This is the time of holidays. Please enjoy the issue of Virtual Physics which is
devoted to the science and art of fractals.
Sincerely yours,
Zbigniew Koziol
Fractals, Multifractals and the Science of Complexity
M. K. Hassan
Describing natural objects by geometry is as old as science
itself, traditionally this has involved the use of Euclidean lines,
rectangles, cuboids, spheres, and so on. But, nature is not restricted
to Euclidean shapes. More than twenty years ago Benoit B. Mandelbrot
observed that "Clouds are not spheres, mountains are not cones,
coastlines are not circles, bark is not smooth, nor does lightning
travel in a straight line". Most of the natural objects we see around
us are so complex in shape as to deserve being called geometrically
chaotic. They appear impossible to describe mathematically and used to
be regarded as the "monsters of mathematics".
In 1975, Mandelbrot introduced the concept of fractal geometry to
characterize these monsters quantitatively and to help us to appreciate
their underlying regularity [1]. Fractals are more than brightly-coloured,
computer generated patterns. The coastline of an island, a river
network, the structure of a cabbage or broccoli, or the network of
nerves and blood vessels in the normal human retina can be best described
as fractals. Yet, more than twenty years after they were first introduced
there is no generally-accepted definition of a fractal, although it can
be defined loosely as a shape made of parts similar to the
whole in some sense.
The simplest way to construct a fractal is to repeat a given operation
over and over again deterministically. The classical Cantor set is a
simple text book example of such a fractal. It is created by dividing a
line into n equal pieces and removing (n-m) of the parts created and
repeating the process with m remaining pieces ad infinitum [1,2].
However, fractals that occur in nature occur through continuous kinetic
or random processes. Having realized this simple law of nature, we can
imagine selecting a line randomly at a given rate, and dividing it
randomly, for example. We can further tune the model to determine
how random is random. Starting with an infinitely long line we obtain an
infinite number of points whose separations are determined by the initial
line and the degree of randomness with which intervals were selected. The
properties of these points appear to be statistically self-similar and
characterized by the fractal dimension, which is found to increase with
the degree of increasing order and reaches it's maximum value in the
perfectly ordered pattern.
Recently, this idea has been extended to two dimensions to
understand fractals in nature that have both size and shape. That is, we
divide a rectangle randomly into four smaller rectangles and randomly remove
one of the pieces. We continue the process ad infinitum
with the remaining pieces. In this case it seems that we cannot
describe the phenomenon by a single fractal dimension - infinitely
many are required [3]. Such
phenomena are called multifractal and have become a very active
research area spanning many disciplines. Physically, it
means that it is possible to partition the resulting system into subsets
such that each subset is a fractal with its own characteristic dimension.
In this process a new feature appears: that the support on
which different subsets can be distributed is itself a fractal with one of
infinitely many possible dimensions. That is, a single
experiment for a longer time will not give any averaged
quantity of interest with a good accuracy, but a large number of
independent experiments are required. Typically, multifractal
patterns appear in systems that develop in far from equilibrium and
that do not yield a minimum energy configuration, such as diffusion limited
aggregation, or a metal foil grown by electro-deposition.
Most of our knowledge about fractals comes from computer simulations, but
creating fractals using models of fragmentation process are simple and
analytically tractable. These models can describe
the patterns that arise in random sequential deposition of a mixture of
particles with a continuous distribution of sizes on a finite substrate.
In the case of the deposition of particles of a definite size the system
clearly reaches a jamming limit when it is impossible to place further
objects without overlapping because of its strong non-Markovian and
non-ergodic nature. With a continuous distribution of sizes the system
does not reach a jamming
limit, but instead creates a scale invariant pattern that can be
described as a fractal [4,5] since the system gains its ergodic nature.
Is it possible to say when we might expect a system to exhibit random
fractal behaviour? So far, there does not seem to be a unique answer
to the question. It appears that, whenever we hopelessly fail to produce
an identical copy of a system under the same initial
condition, but each copy has the same generic form, we find a fractal object.
Two snowflakes never appear the same, but due to their generic form even
a child can recognize them. The conclusion may be that creating a
complex shape is much simpler than it appears at first sight!
