The Mathematical Intelligencer 29 1

60 Pages·2007·4.45 MB·English

by  C. Davis (Editors in Chief)

Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters tu the

editor should be sent tu either of the

editors-in-chief, Chandler Davis or

Marjorie Senechal.

Mathematics and Narrative

Marjorie Senechal's article [1] is a delightful account of a meeting organised
by the group Thales and Friends, attempting to explore the relationship hetween mathematics and narrative. What
could he more beguiling than discussing this on the Aegean? But it
seemed strange to read a paper on
mathematics and narrative that doesn't
mention the most commercially successful hooks ever written by a mathematician: Alice 's Aduentures in Wonderland and Through the Looking Glass.
Neither is there any reference to the
most successful paramathematical hook
of recent years (and here I'm guessing
even more wildly than in the previous
sentence), Mark Haddon's The Curious
Incident of the Dof!, in the Night-Time.
Neither Carroll/Dodgson nor Haddon,
who clearly know how to write something that people actually want to read,
was trying to put mathematics across.
They both produced beautifully written
stories with a mathematical theme. Carroll just couldn't help being playful with
mathematical and logical ideas: the
mathematics in the background shines
through. Haddon, who isn't a mathematician, has written a lovely story
about a strange boy with some mathematical talent that (as it seemed to me)
has captured a little of the feeling of
what doing mathematics is like.
Thales and Friends' website [2] is also
a pleasure to browse through. I was particularly interested in the papers by
Mazur [3] and Chaitin [4]. Mazur has
made a heroic attempt to classify the
different ways in which stories can he
used in "mathematical exposition". Rut
I think there is a problem here in his
use of the word exposition. Stories and
exposition don't seem to go together
naturally: stories are surely more about
exploration than exposition. Carroll is
exploring logic in his Alice hooks: set
up a crazy situation, apply the rules of
logic, and see where we get to. And
Haddon is explorinJ< the relationship hetween mathematics and autism, which,
by the way. is exposed by James in [5].
Chaitin, in his paper, contrasts two
views of mathematics: Hilbert's attempt
to describe it as a closed, formal system of axioms, rules of deduction, and
so on; and the Lakatos-Chaitin approach

to mathematics as quasi-experimental.
The Hilbert viewpoint demands exposition: here is mathematics all wrapped
up in this formal system, now we must
expose it. The Lakatos-Chaitin viewpoint suggests exploration: let's look
around us, move off in an interesting
direction, and see where it takes us. My
suggestion is that, following Carroll and
Haddon, you are much more likely to
write a readable narrative if you can
adopt the Lakatos-Chaitin-exploration
point of view.
The contrast between exposition and
exploration, between a formal system
and a quasi-experimental approach,
seems very similar in spirit to the design/evolution dichotomy discussed in
[6]. If you take a design point of view,
then narrative, if it has any role at all ,
is merely a pedagogical device to sugar
the pill or to set the mathematics in a
wider context: a taxonomy for this is
very well set out by Mazur. But with
an evolutionary/exploration viewpoint,
narrative is at the centre of the action.
How could one provide any understanding of an evolutionary world better than by telling stories' No theory of
economic development is going to give
one a better idea of what happens in a
market-place than the story of the evolution of the internet. And Haddon's
story offers something that no formal
theory of autism will ever give.
Some businesspeople have adopted
the idea of writing stories about the future to help them and their colleagues
to understand possible developments in
the business environment that may be
of vital importance. These stories are
called scenarios. (I would prefer to use
the simple word story under all circumstances, but I have often encountered resistance unless I adopted more
bureaucratic words like narrative, fiction, and scenario.) This approach was
pioneered in the energy company Shell
International (see [7]) in the early 1970s
as a way of opening the minds of senior management to the possibility that
the price of crude oil might one day
rise above $2 per barrel , and has been
used in Shell ever since. It has a number of advantages including:
• Having more than one scenario reminds people that the future is unpredictable. Before reading Chaitin's

© 2007 Springer SCience+ Bus1ness Medle, Inc., Volume 29, Number I, 2007 5
paper, I had thought this might not
be relevant to mathematics, but now
I'm not so sure. His idea of adding
new axioms such as the Riemann
Hypothesis (RH) in a quasi-experimental way seems rather analogous
to the writing of different scenarios
to explore the future. In fact I would
like to challenge mathematicians to
write two narratives about mathematics in one of which RH is true
and in the other of which RH is false.
If this turns out to be not possible,
then I suppose the obvious next
question is: does RH really matter? (I
would also very much like to read a
story set in a world in which the
Continuum Hypothesis is false.)
• The scenarios help to liberate people
from "common sense" and from their
prejudices. The different stories allow
unusual ideas to be put forward and
discussed as pieces of fiction rather
than matters of life and death (which
they can literally be, for example
when this technique was used in a
meeting between warring parties in
South Africa towards the end of the
apartheid era). This idea is reminiscent of the development of nonEuclidean geometries, a story that
often inspires me when I write scenarios.
• Occasionally, the same outcome
crops up in the course of two quite
different scenarios. This strengthens
belief that this outcome might actually happen. Compare and contrast
this with the famous result of Skewes
and Littlewood that can be proved
in two quite different ways depending on whether RH is true: far from
having their belief in the result
strengthened, some mathematicians,
of course, have refused to accept that
this constitutes a proof at all.
The most significant benefit of scenarios, in my view, is in the understanding they bring to thinking about
developments in the business environment, and hence an enormous improvement in the quality of discussions
about unfolding events. (This is why [8]
is called The Art of Strategic Conversation.) As events and trends happen, one
can look at them and say, "Ah, yes, this
is what one would expect in scenario A,
but on the other hand that looks like
scenario B". If you can say something
like that, you have understood what is

