friction acts to slow down the rock. The inductor is
analogous to the mass of the rock. The capacitor is analogous to the
spring - but according to the usual conventions, a capacitor with a
big "capacitance" acts like a weak spring. Finally, the voltage
source is analogous to the external force.
So, here's the equation that governs the RLC circuit:
L q"(t) = - R q'(t) - (1/C) q(t) + V(t)
where
L is the inductance of the inductor.
R is the resistance of the resistor.
C is the capacitance of the capacitor.
V is the voltage of the power source.
As you can see, this equation is the same as the one that governs a
rock on a spring! True, 1/C plays the role of k, since a capacitor
with a big capacitance acts like a spring with a small spring
constant. But this is just a difference in conventions. The systems
are isomorphic!
We could have fun solving the above equation and pondering what the
solutions mean, but that would be the class I teach. Instead,
I want to explain how this famous analogy between mechanics and
electronics is just one of many analogies.
When I started thinking seriously about electrical circuits and
category theory, I mentioned them my student Mike Stay, and he
reminded me of the "hydraulic analogy" where you think of an
electrical current flowing like water through pipes. There's a
decent introduction to this here:
2) Wikipedia, Hydraulic analogy,
http://en.wikipedia.org/wiki/Hydraulic_analogy
Apparently this analogy goes back to the early days when people were
struggling to understand electricity, before electrons had been
observed. The famous electrical engineer Oliver Heaviside pooh-poohed
this analogy, calling it the "drain-pipe theory". I think he was
making fun of William Henry Preece. Preece was another electrical
engineer, who liked the hydraulic analogy and disliked Heaviside's
fancy math. In his inaugural speech as president of the Institution
of Electrical Engineers in 1983, Preece proclaimed:
True theory does not require the abstruse language of mathematics to
make it clear and to render it acceptable. All that is solid and
substantial in science and usefully applied in practice, have been
made clear by relegating mathematic symbols to their proper store
place - the study.
According to the judgement of history, Heaviside made more progress in
understanding electromagnetism than Preece. But there's still a nice
analogy between electronics and hydraulics.
In this analogy, a pipe is like a wire. Water is like electrical
charge. The flow of water plays the role of "current". Water
pressure plays the role of "voltage".
A resistor is like a narrowed pipe. An inductor is like a heavy
turbine placed inside a pipe: this makes the water tend to keep
flowing at the same rate it's already flowing! In other words, it
provides a kind of "inertia", analogous to the mass of our rock.
Finally, a capacitor is like a tank with pipes coming in from both
ends, and a rubber sheet dividing it in two lengthwise. When studying
electrical circuits as a kid, I was shocked when I first learned that
capacitors *don't let the electrons through*. Similarly, this gizmo
doesn't let the water through.
Okay... by now you're probably wanting to have the analogies laid out
more precisely. So that's what I'll do. But I'll throw in one more!
I've been talking about the mechanics of a rock on a spring, where the
motion of the rock up and down is called *translation*. But we can
also study *rotation* in mechanics. And then we get these analogies:
displacement flow momentum effort
q q' p p'
Mechanics position velocity momentum force
(translation)
Mechanics angle angular angular torque
(rotation) velocity momentum
Electronics charge current flux voltage
linkage
Hydraulics volume flow pressure pressure
momentum
The top row lists 4 concepts from the theory of general systems, and
my favorite symbols for them, where the prime stands for time
derivative. (If I could, I'd use a dot.) The other rows list what
these concepts are called in various examples. So, "displacement"
is the general concept which people call "position" in the mechanics
of translation. Similarly, "flow" and "effort" correspond to
"velocity" and "force", while "momentum" is just "momentum".
I found this chart here:
3) Dean C. Karnopp, Donald L. Margolis and Ronald C. Rosenberg,
System Dynamics: a Unified Approach, Wiley, New York, 1990.
I discovered this wonderful book after an intensive search for stuff
that makes the analogies between mechanics, electronics and hydraulics
explicit. It turns out there's a whole theory devoted to precisely
this! It's sometimes called "systems theory" or "network theory".
Engineers use this theory to design and analyze systems made out of
mechanical, electronic, and/or hydraulic components: springs, gears,
levers, pulleys, pumps, pipes, motors, resistors, capacitors,
inductors, transformers, amplifiers, and more!
Engineers often describe such systems using a notation called "bond
graphs". Bond graphs are reminscent of Feynman diagrams... so
they're simply *begging* to be understood as a branch of applied
category theory. In fact, there's an interesting blend of category
theory, symplectic geometry and complex analysis at work here.