References
- [1] Mandelbrot B B, The Fractal Geometry of Nature (Freeman,
San Francisco 1982)
- [2] Hassan M K and Rodgers G J Physics Letters A 208 95
- [3] Hassan M K and Rodgers G J (to appear in Physics letters
A, 1996)
- [4] Brilliantov N V Andrienko Y A Krapivsky P L and Kurths J 1996 Phys.
Rev. Lett. 76 4058
- [5] Hassan M K Comment on Ref. [4] submitted to Phys. Rev. Lett.
Topological and Hausdorff dimension
- special cases
W. Holsztynski
I'll give you just the taste.
We will see that in the case of polyhedra they coincide
with the naive notion of dimension, e.g. non-empty finite sets
have dimension zero (also countable sets), intervals have
dimension 1, solid triangles and their finite unions (in a plane
or in the 3 dimensional euclidean space) have dimension 2 - we are
talking about solid triangles (filled in), while the boundary of
any triangle has dimension 1 (it is just a union of 3 intervals);
solid 3-dimensional polyhedra like tetrahedra and cubes have
dimension 3 - that's their both topological and Hausdorff
dimensions.
Notation: below I will use the common mathematical abbreviation:
iff = if and only if
Also, a ball means always a solid ball (like in bowling, not just
the surface called sphere) of positive radius. And disks are
filled in circles, also of positive radius. Similarly, triangles
and squares are always meant solid (filled in, not just boundaries).
The topological dimension of any set in the 3-dimensional euclidean
space is always an integer: 3 or 2 or 1 or 0 or -1.
In the 3-dimensional euclidean space a set has topological
dimension 3 iff it contains a ball (e.g. every cube contains a ball
hence cubes have dimension 3). Every set in the 3-dimensional
euclidean space which is so thin or so full of holes that it
contains no ball, not even a tiny one (but of positive radius),
such a set has topological dimension equal to 2 or 1 or 0 (or
even -1 if it is the empty set).
In the 2-dimensional euclidean space (or in any plane) a set has
topological dimension 2 iff it contains a disk (a filled in circle).
Otherwise the dimension of a set in a plane is 1 or 0 (or -1 if
it is the empty set).
In the 1-dimensional space (or in any straight line) a set has
dimension 1 iff it contains an interval (of positive length, not
just a single point); otherwise it has dimension 0 (or -1 if it
is the empty set).
Also if set A is contained in set B then the topological dimension
of set A is less or equal to the topological dimension of B.
All the above statements (and many others not mentioned above) are
theorems but let's take them for granted here. They allow us
to determine the topological dimension of many sets. If a set
is contained in a plane, contains an interval and contains no
disk then it's dimension is 1 - you should have easy time to
prove it.
That much about topological dimension for a quick introduction.
Now about Hausdorff dimension. Let's consider
sets which split into pieces similar to itself. Start with
a square. You can partition it into a grid of 3 by 3 squares
which are scaled down 3 times (have sides 3 times shorter than
the original square). Thus the scale of the original square
as compared to the pieces is 3 and the number
of pieces (tiles) is 3*3. When this is the situation than
log(number of pieces)/log(scale) is the Hausdorff dimension
of the given set. Thus the Hausdorff dimension of a square,
any square is log(3*3)/log(3) = 2. The Hausdorff dimension of
any square is 2.
Like in the case of the topological dimension, if a set A is
contained in a set B then a theorem says that Hausdorff
dimension of A is less or equal to that of B. Thus every
set containing a square (or, what is equivalent, a disk)
has Hausdorff dimension greater or equal to 2.
Observe that every triangle contains a square, and every square
contains a triangle. This means that triangles have their
Hausdorff dimension both less or equal and greater or equal than
the Hausdorff dimension of a square. I.e. the Hausdorff
dimension of any triangle is equal 2, just like for squares.
We could divide a square into a grid of 10 by 10 squares.
Then the scale would be 10 and the hausdorff dimension would be
log(10*10)/log(10) = 2 - ufff, what a relief, it's the same
number 2; we get a hint that Hausdorff dimension is well
determined by splitting into similar pieces, that the result
does not depend how the splitting is done.
Now let's construct our first fractal, the so called
Sierpinski Carpet.
From the 3 by 3 grid of squares remove the
interior of the central square. Now we are left with eight
squares forming something like a ring with sharp corners.
Divide each of the eight squares into a 3 by 3 grid and remove
the interiors of the new central squares (all eight of them).
You are left with 64 smaller squares forming a funny pattern.
