Chapter 2
Unsolved Problems and Still-Emerging
Concepts in Fractal Geometry
Benoit B. Mandelbrot
The preceding chapter sketches a striking property of fractal
geometry. Its first steps are, both literally and demonstrably,
childishly easy. But high rewards are found just beyond those
early steps. In particular, forbiddingly difficult research frontiers
are so very close to the first steps as to be understood
with only limited preparation. Evidence of this unique aspect
of fractal geometry is known widely, but scattered among
very diverse fields. It is good, therefore, to bring a few together.
A fuller awareness of their existence is bound to influence
many individuals' and institutions' perception of the
methods, goals, and advancements of fractal geometry.
1 Introduction
"You find fractals easy? This is marvelous." Thus begins my
response to an observation that is sometimes heard. "If you
are a research mathematician, the community needs you to
solve the challenging problems in this nice long list I carry
around. If you are a research scientist, you could help to better
analyze the important natural phenomena in this other long
list."
The first half-answer is elaborated in Section 2. The point
is that fractal geometry has naturally led to a number of compelling
mathematical conjectures. Some took 5, 10, or 20
years to prove, others--despite the investment of enormous
efforts--remain open and notorious. If anything, what slows
down the growth of fractal-based mathematics is the sheer
difficulty of some of its more attractive and natural portions.
The second-half answer is elaborated in Section 3. The
point is that, among other features, fractal geometry is, so far,
the only available language for the study of roughness, a concept
that is basic and related to our senses, but has been the
last to give rise to a science. In many diverse pre-scientific
fields, the absence of a suitable language delays the moment
when some basic problems could be attacked scientifically. In
other instances, it even delays the moment when those problems
could be stated.
2 From simple visual observation
to forbiddingly difficult
mathematical conjecture
A resolutely purist extreme view of art holds that great
achievements must be judged for themselves, irrespective of
their period and the temporary failures that preceded their being
perfected. In contrast, the most popular view attaches
great weight to cultural context and mutual influences, and
more generally tightly links the process and its end-products.
Some works do not survive as being excellent but as being
representative or historically important. For example, a resolutely
sociological extreme view that we do not share holds
that Beethoven's greatness in his time and ours distracts from
the more important appreciation of his contemporaries. Few
persons, and not even all teachers, are aware that a very similar
conflict of views exists in mathematics.
As widely advertised, the key product of mathematics consists
in theorems; in each, assumptions and conclusions are
linked by a proof. It is also well known that many theorems
began in the incomplete status of conjectures that include
assumptions and conclusions but lack a proof. The iconic example
was a conjecture in number theory due to Fermat. After
a record-breaking long time it led to a theorem by Wiles.
Conjectures that resist repeated attempts at a proof acquire an
important role, in fact, a very peculiar one. On occasion the
news that an actual proof has made a conjecture into a theo-
11
M.L. Frame & B.B. Mandelbrot
Washington D.C. Mathematical Association of America, 2002
Fractals, Graphics, & Mathematics Education
12 Chapter 2. Unsolved Problems and Still-Emerging Concepts in Fractal Geometry
rem is perceived as a letdown, while it is suggested that these
conjectures' main value resides in the insights provided by
both the unsuccessful and the successful searches for a proof.
Be that as it may, fractal geometry is rich in open conjectures
that are easy to understand, yet represent deep mathematics.
First, they did not arise from earlier mathematics, but
in the course of practical investigations into diverse natural
sciences, some of them old and well established, others newly
revived, and a few altogether new. Second, they originate in
careful inspections of actual pictures generated by computers.
Third, they involve in essential fashion the century-old
mathematical monster shapes that were for a long time guaranteed
to lack any contact with the real world. Those fractal
conjectures attracted very wide attention in the professions
but elude proof. We feel very strongly that those fractal conjectures
should not be reserved for the specialists, but should
be presented to the class whenever possible. The earlier, the
better. To dispel the notion that all of mathematics was done
centuries ago, nothing beats being able to understand appropriate
problems no one knows how to solve. Not all famous
unsolved problems will work here: the Poincar´e conjecture
cannot be explained to high school students in an hour or a
few. But many open fractal conjectures can.