So in the Weeks to come, I'd like to tell you more about bond graphs
and analogies between different kinds of systems.
(I'll warn you right now that Karnopp, like most experts on systems
theory, use the symbols "f" and "e" for flow and effort, instead of q'
and p'. It's more or less impossible to find a unified notation for
general systems that doesn't conflict with some existing notation
used in the study of some *particular* kind of system. But since I
want to get into symplectic geometry, I want to use some notation that
reminds me of that - and for physicists, symplectic geometry is the
study of "conjugate variables" like position q and momentum p.)
Okay... enough of this for now.
Last week I introduced the differential graded commutative algebra
approach to rational homotopy theory. Next week I'll get into the
differential graded Lie algebra approach, filling in another corner
of the triangle here:
RATIONAL SPACES
/ \
/ \
/ \
/ \
/ \
DIFFERENTIAL GRADED ------- DIFFERENTIAL GRADED
COMMUTATIVE ALGEBRAS LIE ALGEBRAS
But I realized there's some important stuff I can tell you
before we get to that!
Last time I told you how Sullivan defined "rational differential
forms" for any topological space X:
First he converted this space into a simplicial set Sing(X).
Then he defined an algebra A of functions that are polynomials
with rational coefficients on each simplex of Sing(X).
Then he defined his algebra Omega(A) of rational differential forms,
using a general recipe that takes a commutative algebra A and spits
out a differential graded commutative algebra.
But towards the end, I admitted that homotopy theorists feel perfectly
fine about studying simplicial sets rather than topological spaces.
The reason is that both the category of simplicial sets and the category
of topological spaces are "model categories" - contexts where you can do
homotopy theory. Moreover, these model categories are "Quillen equivalent" -
the same in every way that matters for homotopy theory! Don't worry too
much if you don't know about model categories and Quillen equivalence.
The point is that we have a functor that converts spaces into simplicial
sets:
Sing: [topological spaces] -> [simplicial sets]
and its right adjoint going back the other way, called "geometric
realization":
| | : [simplicial sets] -> [topological spaces]
which we also saw last time. And, these let us freely switch
viewpoints between topological spaces and simplicial sets.
So, while in "week286" I defined rational spaces to be specially nice
*topological spaces*, I could equally well have defined them to be
specially nice *simplicial sets*. Taking this viewpoint, we can
forget about topological spaces, and think of Sullivan's innovation as
a recipe for defining rational differential forms on a *simplicial
set*.
This is a good idea. Among other things, it helps us see more simply
what was so new about rational differential forms when Sullivan first
discovered them.
What's new is that they give a functor that takes any simplicial set S
and gives a differential graded algebra that's *commutative* and whose
cohomology is the rational cohomology of S.
(By which I mean: the rational cohomology of the space |S|.)
You see, it's not so hard to achieve this if we drop our insistence
that our differential graded algebra be *commutative*. This has been
known for a long time. You start with your simplicial set S and
define a "rational n-cochain" on it to be a function that eats
n-simplices and spits out rational numbers. This gives a cochain
complex
d d d
C^0(S) ---> C^1(S) ---> C^2(S) ---> ...
where C^n(S) is the vector space of rational n-cochains. This cochain
complex is usually called C*(S), where the star stands for the variable n.
And there's a standard way to make C*(S) into a differential graded
algebra, using a product called the "cup product". But, it's not a
differential graded *commutative* algebra.
Instead, it's only graded commutative "up to chain homotopy". So, we
don't have
v u w = (-1)^{pq} w u v
when v is in C^p(S) and w is in C^q(S). But, we do have
v u w = (-1)^{pq} w u v + da(v,w)
where a(v,w) is something that depends on v and w. This is good
enough to imply that when we take the cohomology of our cochain
complex, we get a graded commutative algebra. This algebra is called
H*(S), and the product in here is also called the cup product. You
can read a lot about it in basic books on algebraic topology. Here's
one that's free online:
4) Allen Hatcher, Algebraic Topology, Section 3.2: Cup Product,
available at http://www.math.cornell.edu/~hatcher/AT/ATch3.pdf
The memorably numbered Theorem 3.14 says the cup product is graded
commutative.
So you might say: "So, who cares if the cup product of cochains is
graded commutative merely up to chain homotopy? When we go to
cohomology, that distinction washes away!"
Well, it turns out there can be lots of interesting information
in this chain homotopy a(v,w). At least, this is true when we do
cohomology using the integers or the integers modulo some prime as our
coefficients - instead of rational numbers, as we've been doing.