Do draw a picture! Divide each of the new squares into a grid...
etc, infinitely many times. What remains is Sierpinski Carpet.
You can see clearly that it consist of 8 separate smaller
Sierpinski Carpets, each being 3 times smaller. Thus the
Hausdorff dimension of Sierpinski Carpet is:
log(8)/log(3) > 1.
Sierpinski Carpet is so full of holes that it contains no disk.
Thus its topological dimension is less than 2. On the other hand
Sierpinski Carpet contains the edge (interval) of the original
square. Thus the topological dimension of the Sierpinski
Carpet is exactly 1. We see that
the topological dimension of Sierpinski Carpet = 1
is less than
the Hausdorff dimension of Sierpinski Carpet = log(8)/log(3)
is less than 2.
The Hausdorff dimension of Sierpinski Carpet is not an integer.
If Hausdorff dimension of a set is not an integer than such a set
is always a fractal. But even when it is an integer but larger
than the topological dimension then it is still a fractal set.
Now you can construct a lot of fractal sets and you can compute
their dimensions. E.g. you may remove in the above construction
not the interiors of the central squares but of the upper right
corners. You will get a very different fractal but it will have
the same dimensions as Sierpinski Carpet. Or you can split
a square into a 2 by 2 grid, remove the interior of the upper
right hand square. Then iterate. This time the Hausdorff
dimension of your fractal will be log(3)/log(2); it will be
smaller than for Sierpinski Carpet but still larger than 1.
Instead of a square you may divide a triangle into 4 similar
triangles. By removing the interior of one, the iterating
the construction you will get still different fractals of
dimension log(3)/log(2).
This method provides you with special fractals only but you
already got plenty variety.
Mathematics and science considers also euclidean spaces of dimension
4,5,... and just any, even infinite (in the infinite dimensional
case they are called hilbert spaces to honor their inventor David
Hilbert). Dimension theory is developed for sets in even more
general spaces, but some statements valid in the euclidean case
might fail in general.
I didn't provide the definition of either dimension because
most of the non-specialist would get bored and specialists
already know the definitions (hopefully :-).
Wlod
Fractal Dimension Estimation With Wavelet Packets
Cameron L. Jones
Centre for Applied Colloid and BioColloid Science
Swinburne University of Technology, School of Chemical Sciences
P.O. Box 218, Hawthorn, Victoria, 3122 Australia
Introduction
Digital images contain large amounts of useful data in addition to visual and perceptual information. The principal aim of the visual image is to describe, characterize, document, or clarify feature relationships. In order to extend the utility of visual information beyond subjective interpretation, one may perform measurements and look for spatial correlations between image features based on size, distance, colour, shape, number, dispersion, angle, or depth of field. This paper presents a general overview of a new technique which can be used to evaluate two dimensional digital images to determine the global Fractal Dimension, D. The Fractal Dimension is a useful method to quantify the complexity of feature details present in an image. For 2 dimensional images: 1<D<2. Several image analysis methods can be used to evaluate fractal complexity (1), although most methods contain inherent problems and biases, such as lack of convergence, insensitivity to upper and lower D boundaries, over and underestimations of the theoretical D or computer intensive algorithms. A new method called: Wavelet Packet Analysis (2-D WPA) can be used to estimate the global Fractal Dimension which does not suffer from the above methodological, algorithmic or implementation problems.
What is a Wavelet?
Wavelets are similar to, but an extension of Fourier analysis and the Wavelet Transform is computationally similar in principle to the Fast Fourier Transform (FFT). The FFT uses cosines, sines and exponentials to represent a signal, and is most useful for representing linear functions. Since many 1- and 2-D signals display non-linear, chaotic or fractal behaviour, Fourier analysis is less suitable for analyzing such signals. Wavelets offer a new method to analyze complex signals using a special filter called a wavelet. Each wavelet function oscillates about zero which provides good spatial localization. The aim of the wavelet transform is to 'express' an input signal into a series of coefficients of specified energy. This partitions the image into a series of multiresolution components, capturing fine and coarse resolution features at different scales. This process detects how spatial features scale in the horizontal, vertical, and diagonal directions. The discrete numbers associated with each coefficient contain all the information needed to completely describe the signal provided one knows which analyzing wavelet was used for the decomposition. The wavelet transform partitions a signal or image with respect to spatial frequency. This is achieved by filtering the signal with a pair of dyadic orthogonal filters called a quadrature mirror filter (QMF). A QMF comes in a pair: termed a 'father' wavelet and a 'mother' wavelet. The father wavelet provides an approximate or blurred version of the signal at successive resolutions, while the mother wavelet captures the detail at each resolution. The wavelet transform of a 2-D signal returns a coefficient matrix which maps all the spatial relationships at multiple scales in the horizontal, vertical and diagonal directions. The coefficient matrix is however a redundant representation, since certain coefficients are large while others are negligible, and practically zero.