For the reasons listed above, the questions raised in this
chapter bear on an issue of great consequence. Does pure (or
purified) mathematics exist as an autonomous discipline, one
that can and should develop in total isolation from sensations
and the material world? Or, to the contrary, is the existence
of totally pure mathematics a myth?
The role of visual and tactile sensations. The ideal of pure
mathematics is associated with the great Greek philosopher
Plato (427?­347 BC). This (at best) mediocre mathematician
used his great influence to free mathematics from the pernicious
effects of the real world and of sensations. This position
was contradicted by Archimedes (287­212 BC), whose realism
I try to emulate.
Indeed, my work is unabashedly dominated by awareness
of the importance of the messages of our senses. Fractal
geometry is best identified in the study of the notion of roughness.
More specifically, it allows a place of honor to fullfledged
pictures that are as detailed as possible and go well
beyond mere sketches and diagrams. Their original goal was
modest: to gain acceptance for ideas and theories that were
developed without pictures but were slow to be accepted because
of cultural gaps between fields of science and mathematics.
But those pictures then went on to help me and many
others generate new ideas and theories. Many of these pictures
strike everyone as being of exceptional and totally unexpected
beauty. Some have the beauty of the mountains and
clouds they are meant to represent; others are abstract and
seem wild and unexpected at first, but after brief inspection
appear totally familiar. In front of our eyes, the visual geometric
intuition built on the practice of Euclid and of calculus
is being retrained with the help of new technology.
Pondering these pictures proves central to a different philosophical
issue. Does the beauty of these mathematical pictures
relate to the beauty that a mathematician rooted in the
twentieth century mainstream sees in his trade after long and
strenuous practice?
2.1 Brownian clusters: fractal islands
The first example, introduced in Mandelbrot (1982), is a wrinkle
on Brownian motion. The historical origins of random
walk (drunkard's progress) and Brownian motion are known
and easy to understand, at least qualitatively. From this, it is
simple to motivate the definition of the Wiener Brownian motion:
a random process B(t) with increments B(t + h) - B(t)
that obey the Gaussian distribution of mean 0 and variance h,
and that are independent over disjoint intervals.
For a given time L, the Brownian bridge Bbridge(t) is defined
by
Bbridge(t) = B(t) - (t/L)B(L),
for 0  t  L. Taking B(0) = 0, we find Bbridge(L) =
Bbridge(0). Combining one Brownian bridge in the xdirection
and one in the y-direction and erasing time yields a
Brownian plane cluster Q. Because we use Brownian bridges
to construct it, the Brownian plane cluster is a closed curve.
See Figure 1. An example of a well-known and fully proven
Figure 1: A Brownian plane cluster; Plate 243 of Mandelbrot
(1982).
2. From simple visual observation to forbiddingly difficult mathematical conjecture 13
fact is that the fractal dimension of Q is D = 2. This result
is important but not really perspicuous, because the big holes
seem to contradict the association of D = 2 with plane-filling
curves. Results that are well known and not perspicuous are
not for the beginner.
Let us proceed to the self-avoiding planar Brownian motion
~Q. It is defined in Mandelbrot (1982) as the set of points
of the cluster Q accessible from infinity by a path that fails to
intersect Q. That is, ~Q is the hull of Q, also called its boundary
or outer edge. The hull ~Q is easy to comprehend because
it lacks double points. The unanswered question associated
with it is the 4/3 Conjecture, that ~Q has fractal dimension
4/3.
An early example of Q, and hence of ~Q is seen in Figure 1.
It looks like an island with an especially wiggly coastline,
and experience suggested its dimension is approximately 4/3.