6 THE MATHEMATICAL INTELLIGENCER

going on. Chaitin says that understanding means compression, "the fact that
you're putting just a few ideas in, and
getting a lot more out". I think he's right
in this, and that in practice the compression is often (always?) in the form
of a narrative. When I say that I understand why somebody became angry, I
mean that there is a narrative starring
certain characters and featuring events
and motivations, and that the anger fits
into this narrative. When we say that we
understand why the planets move round
the sun in ellipses, we mean that this fits
into the narrative of Newton, gravity, calculus, and so on. (This narrative will be
more or less sophisticated for different
people.) Here is an example from mathematics, which I think is archetypal.
In [9], Singer says, "We should not be
too surprised that mathematics has coherent systems applicable to physics. It
remains to be seen whether there is an
already developed system in mathematics that will describe the structure of
string theory. [At present we do not even
know what the symmetry group of string
field theory is.]" Singer is trying to fit
string field theory into a narrative. The
narrative is called "Symmetry Groups",
and Singer might think of this narrative
as starring Galois and Einstein and a cast
of thousands, or he might think of it, as
with Mazur's story about rational points
of elliptic curves, as a narrative of ideas.
But it's a narrative. And if he finds out
what the symmetry group of string field
theory is, he will be justified in saying,
"Now I understand!" in just the same way
as a businessperson faced with the
prospect of new environmental legislation can say, "Yes, I understand what's
happening, it fits into one of my scenarios". If the symmetry group of string
field theory is never found, then either
string field theory will be abandoned or
an entirely new narrative will need to be
written.
I can't stop without taking issue with
a statement in [1]. (I think it is a remark
made by a participant at the Thales meeting rather than necessarily the opinion
of the author.) The statement is, "Popular math books must not mislead. They
must tell the whole truth and nothing
but the truth". If this were taken literally
(and I imagine that advocates of the
whole truth and nothing but the truth
would like to be taken literally) it would
simply mean the death of popular math-

ematics. For the whole truth includes all
the gory details, technical background,
and arcane exceptions. This isn't popular mathematics, it's mathematics. Popular mathematics should certainly not mislead, but it can't afford to be cluttered
up with the "whole truth". So what
should popular mathematics do? It
should be .faitf?ful to the narrative.
The astute reader will have noticed
that I have managed to write 1 ,500
words on mathematics and narrative
without ever saying what I think a narrative is. But I'm in good company because as far as I can tell none of the
Thales people has defined a narrative
either. (Mazur does partly.) So on the
principle of rushing in where angels fear
to tread, here is a stab at a definition.
"A narrative is a sequence over time of
related specific events, emotions, or
ideas designed to hold the attention of
the reader, listener, or viewer. It is not
the laying out of a general situation or
theory in its entirety: but a good narrative will help people to gain a better understanding of the general situation. "

Eric Grunwald

Mathematical Capital

1 87 Sheen Lane

London SW1 4 8LE

UK

e-mail:
The Hidden
Mathematics of
the Mars Exploration
Rover Mission

UFFE THOMAS JANKVIST AND 8J0RN TOLDBOD

0 n January 4, 2004, Mars Exploration Rover (MER) A,
named Spirit, entered the Martian atmosphere. The
spacecraft, weighing 827 kg, was travelling with a
speed of 19,300 km/h. During the next four minutes the velocity of the craft was reduced to 1 ,600 km/h at the meeting between the Martian atmosphere and the aeroshell of
the craft. At this point a parachute was deployed and the
velocity decreased to about 300 km/h. At a point 1 00 m
above the Martian surface, the retrorockets were fired to
slow the descent, and finally the giant airbags were inflated.
The airhag-covered spacecraft hit the surface of Mars with
a velocity of ahout 50 krn/h. The airbag ball bounced and
rolled for about 1 km on the Martian surface just as Mars
Pathfinder had done seven years earlier. When the landing
module finally came to a stop its airbags were deflated and
retracted and its petals were open. After six months in space
the encapsulated rover, Spirit, could at last unfold its solar
arrays. Three hours later Spirit transmitted its first image of
the Gusev Crater to Earth. On January 15 Spirit left its landing module and drove out onto the surface of Mars. Ten
days later, on January 25th, the entire scenario was repeated
at Terra Meridiani with Mars Exploration Rover B, named
Opportunity. 1

Introduction

In March 2005 we spent a week at NASA's Jet Propulsion
Laboratory QPL) as part of our joint master thesis 2 at the

Figure I. Opportunitys heat shield and the shield's place of
landing as seen from the rover. http: I /marsrovers. jpl.

