Ubiquitous

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Ubiquitous

• Mr. Karl Castleton
• Pacific Northwest National Laboratory

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Why discuss recursion?

• Fundamental to many aspects of life
• Something being defined by itself
• and some transformation between levels
• and some criteria to stop recursion

• Simple to state
• Most recursive definitions are very clean
• Intuitive in its definition
• Other than the strangeness of definition
• Difficult to use properly
• Most software engineers shy away from its use
  even though it is very useful

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Fibonacci Sequence as an example

• f(n)f(n-1)
• f(n-2)
• Where
• f(0)1 and f(1)1
• ngt1
• 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

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A software implementation

• f(n)f(n-1)
• f(n-2)
• Where
• f(0)1 and f(1)1
• ngt1
• Generates the series
• 1 1 2 3 5 8 13

• int fib(int n)
• if (ngt1)
• return fib(n-1)
• fib(n-2)
• else return 1

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The Giants 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

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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
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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.

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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.

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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-di-dum 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/features/sync
  opate/

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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/syadegar/MasterThesis/node25
  .html

On to Art
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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
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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
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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
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60s 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

There is an intentional rotation in the alignment
between camera and monitor
On to Nature
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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)

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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.com/CapeCanaveral/Hanga
  r/7959/fractalapplet.html

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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.uni-stuttgart.de/wid
  mer/JASAfinal.pdf

On to Social Structures and Geography
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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.

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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

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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
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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.

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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
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The Mandelbrot Set

• Simple Computation of Complex Numbers
• Zn1Zn2C
• Has unlimited complexity
• Coloring typically done by the number of
  computations before divergence away from 00i
• Julia sets are a peek at the complexity of a
  single point of the Mandelbrot set

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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

-dt-
m
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So where is Godel?

• Godels 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/etext/godel/god
  el3.htm

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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

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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