This comparison with islands made the 4/3 conjecture sensible
and plausible in 1982 and it remains sensible and plausible
to students; that it remained a conjecture for many years
is something they can appreciate. Numerical tests and physicists'
heuristics were added to the empirical evidence and
the conjecture was proved in Lawler, Werner, & Schramm
(2000).
2.2 The Mandelbrot set
Second example: In the past, music could be both popular
and learned, but elitists believe that this is impossible today.
For mathematics, the issue was not raised because no
part of it could be called a part of popular culture. Providing
a counterexample, no other modern mathematical object
has become part of both scientific and popular culture as
rapidly and thoroughly as the Mandelbrot set. Moreover, an
algorithm for generating this set is readily mastered by anyone
familiar with elementary algebra. Thousands of people,
from middle school children to senior researchers and Fields
Medalists, have written programs to visualize various aspects
of the Mandelbrot set.
Recall the simplest algorithm: a complex number c belongs
to the Mandelbrot set M if and only if the sequence
z0, z1, z2, . . . stays bounded, where z0 = 0 and zi+1 = z2
i +c.
For instance, the sequence can stay bounded by converging
to a fixed point or to a cycle. Denote by M0 the set of
all c for which this is true. Of course, M0  M. In fact,
M0 is of interest to the students of dynamics, hence my original
investigations were of M0, not of M. Interest shifted to
M because producing pictures of M is easy. By contrast, to
test if c  M0, we first generate several hundred or thousand
points of the sequence z0, z1, z2, . . . , and test if for
large enough i there is an n for which |zi+n - zi | is very
small. This suggests convergence to a cycle of length n. (An
impractical theoretical alternative is to solve the 2n-degree
polynomial equation f n
c (z) = z, where fc(z) = z2 + c,
then test the stability of the n-cycle by a derivative condition:
-2 -1.5 -1 -0.5 0 0.5 1
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 2: The Mandelbrot set.
1 > | fc(w1)    fc(wn)| = 2n  |w1    wn|. Here the points
w1, . . . , wn of the n-cycles are the sequences of successive
zi for different z0. In general, for each c there are several
n-cycles, but at most one is stable.)
Computer approximations of M0 actually yield a set
smaller than M0, and computer approximations of M actually
yield a set larger than M. Extending the duration of the computation
seemed to make the two representations converge to
each other and to an increasingly elaborate common limit.
Furthermore, when c is an interior point of M, not too close
to the boundary, it was easily checked that a finite limit cycle
exists: the steps outlined above converge fairly rapidly for c
not too close to the boundary. Those observations led me to
conjecture that M is identical to M0 together with its limits
points, that is, M = cl(M0), the closure of M0.
In terms of its being simple and understandable without
any special preparation, this conjecture is difficult to top. But
after almost twenty years of study, it remains a conjecture.
With the proof of Fermat's last theorem, the conjecture M =
cl(M0) may have been promoted to illustrating the shortest
distance between a simple idea (in this case, complete with
popular pictures) and deep, unsolved mathematics. (Not so
simple is the usual restatement of this conjecture: that M is
locally connected.)
2.3 Dimensions of self-affine sets
The first tool for quantifying self-similar fractals is dimension.
For a fractal consisting of N pieces, each scaled in all
directions by a factor of r, the dimension D is given by
D =
log(N)
log 1
r
.
This is easy to motivate, trivial to compute. Working through
several examples, students soon develop intuition for the visual
signatures of low- and high-dimensional fractals. The
generalization to self-similar fractals having different scal-
14 Chapter 2. Unsolved Problems and Still-Emerging Concepts in Fractal Geometry
Figure 3: Top: Osculating circles outlining a Jordan curve limit set from inside and outside, Plate 177 of Mandelbrot (1982).
Bottom: Osculating triangles outlining the Koch snowflake curve from inside and outisde, Plate 43 of Mandelbrot (1982).
ings for different pieces is not difficult. For a fractal consisting
of N pieces, the ith piece scaled by a factor of ri , the
dimension D is the unique solution of the Moran equation
N
i=1
r D
i = 1.