nasa.gov/gallery/press/opportunity/20041227a/

1NN325EFF40CYLA3P0685L000Ml-crop-B330Rl_br.jpg

mathematics department of Roskilde University, Denmark.
The purpose of our stay was to conduct a small investigation of the mathematics in the Mars Exploration Rover (MER)
mission being performed at JPL.
Professor Emeritus Philip ]. Davis had more or less suggested such an investigation in an article published in 2004:
Consider the recent flight to Mars that put a "laboratory
vehicle" on that planet. [ . . . ] Now, from start to finish, the Mars shot would have been impossible without
a tremendous underlay of mathematics built into chips

'This information in great part originates from http: I lnssdc. gsfc. nasa. gov ldatabaseiMasterCatalog?sc=2003-027 A and http: I lnssdcgs fc. nasa.

govldatabaseiMasterCatalog?sc=2003-032A

2The thesis consists of the texts [5], [6], and [7] and can be found in its original Danish version as IMFUFA-text number

imfufateksterlindex.htm

8 THE MATHEMAnCAL INTELLIGENCER © 2007 Spnnger SC1ence+Bus1ness Mecia, Inc

449 at http: 1 lrrunf. rue. dkl
and software. It would defy the most knowledgeable historian of mathematics to discover and describe all the
mathematics that was involved. The public is hardly
aware of this; it is not written up in the newspapers. [2]
Although we are not "the most knowledgeable historians of
mathematics," we nevertheless decided to engage in Philip
Davis's project. Unfortunately, newspapers are not the only
place in which this wasn't written up. In fact, finding extensive literature on the mathematics of the mission was so
difficult that we decided to base our investigation on interviews. Hence the long travel to Pasadena, California. While
in the US we decided also to visit Davis at Brown University in Providence, Rhode Island, to discuss our pending investigation with him. Davis advised us to tly to gain an insight into the employees' personal motivations for working
in the aerospace industry, as well as an understanding of the
nature of the mathematical work performed at JPL [1]. We
also decided to look at what might be referred to as the external influences on the daily work, such as deadlines and
economic limitations-the basic work context.
One of the more interesting aspects of our investigation
quickly turned out to be the invisibility of the mathematics involved in the mission. The fact that the mathematics involved
is hidden from the public may seem natural, hut parts of the
mathematics of MER are also hidden from the scientists participating in the mission. In fact the hiding, or invisibility, of
the mathematics in MER occurs on several levels, some intended and some not. The aim of this a 1ticle is to present
some of the mathematical aspects of the MER mission and to
discuss the way they are hidden in the mission, as well as the
effect the work context had on the process5 Much of the account is built by letting the JPL scientists speak for themselves,
i.e., by frequently quoting from our interviews. 1

JPL Scientists at Work

We found that, in general, JPL's scientists arc people with
the highest educational level who join the institution shortly
after completing their university studies. They are driven by
a desire to he part of the aerospace industry and a passion
for planetary exploration. To some extent, they were also
drawn to JPL by a fascination with the mathematical, physical, and engineering problems involved in space exploration; hut as a motivating factor this seemed secondary.

Among the first to discuss the mathematical aspects of
the work at JPL with us was Jacob Matijevic, a mathematician who had been with JPL for a long time. Particularly
we discussed the modelling aspects of the work, which
takes place before the actual mission is set in motion.
A mission like MER is to a large extent about being able
to predict how the technology onboard the craft is going
to behave in space or in the Martian environment. Once
the craft is flying it is impossible to make adjustments requiring more than a radio signal. Everything must therefore
function as expected. Take for instance the Mars environment's influence on the instruments onboard Spirit and Opportunity. You have to have very precise knowledge about
the distribution of heat inside the rover and how this affects the instruments. To acquire such knowledge, virtual
models of the rovers are built in software so that the thermic conditions can be simulated. Such thermic models are
typically based on a number of differential equations which
are solved within the programs. The work for the JPL employee consists of building the virtual model of the rover.
The exact method of solution which the program implements is secondary, as long as it works and is not too slow.
According to Matijevic [ 1 1 ] you also need models of how
the environment depends on the seasons on Mars to he
able to predict the concrete influence on the instruments.
These models are partly based on data from the different
Mars orbiters and partly on concrete measurements performed on the Martian surface. The correctness of the surface measurements depends on how good the description
of the instrument's behavior in the Martian environment is,
and it cannot be guaranteed. By comparing the data from
the orbiters with the surface measurements, a more accurate picture may arise; this may then be used to modify the
models, so that they slowly become better and better. All
of this is done in software. Regarding the models of how
the seasons affect the Mars environment, it is probably fair
to compare the work at JPL with that performed by an institute of meterology. Matijevic reported,
When I first arrived here over twenty years ago there
were still efforts to hand-implement certain mathematical models for certain applications. And there were specialist applications here for specialists in the applied
mathematical sciences who worked here to make those

3An expanded Danish version of this article with a slightly different angle has also appeared in the Nordic mathematical journal Normal (13].

4Transcripts of these interviews in full, along with our conversation with Ph1hp Davis, can be found in (7].