Wavelet Packets
Wavelet Packets are a subset of the generalised discrete wavelet transform, and offer greater flexibility for the detection of oscillatory or periodic behaviour. Variables which can be manipulated include frequency, position, and scale. This is important since for many 1- and 2-D signals, one may be interested in 'fractal scaling'. Such signals may also combine stationary and non-stationary components. In turn, it has been shown that many stochastic (real-world) fractals display either repetitive (persistent) or alternate (antipersistent) scaling trends (2). This behaviour equates to oscillatory and periodic motion. An example of a symmetrical wavelet filter (s.8) with support length of 7 is shown in Figure 1.
Figure 1. A symmetrical s.8 wavelet filter.
A statistical technique called the best basis is used to find the optimal subset of coefficients (from within the redundant wavelet packet transform) to represent the signal. This subset contains a listing of wavelet packet energies which can be ordered (from highest to lowest) and plotted against index position. A least squares linear regression is plotted through the points to estimate fractal scaling.
Figures 2 shows an example image of a 'box', and will be used to illustrate how wavelets "see" an image.
Figure 2. Example image of a box. 128x128 pixel size.
We will perform the (i) Discrete Wavelet Transform and the (ii) Wavelet Packet Transform on this image using the symmetrical s.8 wavelet and 3 transform levels. The respective wavelet transform or wavelet packet transform matrices are shown in Figure 3 for the box image.
Figure 3. On the left is the Discrete Wavelet Transform coefficient matrix for the box image. On the right is the Wavelet Packet Transform coefficient matrix.
Both transforms clearly show how information is localized with respect to scaling in the horizontal, vertical, and diagonal directions. Furthermore, the Wavelet Packet Transform identifies additional levels of scaling compared with the Discrete Wavelet Transform. The rest of this paper will focus on the Wavelet Packet Transform and how it can be used to estimate fractal scaling in two dimansions. A paper detailing the use of the Wavelet Packet Transform to estimate self-affine scaling, and multifractality for one dimensional signals is also available (2). Downloadable software is available to perform Fractal Analysis on image arrays (3), although users must be running S-Plus and S+Wavelets.
Wavelet Packet Fractal Analysis
The next example shows
how to apply 2-D WPA to analyse the branching complexity seen in a filamentous fungal colony. Different nutrient substrates induce various morphological branching responses (4). Filamentous fungi grow by repeated branching and extension from the hyphal tips. It has been shown that protein expression and enzyme secretion across the cell wall occurs preferentially at the hyphal tip. Our interest focusses on the interrelationship between branching and enzyme expression. Measurement of the Fractal dimension to index branching was used as a rapid test to optimise nutrient levels which caused an increase in branching aginst untreated controls. Figure 4 shows an image of a 24 hour old colony of Pycnoporus cinnabarinus growing on Newspaper medium containing 2% cellobiose. This fungus metabolises the residual lignin, cellulose, and hemicellulose present in the newspaper as a carbon source. The cellobiose acts as a paramorphogen to increase the amount of branching.
Figure 4. Image of P. cinnabarinus after 24 hours growth on a medium containing powdered newspaper (0.2%) and cellobiose (2%). Note the apical and sub-apical branching
We now perform a level 3 Wavelet Packet Transform on this image using the s.12 wavelet filter. This yields the following wavelet packet coefficient matrix. Then we plot selected wavelet packet coefficients which belong to the best basis. The best basis is an algorithm which searches the wavelet packet coeficient matrix and finds the optimal subset of coefficients which best represents the image at all scales and resolutions. This graph is plotted on the right.
Figure 5. Wavelet Packet Transform coefficient matrix for
Figure 4. A pseudocoloring function has been used to highlight
coefficient energy magnitude. Highest energies are shown in yellow.