Often this must be solved numerically, but this is not a difficulty
given today's graphing calculators and computer algebra
packages.
The simplicity of these calculations leads some people to
believe that calculating dimensions is a simple process. This
is a misperception resulting from the almost exclusive reliance
on self-similar fractals for examples. The case of selfaffine
fractals, where the pieces are scaled by different factors
in different directions, is much more difficult. Although
some special cases are known, no simple variant of the Moran
equation has been found. Kenneth Falconer describes the situation
this way, "Obtaining a dimension formula for general
self-affine sets is an intractable problem." (Falconer (1990),
129.) By simply changing the scaling factors in one direction,
a completely straightforward exercise becomes tremendously
difficult, perhaps without general solution.
2.4 Limit sets of Kleinian groups
A collection of M¨obius transformations of the form z 
(az + b)/(cz + d) defines a group that Poincar´e called
Kleinian. With few exceptions, their limit sets S are fractal.
For the closely related groups based on geometric inversions
in a collection C1, C2, . . . , Cn of circles, there is a
well-known algorithm that yields S in the limit. But it converges
with excruciating slowness as seen in Plate 173 of
Mandelbrot (1982). For a century, the challenge to obtain a
fast algorithm remained unanswered, but it was met in many
cases in Chapter 18 of Mandelbrot (1982). See also Mandelbrot
(1983). In the case of this construction, fractal geometry
did not open a new mathematical problem, but helped close a
very old one.
In the new algorithm, the limit set of the group of transformations
generated by inversions is specified by covering the
complement of S by a denumerable collection of circles that
osculate S. The circles' radii decrease rapidly, therefore their
union outlines S very efficiently.
When S is a Jordan curve (as on Plate 177 of Mandelbrot
(1982)), two collections of osculating circles outline S, respectively
from the inside and the outside. They are closely
reminiscent of the collection of osculating triangles that outline
Koch's snowflake curve from both sides (Figure 3). Because
of this analogy, the osculating construction seems, after
the fact, to be very natural. But the hundred year gap before it
was discovered shows it was not obvious. It came only after
respectful examination of pictures of many special examples.
A particularly striking example is seen in Figure 4, called
"Pharaoh's breastplate," Ken Monks' improved rendering of
Plate 199 of Mandelbrot (1982). A more elaborate version of
this picture appears on the cover of Mandelbrot (1999). This
is the limit set of a group generated by inversion in the six
3. "Mathematics is a language": the emergence of new concepts 15
Figure 4: Left: Pharaoh's breastplate. See the color plates. Right: the six circles generating Pharaoh's breastplate, together
with a few circles of the breastplate for reference.
circles drawn as thin lines on the small accompanying diagram.
Here, the basic osculating circles actually belong to
the limit set and do not intersect (each is the limit set of a
Fuchsian subgroup based on three circles). The other osculating
circles follow by all sequences of inversions in the six
generators, meaning that each osculator generates a clan with
its own tartan color.
By inspection, it is easy to see that this group also has three
additional Fuchsian subgroups, each made of four generators
and contributing full circles to the limit set.
Pictures such as Figure 4 are not only aesthetically pleasing,
but they breathe new life into the study of Kleinian
groups. Thurston's work on hyperbolic geometry and
3-manifolds opens up the possibility for limit sets of Kleinian
group actions to play a role in the attempts to classify
3-manifolds. The Hausdorff dimension of these limit sets has
been studied for some time by Bishop, Canary, Jones, Sullivan,
Tukia, and others. The group G that generates the limit
set gives rise to another invariant, the Poincar´e exponent
(G) = inf s :
gG
exp (-s (0, g(0))) < 
where  is the hyperbolic metric. Under fairly general conditions,
the Poincar´e exponent of a Kleinian group equals the
Hausdorff dimension of the limit set of the group. See Bishop
& Jones (1997), for example.