UFFE THOMAS JANKVIST was bom in

BJ0RN TOLDBOD is a native of Roskilde, Den­

Copenhagen. He holds a master's degree from mark He holds a master's degree from Roskilde
Roskilde University. He is now a doctoral can­ Universrty. He is now, as a conscientious objector
didate there in the use of history of mathe­ to military service, doing his a�emative service
matics in mathematics education. worl<ing at the computer department of the Royal

Danish Library.

Roskilde Universitetscenter

Universitetsvej I

Vesterbrogade I 19 A. 3.tv.

4000 Roskilde I 620 Copenhagen V
Denmark Denmark
e-mail:
applications possible. But over time much of that has
been incorporated in fairly standard and available simulation and modelling packages-computer packages.
Expansions have been introduced slowly over time to
these packages and that's basically how the engineers
here do their job. Instead of going back to first principles they apply these tools . . . the foundation theories
are from the eighteenth century to a large degree [ 1 1 ] .
Hence a lot of work involving modelling and simulation is
done at JPL, but all of it is done in software packages. This
might lead one to suspect that JPL has its own staff of mathematicians developing such packages, but Matijevic informed us that the packages mostly come from commercial companies.
A few days after our interview with Matijevic we had the
opportunity to interview Dr. Miguel San Martin, an engineer
with whom we discussed various aspects of the mission. He
told with great enthusiasm about the challenges the scientists must overcome to make the rovers able to figure out
their orientation on the Martian surface. San Martin explained
that the navigation on the surface is based on a well-known
technique which sailors have used for thousands of years on
Earth-you look at the Sun. Together with a vector of gravitation, which can be measured by the rover, the position of
the Sun in the sky provides the information necessary to
determine the rover's position on the surface. San Martin
didn't think of this problem as being very mathematical. In
fact, he claimed that the majority of the mathematics involved
in his work was very simple, and he concluded:
The most important is that you have millions of these
little, simple things. And that's the trick; to make them
all work, and talk to each other and make sure that no
parameter is tightened too much or too little. The complexity of the space problem is keeping it simple. [ 1 2]
Present at this interview was Dr. William Folkner, a
physicist and our main contact at JPL, and he elaborated
on this view:
Well, you hit a lot of mathematics in your descriptions,
right, because you need to know the positions of the axes
of Mars around the Sun as a function of time and you need

to know what the orbits around the Sun and the Earth
were. There is a lot of mathematics hidden in what you
just said. [ . . . l We've worked all that out for us in the tables. So to know where the Sun is now, you just look it
up. Somebody had to figure it out the first time. [ 1 2]
Folkner's answer illustrates why the question of what
mathematics is used in MER is difficult to answer. Knowledge that mathematics previously made accessible can over
time become such an integral part of our conception of the
world that we no longer connect it with mathematics. The
trajectory of Mars is a good example of this: Is it mathematics to look up a table to see the Sun's position relative
to Mars at a given time? Perhaps not, but the making of
such a table is a mathematical problem. Thus the mathematics at JPL is often disguised as "common knowledge."

MER Work Context

Incredibly high reliability is demanded of the work performed at JPL. A single mistake in a piece of technology
or an algorithm may have serious consequences and in the
worst case may result in several years of wasted work for
hundreds of people. All of the work being done at JPL is
therefore subject to careful development and testing.
We asked Jacob Matijevic about the development of the
parachutes for the rovers, partly because we thought there
would be little interaction between the parachutes and other
devices. That is, we thought this would be a "simple" task.
Matijevic, however, revealed more:
We did drop tests. We did wind tunnel tests with the
parachutes. But even before this time it was through
models of the profiles of these devices that we came up
with things like what the entry angles would he, what
sorts of release points should we be looking at, as well
as designing the algorithm that checks for height above
the surface and finding out at which time to deploy the
parachute and at which time to fire the rockets for slowing the descent. All of this was based on what we expected to be the environmental profile that the vehicle
would see as it came down to the surface. So this was
all done in simulation. [ 1 1]

Figure 2. Guided tour at JPL. Left: Uffe and Dr. Albert Haldemann, who showed us some of the facilities. Right: Visiting JPL's
museum for earlier space missions with Dr. William Falkner.

10 THE MATHEMATICAL INTELLIGENCER
Figure 3. The guided tour takes us by JPL's "sandbox" where rovers are test driven. Left: Bjorn in front of the sandbox. Right:
A replica of a Mars Exploration Rover used for test drives at JPL.