In fact this chain homotopy is the tip of an enormous iceberg! For
starters, it satisfies an interesting equation, but only up chain
homotopy... and that chain homotopy satisfies an equation of its own,
but only up to homotopy, and so on. So, we get a differential graded
algebra that's graded commutative up to an infinite series of chain
homotopies. Folks call this sort of gadget an "E-infinity algebra".
And when we work over the integers mod some prime, we can squeeze
interesting information out of all these chain homotopies. They're
called "Steenrod operations". You can use them to distinguish spaces
that would be indistinguishable if you merely used their cohomology as
a graded commutative algebra!
At least, that's what they say. I don't *personally* run around using
Steenrod operations to distinguish weird spaces that shady characters
pull out of their coat pockets on dark streetcorners. Some
topologists actually do. But what fascinates me is the subtle
distinction between equations that hold "on the nose" and equations
that hold only up to homotopy, or up to isomorphism. Sometimes you
can "strictify" a gadget where the equations hold only up to homotopy,
and get them to hold on the nose. But sometimes you can't.
Once I was giving a talk about n-categories and Roger Penrose was
in the audience. I said the most basic fact about n-categories was:
isomorphism is not equal to equality.
He raised his hand and asked:
is isomorphism *isomorphic* to equality?
Very good question! And the answer is: sometimes yes, sometimes no.
This is where things get interesting!
So, it's a famous puzzle whether you can find some functorial way to
turn a simplicial set S into a differential graded commutative algebra
A*(S) whose cohomology is the usual cohomology H*(S). This is called the
"commutative cochain problem".
I haven't said it precisely enough yet, since there's a cheap and easy
way to solve the version I just stated: just A*(S) to be the cohomology
H*(S) itself, with d = 0. What a dirty trick!
To rule out such tricks people demand various extra good properties.
For example, the usual cochains C*(S) are "extendible": any cochain
on a little simplicial set extends to a cochain on a bigger one.
In other words, if
S -> T
is an inclusion of simplicial sets, then the corresponding map
C*(T) -> C*(S)
is onto. This is definitely not true if we replace C* by H*.
This paper gives a bit of history of the commutative cochains problem:
5) Bohumil Cenkl, Cohomology operations from higher products in the de
Rham complex, Pacific J. Math. 140 (1989), 21-33. Available at
http://projecteuclid.org/euclid.pjm/1102647247
It gives a somewhat different statement of the problem, which alas I
don't understand, and it proves that this version has no solution if
we work over the integers. But over the rationals it *does*, if we
take A*(S) to be the rational differential forms on our simplicial set
S.
Just so you don't think this is pie-in-the-sky stuff, I should
emphasize that this problem actually matters in electrical
engineering, where we might triangulate spacetime and study discrete
analogues of famous differential equations on the resulting simplicial
complex! My friends Robert Kotiuga and Eric Forgy have thought about
this a lot. And here's a nice excerpt from the website of a
conference at Boston University. I bet Robert Kotiuga wrote this.
It mentions "Whitney forms", which are simplex-wise linear
differential forms on a simplicial set. These are closely related to
Sullivan's simplex-wise *polynomial* differential forms:
The analysis of electric circuits, using Kirchhoff's Laws, brought
topology into electrical engineering over 150 years ago. Hermann
Weyl's reformulation of Kirchhoff's laws in terms of homology over
80 years ago is an abstraction which is proving to be essential in
the finite element analysis of three-dimensional electromagnetic
fields. It enables computers to be programmed to identify an
electrical circuit in an electromagnetic field problem - a task once
considered the domain of the engineer's intuition. In "control
theory" parlance, circuit theory equations are low frequency model
reductions of distributed parameter electromagnetic systems, and
homology theory yields the key mathematical tools for obtaining robust
numerical algorithms. One aspect of the workshop will deal with
large scale homology calculations and the realization of cycles
representing generators of integral homology groups as embedded
manifolds. The underlying homology calculations involve large sparse
integer matrices with remarkable structure even when the underlying
finite element meshes are "unstructured". One aim of the workshop
is to bring together those performing large scale homology
calculations in the context of dynamical systems and point cloud
data analysis, with those requiring more geometrical applications
of homology groups in electromagnetics.