This is an active area of research: much remains to be done.
3 "Mathematics is a language": the
emergence of new concepts
History tells us that the great Josiah Willard Gibbs (1839­
1903) made this remark at a Yale College Faculty meeting
devoted to the reform of foreign language requirements (some
faculty issues never die!). The context may seem undignified
or amusing, but, in fact, Gibbs's words bring forth a deep
issue. To express subtle scientific ideas, one often needs new
words that are subtler than those of ordinary language.
As background, everyone knows that some great books deservedly
became classics because they provided, for the first
time, a new language in which personal emotions--that the
reader would feel but not be able to express--could be both
refined and made public. This is not at all a matter of coining
new words for old concepts but of making altogether new
concepts emerge.
Advances in the sciences are assessed in diverse ways, one
of which is the emergence of new scientific concepts. Indeed,
the facile precept that the first step is to observe then measure,
16 Chapter 2. Unsolved Problems and Still-Emerging Concepts in Fractal Geometry
sounds less compelling when the object of study is an undescribable
mess and all the measurements that readily come to
mind disagree or even seem self-contradictory. This is why
the point of passage from prescientific to scientific investigation
is often marked by what Thomas Kuhn called change of
paradigm. Sometimes this includes the appearance of a suitable
new language, without which observations could not be
made and quantified.
3.1 Fractals are a suitable language for
the study of roughness wherever
it is encountered
Let us ponder the ubiquity of the notion of roughness and its
lateness in becoming formalized. Many sciences arose directly
from the desire to describe and understand some basic
messages the brain receives from the senses. Visual signals
led to the notions of bulk and shape and of brightness and
color. The sense of heavy versus light led to mechanics and
the sense of hot versus cold led to the theory of heat. Other
signals (for example, auditory) require no comment. Proper
measures of mass and size go back to prehistory and temperature,
a proper measure of hotness, dates to Galileo.
Against this background, the sense of smooth versus rough
suffered from a level of neglect that is noteworthy though
hardly ever pointed out. Not only does the theory of heat have
no parallel in a theory of roughness, but temperature itself had
no parallel until the advent of fractal geometry. For example,
in the context of metal fractures, roughness was widely
measured by a root mean square deviation from an interpolating
plane. In other words, metallurgists used the same tool
as finance experts used to measure volatility. But this measurement
is inconsistent. Indeed, different regions of a presumably
homogeneous fracture emerged as being of different
r.m.s. volatility. The same was the case for different samples
that were carefully prepared and later broken following
precisely identical protocols.
To the contrary, as shown in Mandelbrot, Passoja & Paullay
(1984) and confirmed by every later study, the fractal dimension
D, a characteristic of fractals, provides the desired invariant
measure of roughness. The quantity 3 - D is called
the codimension or H¨older exponent by mathematicians and
now called the roughness exponent by metallurgists.
The role played by exponents must be sketched here. It
is best in this chapter to study surfaces through their intersections
by approximating orthogonal planes. Had these
functions been differentiable, they could be studied through
the derivative defined by P (t) = lim 0(1/ )[P(t + ) P(t)].
For fractal functions, however, this limit does not exist
and the local behavior is, instead, studied through the parameters
of a relation of the form d P  F(t)(dt). Here F(t) is
called the prefactor, but the most important parameter is the
exponent  = lim 0{log[P(t + ) - P(t)]/ log }.
There is an adage that, when you own only a hammer, everything
begins to look like a nail. This adage does not apply
to roughness.
3.2 Fractals and multifractals in finance
Versions of the Brownian motion model mentioned in Section
2.1 are widely used to model aspects of financial markets.
In fact, and contrary to common belief, the first analysis
of Brownian motion was not advanced in 1905 by Einstein.
In 1900 Bachelier had already developed Brownian motion
to study the stock market.
