Ubiquitous
Mr. Karl Castleton
Pacific Northwest National Laboratory
Why discuss recursion?
• Fundamental to many
aspects of life
• Simple to state
– Most recursive definitions are
very clean
– Something being
• Intuitive in its definition
defined by itself
– Other than the strangeness of
– and some
definition
transformation between
• Difficult to “use” properly
levels
– Most software engineers shy
– and some criteria to
away from its use even though
stop recursion
it is very useful
Fibonacci Sequence as an example:
• f(n)=f(n-1)
• + f(n-2)
• Where
• f(0)=1 and f(1)=1
• n>1
Generates the series
1 1 2 3 5 8 13 …
• Defined by itself
• and a
transformation
• and a stopping
condition
– Defines first few
elements and
stopping condition
A software implementation:
• f(n)=f(n-1)
• + f(n-2)
• Where
• f(0)=1 and f(1)=1
• n>1
Generates the series
1 1 2 3 5 8 13 …
• int fib(int n) {
if (n>1)
return fib(n-1)
+fib(n-2);
else return 1;
}
The Giant’s Shoulders
• Douglas R. Hofstadter in 1979 a book
called “Goedel, Escher, Bach: An Eternal
Golden Braid” (GEB for short)
– A book I stumbled across but in its day was
quite a sensation
– Many of the concepts presented here stem from
extensions of ideas presented in this book
– By the way this book pre-dates the modern
personal computers and software tools
But lets not focus on Math and
Computer Science
• Recursion occurs in many other aspects of
life
– We will start as far from Math and Computer
Science as we can get and then return to them
for some interesting examples
– I have focused on cultural artifacts more than
behaviors. So I do not show any recursive
behaviors but we will find artifacts that seem to
have recursive structures.
On to Music and Limericks
Does music contain recursion
• A key is a series of notes that start and stop on the
same note (although the note could be in a
different octave)
• In GEB, Hofstadter explains that “we hear music
recursively-in that we maintain a mental stack of
the keys”
– This is to say when we start hearing a melody you tend
to wait for the starting note to come again
– If we diverge from the melody our brains wait for a
return to the start of diversion before we expect to hear
the original melody end
– An experiment: start humming a scale and stop
somewhere in the middle. It does not feel quite right
does it.
Self Defined, Transformed, and
Stopped
• Key is a series of keys that we expect to
hear
• They can contain time-compressed copies
of itself, or shifted in frequency
• and is expected to end at each level of
recursion
• Some of the most challenging music to play
comes from this nesting a key in a key.
What about the words?
• Lets look at the typical
limericks
– There was an Old Man with a
beard,
Who said, "It is just as I
feared! Two Owls and a Hen,
Four Larks and a Wren,
Have all built their nests in my
beard!"
[Edward Lear]
From: Self-similar syncopations:
Fibonacci, L-systems, limericks
and ragtime by Kevin Jones
• Directly correlates the structure
to the Fibonacci sequence
– There is a “tree” that shows a
recursive structure that in the
end generates the di-dum-didum rhythm of the limericks
• Author has degrees in
Mathematics, Computer
Science and Music
• He then goes on to show that
ragtime music sometimes
shows the same characteristics.
• Visit:
http://plus.maths.org/issue10/fe
atures/syncopate/
For a more complete review
• Self-Similarity (another
name for recursion) is the
bridge between sound and
music.
• Self-similar Synthesis:
On the Border Between
Sound and Music Masters
Thesis at MIT for
Shahrokh D. Yadegari
• The coherencies which
exist in music have to
agree with each other in
any scale and dimension
in which they are being
perceived.
– http://crca.ucsd.edu/~syade
gar/MasterThesis/node25.ht
ml
On to Art
Lets start with older cultural art.
• Rangoli: Ritual
Patterns of Rural India
– For sacred and festive
places
– Drawn by tracing lines
around a grid of dots
• Dr. Ektare pointed out
this example
Pattern is repeated on
a smaller scale
Sometimes it is distorted
The Mandala is also recursive
• A mandala is an
imaginary palace that
is contemplated during
meditation in Tibetan
Buddhism
• A very complex
design repeats itself in
four rotations but also
in the nested palace in
a palace
The border of this mandala also has recursive properties
A more modern example (Escher)
• Image below was
explicitly constructed
for its recursive
properties.
• M.C. Escher has a
number of examples
of recursion.