Over two decades ago, boundary value problems arising in the
analysis of quasistatic electromagnetic fields were reinterpreted
in terms of Hodge theory on manifolds with boundary. This observation
is quite natural when Maxwell's equations are viewed in the context of
differential forms and the problem of defining potentials is
phrased in terms of de Rham cohomology. This observation, along
with the variational formulation of Hodge theory on manifolds with
boundary, created a revolution in the finite element analysis of
electromagnetic fields. When phrased this way, the most difficult
theoretical problems were actually solved in the 1950's by Andre Weil
and Hassler Whitney who were concerned with problems in algebraic
topology. They had an explicit interpolation formula for turning
simplicial cochains into piecewise linear differential forms. This
formula gives a chain homotopy between the algebraic complexes
involved, and an isomorphism of cohomology rings. Although it took
30 years for Whitney forms to impact engineering practice, once the
genie was out of the bottle, there was no way to put it back in.
In the early 1990s, Whitney form techniques solved the problem of
"spurious modes" appearing in electromagnetic cavity resonator
calculations and soon after became widely accepted as an essential
tool which is only recently being appreciated in the context of
nanophotonics.
It is important to re-examine this Whitney form revolution in the
context of recent attempts to develop "discrete exterior calculus,"
"mimetic discretizations," "compatible discretizations" etc. For
example, in algebraic topology it is well known that simplicial
cochains do not admit a graded-commutative, associative product
analogous to the wedge product on differential forms. This classical
result, known as "the commutative cochain problem," is surprising and
unintuitive in light of the fact that simplicial cochains admit a
graded-commutative, associative product on the level of cohomology,
analogous to the one induced by the wedge product in the de Rham
complex. The bottom line is that these types of classical results
are often ignored by newcomers trying to develop a discrete
approach to calculus. Obviously, there is still some important
technology transfer to be performed between algebraic topology and
numerical analysis! Much of the mathematical work was done by Patodi,
Dodziuk and Muller in the 1970's, has been exploited by electrical
engineers, but has been largely ignored by applied mathematicians.
Although the multiplicative structure on differential forms does
not seem to be very important in the context of linear boundary value
problems, it seems to play an important role in magnetohydrodynamics.
Magnetohydrodynamics, in turn is an essential tool in space
physics, in particular, in the growing field of space weather.
So you see, everything is related. The difference between equality
and isomorphism matters when you're trying to simulate the weather in
space! That's the kind of thing that makes math so fun. Here's the
conference webpage:
5) Advanced Computational Electromagnetics 2006 (ACE 'O6), Boston
University, http://www.bu.edu/eng/ace2006/
You can learn more here:
6) P. W. Gross and P. Robert Kotiuga, Electromagnetic Theory and
Computation: A Topological Approach, Cambridge University Press,
2004.
Someday my discussion of electrical circuits may expand to include
some algebraic topology. But all I want to explain now is the usual
cup product on the cochains C*(S) for a simplicial set S. We'll
need this in the Weeks to come!
Actually, it'll be easier if I work with chains instead of cochains.
For us a chain complex will be a list of vector spaces and linear maps
d d d
C_0 <--- C_1 <--- C_2 <--- ...
with d^2 = 0. We call the whole thing C_*, where now the star is a
subscript. I'll show you the usual way to get a chain complex C_*(S)
from a simplicial set, and then show you a way to *comultiply* chains.
Then you can get the cochains by taking duals:
C^n(S) = C_n(S)*
This will give a way to multiply chains.
Here's how it goes. The idea is that we define the comultiplication
directly at the level of simplicial sets and then feed it through a
couple of functors. There's a functor
F: [simplicial sets] -> [simplicial vector spaces]
and a functor
N: [simplicial vector spaces] -> [chain complexes]
Composing these will give the chains C_*(S) for a simplicial set S.
The first functor
F: [simplicial sets] -> [simplicial vector spaces]
creates the free simplicial vector space on a simplicial set.
This functor is symmetric monoidal: it carries products of simplicial
spaces to tensor products of simplicial vector spaces. The second
functor
N: [simplicial vector spaces] -> [chain complexes]
is called the "normalized chain complex" or "normalized Moore
complex" functor. This functor is an equivalence of categories!
It's *almost* symmetric monoidal, but not quite, and this is
where all the subtlety lies.
The category of simplicial sets has finite products. So, every
simplicial set has a diagonal map:
Delta: S -> S x S
It also a unique map to the simplicial set called 1, which
consists of single 0-simplex:
epsilon: S -> 1
These two maps satisfy the usual axioms of a commutative monoid,
written out as commutative diagrams, except with the arrows pointing
backwards. So, S is a "cocommutative comonoid" in the category of
simplicial sets.