Despite this historical precedent, successive differences of
real data sampled at equal time intervals reveal even on cursory
investigation that Brownian models are very far from being
tolerable. Most visibly, (1) the width of the central band
is not constant, but varies substantially, (2) the excursions
from the central band are so large as to be astronomically
unlikely in the Brownian case, and (3) the excursions are not
independent, but occur in clumps, often when the underlying
band is widest. Figure 5 illustrates these differences.
Ad hoc fixes can account for each of these failures of the
Brownian model, but very rapidly become far too complicated
for anybody, especially for courses not addressed to experts.
The fractal/multifractal approach of Mandelbrot (1997)
is much more elegant. It provides a unified way to synthesize
all, and moreover introduces a family of parameterized cartoon
models suitable for student exploration.
Let us dwell on what is happening. Compared with welldeveloped
standard mathematical finance, the fractal cartoons
are incomparably more satisfactory. But they are far simpler
than the first stages of standard finance, so simple that
they have been immediately incorporated into both Fractal
Geometry for Non-Science Students (a course primarily for
humanities students) and Fractal Geometry: Techniques and
Applications (a course for sophomore-junior math and science
students) at Yale. In effect, students are invited to participate
in discussions between experts. They are amazed by the
realistic appearance of forgeries made with these cartoons.
Showing the class a collection of real data and forgeries always
produces interesting results. Students disagree, sometimes
with great animation, about which are real and which
are forgeries. The inverse problem, finding a cartoon to create
a forgery of a particular data set, has been a source of
interesting student projects, some quite creative. After studying
background in the different visual signatures of long tails
and global dependence, students are amazed at how slight
changes in the cartoon generator can achieve both effects.
The basic construction of the cartoon involves an initiator
and a generator. The process to be iterated consists of replacing
each copy of the initiator with an appropriately rescaled
copy of the generator. For a first cartoon, the initiator is the diagonal
of the unit square, and the generator is the broken line
with vertices (0, 0), (4/9, 2/3), (5/9, 1/3), and (1, 1). Fig-
3. "Mathematics is a language": the emergence of new concepts 17
Figure 5: Left: differences in successive daily closing prices for four years of EMC data. Right: successive differences of the
same number of steps in one-dimensional Brownian motion.
Figure 6: The initiator, generator, and first iterate of a non-random Brownian motion cartoon.
Figure 7: Left: the 6th iterate of a non-random Brownian cartoon. Right: the 6th iterate of a randomized Brownian cartoon.
ure 6 shows the initiator (left), generator (middle), and first
iteration of the process (right).
To get an appreciation for how quickly the jaggedness of
these cartoons grows, the left side of Figure 7 shows the 6th
iterate of the process.
Self-affinity is guaranteed because it is built into the process;
each piece is an appropriately scaled version of the
whole. In this case, the scaling ratios have been selected to
satisfy the square root condition of Brownian motion. The
horizontal axis denotes time t, the vertical denotes price x.
The first and third generator segments have t1 = t3 = 4
9
and x1 = x3 = 2
3 ; the middle segment has t2 = 1
9
and x2 = -1
3 . So for each generator segment we have
| xi | = ( ti )1/2.
A cartoon is unifractal if there is a constant H so that for
each generator segment | xi | = ( ti )H . If different H are
needed for different segments, the cartoon is multifractal.
The left side of Figure 7 is far too regular to mimic any real
data. But it can be randomized easily by shuffling the order
in which the three pieces of the generator are put into each
scaled copy. The right side of Figure 7 shows the result of
this shuffling, for the sixth stage of the construction.
Instead of the graph itself, it is less common but far better
to look at the increments. The cartoon sequence we have
produced has jumps at uneven intervals: some at multiples of
1/3n, some at multiples of 1/9n. Because we rarely have
detailed knowledge of the underlying dynamics generating
real data, measurements usually are taken at equal time steps.
To construct a sequence of appropriate increments, we sample
the graph at fixed time intervals and subtract successive
values obtained. Operationally, first make a list of time values
for the sampling, then find the cartoon time values between
which each sample value lies, and linearly interpolate
between the cartoon values to find the sample value at the
sample time.