Reliability is paramount for any mission. If the choice
stands between two different approaches to a problem, a
space scientist will be inclined to choose a well-known,
well-tested solution over a new-perhaps more efficientsolution which has not been thoroughly tested in the context of the mission. Our interview with Dr. Jon Hamkins,
one of JPL's leading coding theoreticians, also confirmed
this for the error-correcting codes used in MER [4]. Mathematics and technology which had been onboard an earlier
mission is considered to be safer and therefore makes a
more attractive choice. This approach is taken in all aspects
of the missions. Of course some development takes place
from mission to mission hut only at a pace that makes extensive testing possible. Jacob Matijevic called this ··steady
progress'' [ 1 1 ] . New ideas introduced into the missions will
he at least '5-10 years old at launch time, because they must
he laid down when the missions are first planned. In the

Figure 4.

case of error-correcting codes, a lot of the mathematics involved has to he implemented in hardware for speed. Generally hardware is much more expensive to replace than
software, so the gain of introducing a new error-correcting
code has to he considerable in order to balance the expense of the substitution .
The grand scale of a project like MER also means that
the work performed by different departments must be completed at specific deadlines. Not surprisingly, deadlines may
serve as a stop block for the development of new ideas.
You cannot promise to use a new method if you are not
sure that there will be enough time to test it.
There is another factor working against the introduction
of new methods: In recent years .JPL has gone from a small
number of large and expensive missions to a large number of small hut less expensive missions. For instance, the
Pathfinder mission of 1996 had a total budget of 265 mil-

Left: Wind tunnel test of the MER landing module parachute. http: //www.nasa.gov/centers/ames/images/
Right: MER landing rnoduit: airhags. http: //photojournal. jpl.nasa. gov/jpegMod/

content/79641main_picture_2. jpg

PIA04999_modest. jpg

© 2007 Spnnger Sc19nce+Bus1ness Media, Inc. Volume 29, Number 1. 2007 11
Figure 5. Guided tour at JPL. Left: A JPL photo of a MER rover testing prior to launch. Right: One of JPL's laboratories.
lion dollars 5 whereas the Viking missions of the seventies
had a budget of around 8 billion dollars. The MER mission
was more expensive than Pathfinder, hut still nowhere near
the Viking budget. This means that if anything from a previous mission can be used again, there are huge amounts
of money and time to be saved. Many of the cheaper missions must necessarily rely on reuse from earlier missions.
The nature of the space missions in general was summed
up in the following way by Falkner:
Everything is a cost-benefit analysis. The whole space
system is a cost-benefit analysis. [3]
To this point we have mostly focused on how external
factors influence the mathematics of a space mission. We
now turn to two examples of concrete mathematical problems in the MER mission. Still the purpose of our selection
will be to illustrate common features of the mathematics in
a space mission. coded, then it goes through a block interleaver and then
it's convolutionally encoded [4].
Thus MER's coding system consisted of two combined, or
concatenated, error-correcting codes; a Reed-Solomon code
and a convolutional code.
Reed-Solomon codes are so-called algebraic codes,
whose code symbols come from a Galois field. ReedSolomon codes are linear and cyclic. If the data words to
be encoded are of length k and code words are of length
n, an (n,k)-cyclic code, with code symbols from the Galois
field F q• is a cyclic subspace of the vector space Fq. Reed­
Two Selected Mathematical Problems of MER Solomon codes can either be defined as special kinds of
cyclic codes with specific generator polynomials or by
means of Fourier transforms. On the one hand, convolutional codes are not as mathematically well understood as
the Reed-Solomon codes and not as easily defined either.
They can, however, be defined by the use of formal Laurent series. On the other hand, convolutional codes are very
efficient and therefore very often used in technology. They
are excellent for correcting single-bit errors, the kinds of
errors which most often occur from interference in deep
space. Unfortunately the decoding of convolutional codes 6
often results in a run of consecutive errors, so-called burst
errors. Fortunately Reed-Solomon codes excel exactly in
correcting burst errors, hence the concatenated system. The
reason for first encoding the data with the Reed-Solomon
code and then with the convolutional code is that the decoding procedure must be the reverse of the encoding procedure. Block interleaving is a technique used to ensure
that the hurst errors from the convolutional decoding are
no more severe than what the Reed-Solomon decoder can
handle.
One of the main purposes of the MER mission was to
take photographs of Mars. Before these photos were transmitted to Earth they had to he compressed. The imagecompression technique primarily used in MER is called The
/Ct-1? Progressive Wavelet Image Compression, in short just
ICER, and was developed at JPL by Drs. Aaron Kiely and

The mathematical emphasis of our investigation was the
two coherent mathematical theories called channel coding
and source coding, which deal with reliable communication and compression of data (including images), respectively. Formal definitions of the specific codes involved
would take us beyond the scope of this article. We therefore merely indicate the areas of mathematics involved.
The signals transmitted to and from Mars are subject to
interference during their travel through deep space. Such
interference of a binary signal may result in bits becoming
altered. The communication between Earth and the rovers
needs to be reliable. The problem of interference is solved
by way of channel codes, which make it possible to correct altered bits in a message; hence the codes are also
called error-correcting codes. The system used in MER depends on two different codes used in combination. Jon
Hamkins explained:
The majority of the missions flying now are concatenated. So the data comes in and is Reed-Solomon en-

5http://nssdc.gsfc.nasa.gov/database/MasterCatalog?sc�1996-068A

6JPL

12

uses the so-called Viterbi algorithm in its convolutional decoder.