Indeed, whenever you have any category with finite products, every
object in it becomes a cocommutative comonoid - and in a unique way!
So far, this is a completely bland fact of life.
If we feed our simplicial set S through the functor
F: [simplicial sets] -> [simplicial vector spaces]
what happens? Well, because this functor is monoidal it sends
comonoids to comonoids. And because it's *symmetric* monoidal, it
sends *cocommutative* comonoids to *cocommutative* comonoids. And a
cocommutative comonoid in simplicial vector spaces is the same as a
"simplicial cocommutative coalgebra".
(I love this kind of stuff, but not everyone does: that's why I
save it for the very end of each Week's Finds.)
So, we've turned our simplicial set S into a simplicial cocommutative
coalgebra F(S). Now feed this gizmo into the next functor:
N: [simplicial vector spaces] -> [chain complexes]
By definition, we get the chains on S:
N(F(S)) = C_*(S)
And thanks to the wonders of functoriality, these chains are blessed
with a comultiplication
C_*(Delta): C_*(S) -> C_*(S) tensor C_*(S)
and counit
C_*(S) -> Q
where Q is our ground field, the rationals.
And *if* the functor N were also symmetric monoidal, C_*(S) would
also be a cocommutative comonoid, but now in the world of chain
complexes. In other words, it would be a "differential graded
cocommutative coalgebra". Then, taking duals, the cochains C*(S)
would be a DGCA!
But I warned you: things aren't quite so simple.
I said the functor N is *almost* a symmetric monoidal functor. But
not quite.
For starters, it's a "lax monoidal functor", which implies among other
things that there's a natural transformation
EZ: N(X) tensor N(Y) -> N(X tensor Y)
This is called the Eilenberg-Zilber map. The word "lax" means that
this map isn't necessarily an isomorphism. A lax monoidal functor is
good enough to send monoids to monoids. That's important - but it's
no use to us now, since we've got a comonoid on our hands!
On the other hand, N is also an "oplax monoidal functor", which implies
among other things that there's a natural transformation going the other
way
AW: N(X tensor Y) -> N(X) tensor N(Y)
This is called the Alexander-Whitney map. The word "oplax" means that
this map isn't necessarily an isomorphism - but now it's going the
opposite way. An oplax monoidal functor is good enough to send
comonoids to comonoids. Yay!
So, our cochains C_*(S) do indeed form a comonoid in the world
of chain complexes - that is, "differential graded coalgebra".
However, it's not cococommutative!
To dig deeper into this, I'd need to draw lots of pictures or write
lots of formulas, and I don't feel in the mood for that. So, I'll
just say what I hope you're thinking: the passage from simplicial
vector spaces to chain complexes is quite tricky.
For example, it's sort of frustrating that we have these EZ and AW
maps going both ways, but they're not inverses! In fact they come
very close. Eilenberg-Zilber followed by Alexander-Whitney is the
identity on N(X) tensor N(Y). Alas, Alexander-Whitney followed by
Eilenberg-Zilber is not the identity on N(X tensor Y). But, it's chain
homotopic to the identity!
You can read more about the "normalized Moore complex" functor here:
7) nLab, Moore complex, available at
http://ncatlab.org/nlab/show/Moore+complex
The fact that it's an equivalence of categories is called the
"Dold-Kan correspondence". You can read more about this here:
8) nLab, Dold-Kan correspondence, available at
http://ncatlab.org/nlab/show/Dold-Kan+correspondence
And I should point out that while I've been working with vector spaces
over the rational numbers, everything I've said about the functors F
and N generalize to R-modules for an arbitrary commutative ring R.
So, we have
F: [simplicial sets] -> [simplicial R-modules]
N: [simplicial R-modules] -> [chain complexes of R-modules]
with all the good (or frustrating) properties that I just described.
The nLab pages focus somewhat on the case where R = Z, where we get
F: [simplicial sets] -> [simplicial abelian groups]
N: [simplicial abelian groups] -> [chain complexes of abelian groups]
This is indeed the most fundamental case.
Finally, what about that picture at the beginning? If you're smirking
just because you can guess what *planet* it was taken on, wipe that
smile off your face! If I showed you a picture of a city and asked
you where it is, would you say "Earth"?
These pictures show linear dunes on the north polar region of Mars:
latitude 78 degrees north, longitude 209 degrees east.
(Hmm, on Earth there was a big battle between the British and French
to say what longitude counts as "0 degrees". How did it work on Mars?)