Figure 8 illustrates how the statistical properties of the differences
can be modified by making a simple adjustment in
the generator. Fixing the points (0, 0) and (1, 1), we keep
the middle turning points symmetrical: (a, 2
3 ) and (1 - a, 1
3 ),
where a lies in the range 0 < a  1
2 . All pictures were constructed
from the tenth generation, hence consist of 310 =
18 Chapter 2. Unsolved Problems and Still-Emerging Concepts in Fractal Geometry
Figure 8: Generators, cartoons and difference graphs for symmetric cartoons with turning points (a, 2
3 ) and (1 - a, 1
3 ), for
a = 0.333, 0.389, 0.444, 0.456, and 0.467. The same random number seed is used in all graphs.
59,049 intervals. The difference graphs are constructed by
sampling at 1000 equal time steps.
Certainly, correlations are introduced as the point
(4/9, 2/3) is moved to the left. Of course, this is just the
beginning. More detailed study reveals relations between the
H¨older exponents and the slopes of the generator intervals,
and properties of the multifractal measure can be extracted
from the cartoons (Mandelbrot (1997)). The H-exponents
and the f () curve are much too technical for Fractal Geometry
for Non-Science Students, but are appropriate topics
for the more mathematically sophisticated Fractal Geometry:
Techniques and Applications. Even for this audience, these
are challenging concepts. Yet these simple cartoons provide
accessible introductions to some of the subtle mathematics of
multifractals.
As a last example, we mention a fascinating theorem
and a visual representation of its meaning. The Yale students
taking Fractal Geometry for Non-Science Students
in autumn of 1998 followed the development of Figure 9
with passion and helped improve it. The generator increments
t represent clock time. Viewed in clock time,
prices sometimes remain quiescent for long periods, and
sometimes change with startling rapidity, perhaps even discontinuously.
For these cartoons, clock time can be recalibrated
to uniformize these changes in price variation.
Basically, slow the clock during periods of rapid activity
and speed it during periods of low activity. Students
found the VCR a useful analog. Fast-forward through the
commercials (low activity) and use slow-motion through the
interesting bits (rapid activity).
For the cartoon generators, this is achieved by first finding
the unique solution D of | x1|D +    + | xn|D = 1,
then defining the trading time generators by Ti = | xi |D.
By changing to trading time, every multifractal price cartoon
can be converted into a unifractal cartoon in multifractal time.
Global dependence and long tails are unpacked in different
ways by converting to trading time. Specifically, global dependence
remains in the price vs. trading time record, but the
long tails are absorbed into the multifractal nature of trading
time.
Figure 9 shows a three-dimensional representation of this
conversion. Note how the clock time-trading time curve compresses
the flat regions and expands the steep regions of the
price-clock time graph. Thus the long tails of the price-clock
time graph are absorbed into the multifractal time measure.
In addition, the dependence of increments is uniformized to
fractional Brownian motion in the price-trading time graph.
That is, the conversion to trading time decomposes long tails
and dependent increments into different aspects of the graph.
Starting from a rough idea of such a representation, this
picture evolved over about a week, through discussions with
the class. Few things have excited the class as much as being
4. Conclusion 19
Clock
Time
Price
Trading
Time
Figure 9: Converting the price-clock time graph to the price-trading time graph by means of the clock time-trading time graph.
involved, as a group, in the production of a figure to explain
current research in the field.
4 Conclusion
A famous tongue-twister and test in Greek and evolution, due
to E. H. Haeckel, proclaims that "ontogeny recapitulates phylogeny."
In plainer English, the early growth of an individual
repeats the evolution of its (his, her) ancestors. As argued
elsewhere in this book (Chapter 3), this used to be the BIG
PICTURE historical justification of old math--not a welldocumented
one. But teachers ought to welcome any welldocumented
small picture version that happens to come their
way. Fractals deserve to be welcomed.