On the right, is a plot of selected coefficient energies belonging
to the best basis. These are sorted from highest to lowest and a
least-squares linear regression is plotted to determine the slope.
From the slope, the global or mean Fractal Dimension of branching
complexity can be determined. For this image the global Fractal
Dimension, D=1.549 r2=0.938. Deviation from linearity
(i.e. a low r2 value) is expected since the graph is
similar to a Power Spectrum FFT approach.
Conclusions
2-D WPA is a useful method to estimate the global Fractal Dimension of object features in digital images. This method has been applied principally to binary images, although preliminary work suggests that it is also appropriate for measuring self-affine scaling in grey-scale images. Furthermore, the technique is valid for Sierpinski fractal carpets. This provides a new tool to estimate connectivity and percolation phenomena in digital images (5).The algorithm can also be modified to provide a multifractal interpretation following (2). Further information about wavelets and their various applications can be found in (6).Acknowledgements
This work was supported by an ARC Collaborative Industry grant with Visy Paper Recycling
References
- Jones, C.L., Lonergan, G.T. and
Mainwaring, D.E. (1993) A Rapid Method for the Fractal Analysis of Fungal Colony Growth using Image Processing. Binary. 5: 171-180. Abstract available online.
- Jones, C.L., Lonergan, G.T. and Mainwaring, D.E. (1996) Wavelet Packet Computation of the Hurst Exponent. J. Phys. A: Math. Gen. 29: 2509-2527. Online preprint minus color images in Word for Windows version: 2 or 6 or Adobe Acrobat format.
- Jones, C.L. (1996) Wavelet Packet Fractal Analysis - Software Operating Instructions. A series of 7 software functions which calculate the mean or global Fractal Dimension of 2-D objects in digital images. Requirements include: S-Plus and S+Wavelets. Programs available for the following image sizes: 64x64, 128x128, 192x192, 256x256, 384x384, 512x512, and 300x200.
- Jones, C.L. (1996) 2-D Wavelet Packet Analysis of Structural Self-Organization and Morphogenic Regulation in Filamentous Fungal Colonies. Complex Systems Conference - "From Local Interactions to Global Phenomena" July 14-17, 1996 Charles Sturt University; Albury - Wodonga, N.S.W. Australia. Also in: Complex Systems 96 - From Local Interactions to Global Phenomena. (Stocker, R., Jelinek, H., Durnota, B. and Bossomaier, T. eds.) IOS Press, Amsterdam, 1996. pp. 12-23.
- Jones, C.L., Lonergan, G.T. and Mainwaring, D.E. (1995) Sierpinski Fractal Analysis and Percolation Threshold Implications in Fungal Colonies. Bioimages 3(2): 71-84. Abstract available online.
- Hubbard, B.B. (1996) The World According to Wavelets - The Story of a Mathematical Technique in the Making. (Massachusetts: A.K. Peters).
Hunting for Fractals
A popular representation of fractal geometry lies within the
Mandelbrot set, named after its
creator
Benoit B. Mandelbrot
who coined the name "fractal" from the Latin fractus
or to break.
At a web page of
Department of Atomic Physics of Eötvös University, Budapest,
we read:
-
Fractals are complicated geometrical objects which can be described in terms of non-integer (fractal)
dimensions. For the last decade fractals have been shown to represent the common aspects of many
complex processes occurring in an unusually diverse range of fields including physics, mathematics,
biology, chemistry and earth sciences.
- Using fractal geometry as a language in the related theoretical, numerical and experimental
investigations, it has been possible to get a deeper insight into previously intractable problems.
Through the application of such concepts as scale invariance, self-affinity and multifractality a better
understanding of such phenomena as aggregation, turbulence, percolation, biological pattern
formation and granular flows has emerged.
Hence, lets surf the net in hunting for fractals!.
Searching the Lycos internet catalog for WWW documents containing words
fractal AND physics
gives a list of 104 URL addresses. Searching condensed matter abstracts of LANL archive for
occurrences of
fractal yields 28 articles submitted there during this year only.
We realize quickly that finding some sort of order and good quality materials
is by no means an easy task. Some people try, however, to accomplish that.
Here is an example. A large collection of links to fractal resources is maintained by
Tim Norman.
