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Mountain Video

Showed video of mountain landscape generated by the 4k file on this page: http://pouet.net/prod.php?which=52938. Mountain Video. A Random Talk About Random. Dave Feinberg. Which has more information?. The outcome of 2 coin flips: heads heads The outcome of 3 coin flips: tails tails heads.

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Mountain Video

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  1. Showed video of mountain landscape generated by the 4k file on this page: http://pouet.net/prod.php?which=52938 Mountain Video

  2. A Random Talk About Random Dave Feinberg

  3. Which has more information? The outcome of 2 coin flips: heads heads The outcome of 3 coin flips: tails tails heads

  4. Which has more information? The outcome of 10 coin flips: "HHTTHTHTTT" A strand of 10 DNA bases: "ATTGACATGG" 10 decimal digits: "7523104698"

  5. Units of Information bit = binary digit a single "0" or "1" outcome of 1 binary decision like a coin flip ten coin flips -> ten bits

  6. Bits • How many bits do I need to store the outcome of one home pregnancy test? • A home pregnancy test reveals 1 bit of information. It may be the most important life-changing bit of information, but it's still just 1 bit.

  7. One Itty Bitty Guy

  8. Representing in Binary "HHTTHTHTTT" "H" = 0 "T" = 1 0011010111 How can we represent "ATTGACATGG" in binary?

  9. Representing in Binary 2 bits to represent 1 DNA base "A" = 00 "C" = 01 "G" = 10 "T" = 11 ATTGACATGG 00111110000100111010 How many bits?

  10. Representing in Binary bits per symbol = log2(possible symbols) 4 symbols: ACGT log24 = 2 bits per symbol

  11. Representing in Binary How much information in "7523104698"? 10 possible symbols log210 = approx 3.32 bits per symbol 10 decimal digits = approx 33.2 bits Bogus! How many bits does it really take to represent 10 possible symbols?

  12. decimal digits in 4 bits each 0 = 0000 5 = 0101 1 = 0001 6 = 0110 2 = 0010 7 = 0111 3 = 0011 8 = 1000 4 = 0100 9 = 1001 Seems wasteful. Can we do better?

  13. decimal digit = 3 or 4 bits 0 = 000 5 = 101 1 = 001 6 = 1100 2 = 010 7 = 1101 3 = 011 8 = 1110 4 = 100 9 = 1111 What number is "0101011100"? On average: 3.4 bits per digit.

  14. Which contains most information? • Outcome of 6 coin flips: "HHTHTT" • 3 DNA bases: "ATG" • 2 decimal digits: "74"

  15. Information • more bits = more information • Right?

  16. Memorizing Volunteer to memorize 8 bits • 00011110 Volunteer to memorize 50 bits • 00000000000000000000000000000000000000000000000000 Which is easier to memorize? Which contains more information?

  17. Memorizing 00000000000000000000000000000000000000000000000000 Another volunteer to memorize 50 bits 00100011001110100110110001010001100010101001111001 Which is easier to memorize? Which contains more information? Why?

  18. Information: another definition • Amount of information = length of shortest program that outputs those bits.

  19. Write a program to print 00000000000000000000000000000000000000000000000000 for i in range(50): print "0",

  20. Write a program to print 00100011001110100110110001 010001100010101001111001 print "00100011001110100110110001 010001100010101001111001"

  21. Memorizing Programs print "00100011001110100110110001 010001100010101001111001" is harder to memorize than for i in range(50): print "0",

  22. Therefore 00100011001110100110110001 010001100010101001111001 contains more information than 00000000000000000000000000000000000000000000000000

  23. Information • More random = more information

  24. Pi and Information • 3.14159265358979323846264338327... • How much information is stored in the digits of pi?

  25. Calculating Pi pi = 4/1 - 4/3 + 4/5 - 4/7 + ...

  26. Calculating Pi sign = 1 n = 1 pi = 0 while True: pi = pi + sign * 4.0 / n print pi sign = 0 - sign n = n + 2

  27. Information in Pi The digits of pi are output by a short program. Therefore, pi does not contain very much information. (Then what does???)

  28. Programs That Print Numbers Write a program to print 0.1 print "0.1"

  29. Programs That Print Numbers Write a program to print 0.11111... print "0.", while True: print "1",

  30. Programs That Print Numbers Write a program to print 0.121212... print "0.", while True: print "12",

  31. Programs That Print Numbers Write a program to print 0.12112111211112111112... print "0.", ones = 1 while True: for i in range(ones): print "1", ones = ones + 1 print "2",

  32. Programs That Print Numbers • Although 0.121121112... requires an infinite number of decimal digits, • It can be printed by a program of finite length.

  33. Programs That Print Numbers • Are there any numbers that cannot be printed by a computer program? • Yes!

  34. Programs That Print Numbers • A number that can be printed by a computer program is called a computable number. • A number that cannot be printed by a computer program is called an uncomputable number. • Are there a lot of uncomputable numbers?

  35. If I point to a random point on the number line, what is the probability that I will point to an integer? A rational (fraction)? A computable number? The Number Line -1 0 1 2

  36. Information and Science • more random = more information. • randomness = disorder. • In science, what do we usually call the measure of the amount of disorder in a system? • information = randomness = disorder = entropy • 2nd Law of Thermodynamics: In a system, a process that occurs will tend to increase the total entropy of the universe. • Does this tell us anything about information?

  37. Something Different ... • We'll come back to random/information ... • Let's make some pictures

  38. Tiling Squares Rewrite rule: Add square to long side.

  39. Another Redraw Rule Shrink and arrange 3 copies

  40. Redrawing ...

  41. Redrawing ...

  42. Redrawing ...

  43. Redrawing ...

  44. Redrawing ...

  45. Fractals • A fractal is: • "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"--a property called self-similarity. --Wikipedia • Fractals are recursive structures.

  46. Triangles? • Why is the sierpinski triangle full of little triangles? Where did the triangles come from?

  47. Squares ... Shrink and arrange 3 copies

  48. Squares ... Shrink and arrange 3 copies

  49. Squares ... Shrink and arrange 3 copies

  50. Squares ... Shrink and arrange 3 copies

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