THE MATHEMATICAL INTELLIGENCER
Matthew Klimesh. The word "progressive" refers to progressive fidelity compression. In such compression a lowquality approximation of the photo is first transmitted . Afterwards hits are transmitted in such a way that the quality
of the photo is gradually improved. When all hits are transmitted the reconstructed image equals the original image.
By stopping the transmission before it is complete, lossy
compression (i.e., compression with loss of data) can be
obtained. In this way ICF.R supports lossless as well as lossy
compression, even though it was entirely used for lossy
compression. For lossless compression MER relied on the
commercial compression algorithm LOCO. The ICER algorithm, like many other image-compression techniques. overall consists of three stages: preprocessing, modelling, and
entropy encoding of data. Kiely explained:
We got data coming in, an image or whatever it is, and
then some sort of preprocessing stage, for example a
wavelet transform plus quantization or a discrete cosine transform or something. The goal is that it doesn't perform any compression, and in fact it is often a
lossy process, it might throw out some of the data. but
the idea is to process the data in a way that makes it
more receptive to compression through the entropy encoder. The entropy encoder is sort of the engine. Given
some sort of probabilistic model of the source, it compresses data or represents it in a more efficient way
through something like a variable-length code. That is
sort of the big picture of what is going on . So for example for ICER what is going on is mostly a probabilistic transform. For LOCO it is in essence trying to
project a probability distribution on the next pixel that
it is about to encode based on what it has seen in the
nearby neighbors. (9]
ICER uses a wavelet transform that closely resembles a Haar
tran.iform, a context model also known as a Markov model,
and the majority of the entropy codes used by the entropy
encoder are the Golomh codes [R]. The LOCO algorithm is
a hit different from ICER in that it does not have a preprocessing stage which is typical for losslcss compression

techniques. It does use context modelling, and it uses both
Golomb codes and Huffman codes [14] .
The Haar transform i s an invertible transform which makes
hoth lossy and lossless compression possible. An image with
n X n pixels is naturally represented by a n X n matrix M
containing integer values. By the invertible transform, say
T: R" � R", which is first performed on rows and then on
columns of M, a shift of basis from the euclidian basis to another orthonormal basis is made. The elements of this basis
are called wauelets. The transform splits the image into a lowfrequency suhimage and a high-frequency subimage, concentrating the majority of the energy in the low-frequency
area, thus making efficient lossy compression possible.
In a context or Markov model the likelihood of a symbol's being encoded depends on the previously encoded
symbols; i . e . , the model is said to have a memory. This is
a very common situation in digital images, where the value
of a certain pixel may indeed depend on the values of the
surrounding pixels. ICER's context model maintains a statistical model with the purpose of estimating the likelihood
of the next hit's being a zero bit.
Huffman codes are variable-length prefix codes, which
means that they can assign shorter codewords to more frequent data symbols and that no codeword is a prefix of
another codeword. Golomb codes make u p another family
of codes which are parametrized by an integer m > 0 and
therefore are often written as <§m· A Golomb code encodes
integers under the assumption that the larger the integer
the less likely its occurrence is. It can be shown that Golomb
codes make optimal encodings for geometric probability
distributions of non-negative integers, which make them attractive in many different contexts.
The above presentation of mathematics in MER is, of
course, merely scratching the surface. Other mathematical
theories that we came across during our interviews at JPL
include the "Lost in Space" problem, pinpointing the craft's
position in space Jt a given time; Kalman filtering, estimating incomplete and interfered-with data from the craft's
different sensors; the Hohmann trajectory, the trajectOiy hetween Earth and Mars which calls for the least amount of
energy at launch; and of course control theory. Discovering every little piece of mathematics put to use in the MER
mission probably is an impossible task, as Folkner im!icated:
There is mathematics in everything. There is control theory, aerodynamics, orbital dynamics, Newtonian gravity ,
bodies going around the Sun. We use general relativity,
that's mathematics of physics. [ . . . ] Linear algebra is a
field of mathematics we use all the time. Matrices. That's
in the control theory all the time. There is Riemannian
geometry in the general relativity. Calculus. [ 1 2]
Instead, we shall now turn to the theme of this article, the
hidden mathematics of MER.

Hidden Mathematics of MER

Figure 6. Spirit's landing module as seen from the rover.

http://www.seds. org/-spider/spider/Mars/Hi-res/

mer a lander. jpg

When you view a space mission from the outside, it is clear
that the mathematics involved is to a certain degree hidden (or invisible). We had imagined that once we entered
the JPL facilities the mathematics of MER would suddenly
become more visible. Our investigations showed otherwise.