What would the
close-up of his
eye reveal
60’s Music Videos
• Video feedback was used to
produce many.
– Basically you point a video
camera at a monitor and then
add some light source
– This was produced (by me)
using a net cam and a monitor.
The light source is the mouse
• The frame you see is the frame
you saw plus the distortion of
the alignment of the camera,
and processing it stops at the
limit of the camera resolution
On to Nature
There is an intentional rotation
in the alignment between camera
and monitor
Nature uses recursion frequently
• Nature has many fractals
– Fractals are self-similar
structures
• This shell is related to the
example video because the
shell is a simple
enlargement and rotation
of the previous shell.
• The shell started as small
as possible (for the
creature)
Flowers and plants also have this
self-similar behavior
• Plants are frequent
examples
• Fractal ferns are a
classic examples of
recursive definition
• Vist:
http://www.geocities.c
om/CapeCanaveral/Ha
ngar/7959/fractalapple
t.html
Even Sensing Nature Needs
Recursion
• Sensing subaudible sound
requires a special instrument
• Notice the eight sided star with
the eight sided stars…
• That is the sensor used to sense
sub-audible sound that
elephants and weather produce
• An optical fiber infrasound
sensor: A new lower limit on
atmospheric pressure noise
between 1 and 10 Hz
• Visit: http://klops.geophys.unistuttgart.de/~widmer/JASAfinal
.pdf
On to Social Structures and Geography
Nesting of Elected Officials with
Elected Officials
• The Federal representation
structure (Executive,
Legislative, Judicial,
Press) is repeated at the
state and local levels.
– With some minor
transformations at each
level (term limits, required
age, etc.)
• This basically gives a
structure that can govern
many people with
relatively few individuals
• It is not unlike the flower
in the nature section
• Does the structure even
extend into your
household?
• Scientific and technical
communities tend to have
nested leadership
organization as well.
Structure of the Electrical Grid
• The power grid is
essentially a binary
tree from the power
plant to the consumer
• Here the
transformation steps
voltage up as losses
occur over the
transmission wires
How much coast line do we have to
protect/govern?
• A rather simple question
posed to Benoit B.
Mandelbrot
• The solution he came up
with was it depends
– What scale do you want to
measure on
– Can really be any answer
you wish
– Any structure at one scale
has equivalent structures at
a smaller scale
Back to Computer Science
Many Divide and Conquer
Algorithms are Recursive
• Binary Search
• Fast Fourier
Transformation
• Recursive Decent
Parsers
– Backus Naur Form is a
way of encapsulating
recursive language
structures
• And many many more.
Web Page Design
• Nesting of
– Styles within styles
– Tables within tables
– Lists within lists
• A fundamental
concept to the layout
of web-pages
Back to Mathematics
The Mandelbrot Set
• Simple Computation of
Complex Numbers
– Zn+1=Zn2+C
• Has unlimited complexity
• Coloring typically done by
the number of
computations before
divergence away from
0+0i
• Julia sets are a peek at the
complexity of a single
point of the Mandelbrot
set
Another Approach to the Integration
of a Function
• Consider the definition of
integration where you take
smaller and smaller dt
until you reach the limit
• What “integrate” means
you simply compare the
area of one trapezoid
including a and b to total
area of two trapezoids a
and m plus m and b. If the
difference is “significant”
“integrate” a to m add to
“integrate” m to b stop
when you hit the precision
required
a
b
a
b
-dt-
a
a
b
m
b
So where is Godel?
• Godel’s incompleteness
theorem is a proof about proofs
that uses recursion
• The ability to be comfortable
with the concept of one level
addressing something about a
subsequent level is key
• I will Math professors explain it
more thoroughly or visit:
http://home.ddc.net/ygg/et
ext/godel/godel3.htm
Some interesting contrasts
• The recursive nature of the physical world may be
the link Wolfram should have used to make more
clear the connection between simple programs and
larger physical effects
– This is a frequent criticism of ANKOS
• Recursive structures can typically be done in an
iterative (step by step) structure but sometimes
looses the essence of the concept.
– The integration example is like this
– The recursive implementation is much easier to
implement in a computer
Conclusions
• Recursive relationships are “everywhere”
– If you are mathematician you should consider
recursive approaches and definitions (Godel
did)
– If you are a computer scientist you should not
be intimidated by a recursive algorithm
• I hope you enjoyed the tour of just a few
examples of