Anyway, according to Maria Banks:
This observation shows linear dunes in the north polar region of
Mars. Linear or longitudinal sand dunes are elongated, sharp
crested ridges that are typically separated by a sand-free surrounding
surface.
These features form from bi-directional winds and extend parallel
to the net wind direction. In this case, the net wind direction
appears to be from the west-southwest. Linear sand dunes are found
in many different locations on Earth and commonly occur in long
parallel chains with regular spacing.
Superimposed on the surface of the linear dunes are smaller
secondary dunes or ripples. This is commonly observed on
terrestrial dunes of this size as well. Polygons formed by
networks of cracks cover the substrate between the linear dunes and
may indicate that ice-rich permafrost (permanently frozen ground)
is present or has been present geologically recently in this
location.
9) HiRISE (High Resolution Imaging Science Experiments), Linear dunes
in the north polar region, http://hirise.lpl.arizona.edu/PSP_009739_2580
-----------------------------------------------------------------------
Quote of the Week:
"If you haven't found something strange during the day, it hasn't been
much of a day." - John Wheeler
-----------------------------------------------------------------------
Addenda: I thank Richard Lozes and Jonathan vos Post for catching typos.
Jesse McKeown pointed out a NASA website that addresses my puzzle about how people settled on a definition of longitude on Mars:
10) NASA, The Martian Prime Meridian - longitude zero,
http://www.nasaimages.org/luna/servlet/detail/nasaNAS~4~4~16934~120671:The-Martian-Prime-Meridian----Longi
On Earth, the longitude of the Royal Observatory in Greenwich,
England is defined as the "prime meridian," or the zero point of
longitude. Locations on Earth are measured in degrees east or west
from this position. The prime meridian was defined by international
agreement in 1884 as the position of the large "transit circle," a
telescope in the Observatory's Meridian Building. The transit
circle was built by Sir George Biddell Airy, the 7th Astronomer
Royal, in 1850. (While visual observations with transits were the
basis of navigation until the space age, it is interesting to note
that the current definition of the prime meridian is in reference to
orbiting satellites and Very Long Baseline Interferometry (VLBI)
measurements of distant radio sources such as quasars. This
"International Reference Meridian" is now about 100 meters east of
the Airy Transit at Greenwich.) For Mars, the prime meridian was
first defined by the German astronomers W. Beer and J. H. Mddler in
1830-32. They used a small circular feature, which they designated
"a," as a reference point to determine the rotation period of the
planet. The Italian astronomer G. V. Schiaparelli, in his 1877 map
of Mars, used this feature as the zero point of longitude. It was
subsequently named Sinus Meridiani ("Middle Bay") by Camille
Flammarion . When Mariner 9 mapped the planet at about 1 kilometer
(0.62 mile) resolution in 1972, an extensive "control net" of
locations was computed by Merton Davies of the RAND Corporation.
Davies designated a 0.5-kilometer-wide crater (0.3 miles wide),
subsequently named "Airy-0" (within the large crater Airy in Sinus
Meridiani) as the longitude zero point. (Airy, of course, was named
to commemorate the builder of the Greenwich transit.) This crater
was imaged once by Mariner 9 (the 3rd picture taken on its 533rd
orbit, 533B03) and once by the Viking 1 orbiter in 1978 (the 46th
image on that spacecraft's 746th orbit, 746A46), and these two
images were the basis of the Martian longitude system for the rest
of the 20th Century. The Mars Global Surveyor (MGS) Mars Orbiter
Camera (MOC) has attempted to take a picture of Airy-0 on every
close overflight since the beginning of the MGS mapping mission. It
is a measure of the difficulty of hitting such a small target that
nine attempts were required, since the spacecraft did not pass
directly over Airy-0 until almost the end of the MGS primary
mission, on orbit 8280 (January 13, 2001). In the left figure
above, the outlines of the Mariner 9, Viking, and Mars Global
Surveyor images are shown on a MOC wide angle context image,
M23-00924. In the right figure, sections of each of the three
images showing the crater Airy-0 are presented. A is a piece of
the Mariner 9 image, B is from the Viking image, and C is from the
MGS image. Airy-0 is the larger crater toward the top-center in
each frame. The MOC observations of Airy-0 not only provide a
detailed geological close-up of this historic reference feature,
they will be used to improve our knowledge of the locations of all
features on Mars, which will in turn enable more precise landings
on the Red Planet by future spacecraft and explorers.
For more discussion, visit the friendly and welcoming n-Category Cafe
at:
http://golem.ph.utexas.edu/category/2010/01/this_weeks_finds_in_mathematic_49.html
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