© 2007 Spnnger Science+Bus1ness Media, Inc , Volume 29, Number 1. 2007 1 3
The mathematics of a mission like MER is hidden on
several different levels. To most people reading about space
missions in a newspaper, the mathematics is hidden in a
lot of technology whose mode of operation is seldom discussed. Partly, the media fear boring their readers; but even
when they try, it is difficult to communicate abstract mathematics to the uninitiated. This type of invisibility seems
obvious.
More interesting, the mathematics to some extent also is
hidden inside the walls of JPL. As we pointed out, a large
part of the mathematics for solving the mission's problems
is embedded in software packages-and therefore hidden.
The packages are often commercial packages, developed outside JPL. To users of such packages it is important to know
how to use the packages correctly, and this most certainly
requires mathematical knowledge, but it is not necessary for
a user to understand the specific mathematics in detail. One
might say that the mathematics is outsourced, and that the
implementation in software packages contributes to hiding
the mathematics from the people at the ]PL. As the conversation between Falkner and San Martin reported above illustrates, a consequence of the mathematics being hidden is
that to some degree it is being ignored.
The virtual m o dels of the rovers and the models of the
Martian environment of which Matijevic speak above rely
also on the mathematical technique of modelling. If the scientists of JPL who prepare these simulations do not regard
the modelling as mathematics-and our investigation suggests that to some degree they may not-then this mathematics will also be hidden from them.
Anyway, the JPL projects are so vast that a distinction
between being inside or outside is somewhat meaningless.
A scientist who works with a specific part of the mission
of course has an overall perception of the mathematics involved in other parts of the mission, but as the following
quote from one of our interviews with Falkner shows, the
details are sometimes quite hidden. Falkner's first remark
was a joke intended to illustrate that the use of mathematics in a mission like MER is so extensive that it would be
easier to expose the mathematics not involved.
We don't do any map theory for instance. Four-color
map problem.
You know that? We don't do any of that. We don't really use any abstract algebra, group theory, and that sort.
Except in the channel coding.
They use abstract algebra and group theory in that?
The Reed-Solomon codes are based on Galois Fields.
That's news to me. I didn't know that; all right, interesting. [ 1 2)
Worth noticing is that Falkner says, " They use abstract algebra and group theory in that?" This suggests that the different departments of JPL may not have a large degree of
interaction. In fact William Falkner and Miguel San Martin
had never met each other before we interviewed them together. The point here is that the vastness of the project
contributes to hiding the mathematics from the employees.
One last kind of invisibility which must be mentioned
is that certain areas of the missions are classified: some
things are deliberately hidden. Our experience at JPL was
that the institution generally was very open to giving in-

14 THE MATHEMATICAL INTELLIGENCER

formation and answering questions, but also that security, control, and classification were part of a normal work
day. Thus when we talked to Mark Maimone about the
amount of control theory used for steering the rovers, he
replied at one point, "I don't know what has been published about the details of those algorithms, so I can't tell
you anything about that" [ 10). We have not examined the
motives for such classification any further, but it seems
fair to say that classification makes it harder to discover
the mathematics involved and therefore contributes to the
hiding of it.
When discussing the hidden mathematics of a project
like MER, one question springs to mind: Would the applied
scientists of MER have been able to benefit from knowing
more about the mathematical tools they use in their work?
Perhaps. Our investigation does not imply a unique answer
to this question. If we restrict our attention to the modelling and computer simulation aspects of MER, there might
be something to say on this topic. Whenever working with
commercial software and standard packages users may have
only a general idea of the (mathematical) elements involved.
In a large-scale project like MER, different people, and different departments work on different aspects of a computer
simulation. Indeed, scientists do not always know in detail
what is going on 'on the other side of the fence. ' In recent
years the engineering industry has been experiencing a 'paradigm shift' in the way engineers work: the testing of scale
models has now been replaced by computer simulations,
and the engineer most likely does not know anything about
the model(s) on which the simulation is based. Now, this
is a scary scenario, for wrong applications or wrong interpretations of the simulations may lead to disastrous mistakes. In such situations the applied scientists would most
definitely benefit from knowing more of the ramifications
of the (hidden) mathematical tools they use. That they don't
is quite paradoxical: the space missions are devoted to minimizing mistakes, yet the nature of the space missions may
itself inflict errors-just because the mathematics in use is
hidden.

Conclusions

A lot of the mathematics of MER is hidden and not only
from the public but even from the applied scientists working on the mission. As briefly sketched above, for the scientists, this could be disastrous in a worst-case scenario.
The hiding of mathematics, both in our everyday life and
within science itself, is a matter not often discussed in public-which in itself is a disaster, taking into account the
consequences the hiding of mathematics might have for the
public. We like to think that this article may help let in
some light.
Another question raised by our work is that of beliefs in
mathematics. Only occasionally are the beliefs of mathematicians discussed. We found repeatedly that mathematical elements of MER are not actually considered to be mathematics among the applied scientists themselves, not on
first hand anyway. Is this due to the fundamentally different views of what mathematics is between applied scientists (including engineers) and pure scientists of the 20th
century? We do not know.
Finally, we comment on the nature of the mathematics
involved in MER. Because of the extreme nature of a Mars
mission, one might expect "extreme'' mathematics, mathematics developed for the sole purpose of this mission. This
does not seem to he the case. We did not come across any
basic innovations in mathematics as a result of the MER
mission. The MER mission is based on well-established
mathematical theories and disciplines. Some of these date
back to the 18th century, but a lot of them arc also from
the 20th century: the Hohmann trajectory is from 192'i, the
convolutional error-correcting codes used arc from 19'i4,
and the Reed-Solomon codes are from 1 960, the Lost in
Space problem and Kalman filtering arc also from the 1 960s.
Image compression relies on the Haar transform from 1 910,
Huffman codes from 19'i2, Golomb codes from 196(J, and
wavelet theory from the 1990s.
Now, this is not to say that basic research in mathematics is not used. Rather it suggests that technological and mathematical developments seem to he on different timelincs so
that a stop in research in pure mathematics would damage
technological development. And since so many of the mathematical theories in MER are from the 19(JOs, one might think,
that the timelines in the case of the space industry arc no
more than fifty years apart. Feedback also goes the other
way: basic research in mathematics can he inspired by problems in applied sciences and technology [lJ-therefore also
from missions like MER and institutions like JPL. But the major role of JPL seems to he that of consumer of already developed applied mathematics. From start to finish MER is an
example of the application of mathematics. But not just any
example. It is an extraordinary example of what can he accomplished with the mathematics at our disposal.