Applications
Science of Nanotechnology
Charles Ostman from Nanothinc Inc., writes in
Fractal Dimensions about nanoscale applications:
-
Fractal geometries and fractal-based processes can be utilized for a variety of
applications, including image compression, content addressable feature recognition, and
other forms of image analysis and processing. Further, fractal geometries can be useful as
data sets for complex three dimensional modeling and content creation. A discussion of
various fractal-related applications relevant to nanotechnology follows.
Content-addressable feature recognition, spectral/topographical feature cue
sets for comparative analysis, and image data compression and enhancement
of AFM and STM image data via fractal image compression techniques.
Science and Art
-
Refractal Design produces limited edition, investment quality fractal jewelry in gold,
platinum, diamond, sapphire, silver, and ruby. Our patents-pending MicroArt® process
is a reapplication of semiconductor science and technology for the design and
manufacture of art.
Education
-
The Fractal Microscope
is an interactive tool designed by the
Education Group at the National Center for
Supercomputing Applications (NCSA) for exploring the Mandelbrot set and other
fractal patterns. By combining supercomputing and networks with the simple
interface of a Macintosh or X-Windows workstation, students and teachers from
all grade levels can engage in discovery-based exploration. The program is
designed to run in conjunction with NCSA imaging tools such as DataScope and
Collage. With this program students can enjoy the art of mathematics
as they master the science of mathematics. This focus can help one
address a wide variety of topics in the K-12 curriculum including scientific
notation, coordinate systems and graphing, number systems, convergence,
divergence, and self-similarity.
Designing Backgrounds for WWW pages
This is a well without a bottom. Please have a look to my
Home Pages of SuperPhysicists
and hundreds of other pages created and maintained by me.
The backgrounds of nearly all of them are based on some sort of
fractal images. There are more designers of web pages who have noticed
the opportunities:
Neal A. Kettler,
Ruslan Prozorov, and many more...
Pleasure
Fractal Tools
Compiled by Zbigniew Koziol
Do not complain
That I am insane
The things I say
| | |
Fractal Curve
|
|
You're working hard
And you like to play
And you also drew
A fractal line
| |
| |
It fooled me once
From nine to five
And fooled again
From five to nine
| |
H. Florida
1985
| | |
The lines revive
Between the ocean and the sky
Between the day and night
And you claim
That I am insane
|
Virtual Physics:
a forum for virtual meetings of scientists and students involved in a research activity on
THE SOLID STATE PHYSICS AND SUPERCONDUCTIVITY
Editors:
Marcel Ausloos, ausloos@gw.unipc.ulg.ac.be, Institut de Physique B5,
Université de Liège, Sart Tilman, B-4000 Liège, Belgium, tel. (+32 41) 66 37 52
Cameron L. Jones, CJONES@swin.edu.au, Centre for Applied Colloid and BioColloid Science
Swinburne University of Technology, P.O. Box 218 Hawthorn, Victoria, 3122 Australia, tel. +613 9214 8935
Zbigniew J. Koziol (Editor-in-Chief), WebEx@ra.isisnet.com, WebExperts Inc.,
2-6032 Compton Ave., Halifax, Nova Scotia, B3H 1E7 Canada, tel. (902) 423 2149
Michal Spalinski, Michal.Spalinski@fuw.edu.pl, Institute of Theoretical Physics,
Warsaw University, Hoza 69, 00-681 Warsaw, Poland, tel. (+48 2) 628 3031
Virtual Physics URL address: http://www.isisnet.com/MAX/vp.html
To subscribe a F R E E e-mail
version or submit materials for publication, write to Zbigniew Koziol.
Copyright (C) 1996 by Zbigniew Koziol.
this copyright notice concerns the whole of the Virtual Physics
edition but not specific articles published there which are
property of their respective copyright holders
No responsibility is assumed by the publisher for any damage to
persons or property as a matter of the product liability, negligence
or otherwise, or from any use of methods, instructions or ideas
contained in the material herein. The opinions expressed in this
publication do not necessarily reflect the opinions of the Editor
and certainly they have nothing to do with WebExperts Inc.
![Back to the main page of Virtual Physics ]](../images/vphome.gif)
![[ PhysicsPage ]](../images/v5.gif)
![[ SuperPage ]](../images/v1.gif)
![[ WebExperts ]](../images/we5.gif)
![[Virtual Physics]](../images/vp.gif)
Letter from the Editor