BooiS-Bavnhek for suggestions and comments on the original
thesis.

BIBLIOGRAPHY

[ 1 ] P. J. Davis. Interview with Professor Emeritus Philip J . Davis, March

6th, 2005. Recorded at Brown University, Providence, R l .

[2] P. J. Davis. A Letter t o Christina of Denmark. EMS, pages 21-24,

March 2004.

[3] W. Falkner. Interview with Doctor William Falkner, March 1 7th,

2005. Recorded at JPL, Pasadena. CA.

[4] J. Hamkins. Interview with Doctor Jon Hamkins, March 1 4th, 2005.

Recorded at JPL, Pasadena, CA.

[5] U. T. Jankvist and B. Toldbod. Mathematikken bag Mars-missio­

nen - En em pi risk unders0gelse af matematikken i MER med fokus

pa kildekodning og kanalkodning. Master's thesis, Roskilde Uni­

versity, October 2005. Tekster fra IMFUFA, nr. 449a.

[6] U. T. Jankvist and B. Toldbod. Matematikken bag Mars-missio­

nen - lndfmelse i den grundlceggende teori for kildekodning og

kanalkodning i MER. Master's thesis, Roskilde University, October

2005. Tekster fra IMFUFA, nr. 449b.

[7] U. T. Jankvist and B. Toldbod. Matematikken bag Mars-missio­

nen -Transskriberede interviews fra DTU, Brown University, MIT

og JPL. Master's thesis, Roskilde University, October 2005. Tek­

ster fra IMFUFA, nr. 449c.

[8] A. Kiely and M. Klimesh. The ICER Progressive Wavelet Image

Compressor. lPN Progress Report, 42(1 55) : 1 -46, November 2003.

[9] A. Kiely. Interview with Doctor Aaron Kiely, March 1 4th, 2005.

Recorded at JPL, Pasadena, CA.

[1 0] M. W. Maimone. Interview with Doctor Mark W. Maimone, March

1 7th, 2005. Recorded at JPL, Pasadena, CA.

[1 1 ] J. Matijevic. Interview with Doctor Jacob Matijevic, March 1 4th,

2005. Recorded at JPL, Pasadena, CA.

ACKNOWLEDGMENTS

First of all we would like to thank all the employees at JPL
who set aside their duties to talk to us, especially Bill Folkner.
who arranged most of our meetings. We also thank Phil Davis
for taking such an extraordinary interest in our investigation
and for commenting on this article. Furthermore we thank
Tinne Hoff Kjeldsen, Man-Keung Siu, and Chandler Davis for
helpful suggestions about how our investigation should he
presented here. Thanks also to Anders Madsen and Bernheim

[1 2] M. San Martin and W. Falkner. Interview with Doctor Miguel San

Martin & Doctor William Falkner, March 1 4th, 2005. Recorded at

JPL, Pasadena, CA.

[1 3] B. Toldbod and U. T. Jankvist. Reportage fra en Mars-mission.

Normat, 54(3): 1 0 1 -1 1 5 and 1 44, 2006.

[1 4] M. J. Weinberger, G. Seroussi, and G. Sapiro. LOCO-I: A Low

Complexity, Context-Based , Lossless Image Compression Algo­

rithm. Proceedings of the IEEE Data Compression Conference,

pages 1 -1 0, March-April 1 996.

"Celebrated Turtle"

Plea�ed and proud as \'d.' were to present the mono-monostatic

and P. L. Ya rkon ·i, a nd gratefu l as we are to the artbt for ih

we ''

nml • ( ,-ol. 2H, no. "!. 3-1-3R) of Gabor Domokos
beautiful pr •s ·ntation on the cover of th ' issue.

cr •n't quite prt'pared for the media frenzy it excited in H ungary and far herond. Th ·re ar • dozens of ar­

tid ·s ,1hout it-sec. for example,

http; WW\\ .lahoodlc.hu nc news ne,,-s_archi,-e singlc_page article 1 1 hungarian_ sc ?cHash� i2Sf3fct7b

The story has h L'n pick •d u p h} Reuters and broadcast around the world: see. for e. ample,

http:/ WW\\ . msnlx.msn.com id 1 - 1 20210/. And

go to

Imp:/ ww\v .sciam .com

and d ick on Biolog} .

Gabor comments. "I am tell i ng vou this to sho\\' hem much impa t your \York ha., in u nli ke! corners of the

world." \\'e r •ply . with all due t hanks, that the \\'ork '' .1s the authors . not our. , and that Hu ngary, if indeed it

is a corner of the world, is a rel::nin:l} likely corner.

-The Editors

© 2007 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 29, Number 1 , 2007

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