# Position of Inequality Or POI Metrics

Position Of Inequality or POI metrics

------

also called

Hawaiian Metrics or HOBBit metrics

======

Any time there are weighted categorical bands

(There are always "categorical" bSQ bit bands

after fully decomposing)

Rank order them in decreasing order of weight

E.g.,

Hobbit dist. = Max Position Of Inequality (MaxPOI)

Manhattan = Sums Weights of Pos Of Ineq (SumPOI or L1-POI)

Euclidean = SQRT(Sum SQRs of Wts of Pos Of Ineq) (L2-POI)

Minkowski-q = q-RT(Sum (Wts of Pos Of Ineq)^q) (Lq-POI)

Maximum = ??? Hobbit

These are the Hawaiian Metics.

======

When there are more than 2 categories (more than just 0 or 1),

1. Attributize the categories (ala MBR) or code them numeric?

2. Use Hawaiian Distance?

How might we relieve some of the "problems" of HMs?

- eccentricity of Hobbit rings?

- is it a problem? (see below under hobbit rings)

- thickness of Hobbit rings?

- is it a problem? (see below under hobbit rings)

assuming both are problems, how can we relieve them:

FIBONACCI HAWAIIAN METRICS and HOBBIT RINGS

======

If we think of binary (and decimal) digital coding of a number:

Start with binary base sequence, B = {..., 2^n, ..., 2^1, 2^0 }

(decimal base sequence, D = {..., 10^n, ...,10^1, 10^0}

Remove the largest base <= number (digit = # of copies removed)

Repeat with number := remainder until remainder = 0.

Code using Fibonacci sequence as base sequence (not B or D)

Fibonacci base sequence: ...233 144 89 55 34 21 13 8 5 3 2 1 1

( ni = n(i+1) + n(i+2) )

For byte data:

Index:13 12 11 10 9 8 7 6 5 4 3 2 1 0

Pos: 11 10 9 8 7 6 5 4 3 2 1 0

Fib: 233 144 89 55 34 21 13 8 5 3 2 1 _1__ 0

NUM seed

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 1

2 0 0 0 0 0 0 0 0 0 0 1 0

3 0 0 0 0 0 0 0 0 0 1 0 0

4 0 0 0 0 0 0 0 0 0 1 0 1

5 0 0 0 0 0 0 0 0 1 0 0 0

6 0 0 0 0 0 0 0 0 1 0 0 1

7 0 0 0 0 0 0 0 0 1 0 1 0

8 0 0 0 0 0 0 0 1 0 0 0 0

9 0 0 0 0 0 0 0 1 0 0 0 1

. . . (find the rest in the appendix)

More hobbit rings, thinner and better centered ;-))))

"Thin-ness of rings" needs to be studied and quantified.

To push the idea a little further, consider a Fibonacci starter

value of .1 rather than 1 (results in 16 bit representations and

results in more plateaus which should be even thinner ;-)))

159 98 61 37 23 14. 8.9 5.5 3.4 2.1 1.3 .8 .5 .3 .2 .1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0

2 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0

3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1

4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1

5 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0

6 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0

7 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0

8 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1

. . . (find the rest in the appendix)

Taking seed to be 1/B where B is any of Fibonacci {1,2,3,5,..}

gives a representation base which will always include 1

(include both copies of seed??).

******75 46 28. 17. 11 6.8 4.2 2.6 1.6 1 0.6 0.4 0.2 0.2

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

num_

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1

2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1

3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1

4 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1

5 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1

6 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1

7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

. . . (find the rest in the appendix)

Fibonacci base provides multiple representers for most numbers.

For seed=s defines s-Canonical Fibonacci representation (sCF)

======

(when s=1 we will drop the s)

s-Packed Fibonacci (sPF) will be the representation with 1-bits

======moved as far right as possible, for s=1:

(see appendix for listing and CF-to-PF conversion)

Data Mining classification method based on Hawaiian Metrics

------

1. Form basic CFPtrees and basic PFPtrees (canonical and packed)

2. For a unclassified sample, x, form each hobbit ring mask as

CFring-i OR PFring-i = Hring-i

3. Apply Hring-i to both PFtrees and CFtrees

(OR the results together)

4. Vote weighting ratios should be according to the fibonacci

index of the ring (i.e., inner ring has index = 1, next ring

has index = 2 ...)

- This is a matter of using both represetations (CF and PF)

for both the sample and the training vectors.

- This will be more accurate than just applying the ORed

ring mask to CFtrees, but will be slower. That's the

tradeoff. "How much slower?" and "How much more accurate?"

General Note on Classification:

------

1. Most lazy classifiers (ours for example) may not be good

for finding edges! or separation boundaries (ala SVMs).

2. Most lazy classifiers are good for continuous classification

(Class Attribute is continuous or approximately so -

many classes, numeric)

- Most lazy classifiers are good at fitting, not edge

detection because they look for "fit" based on

continuity, not sudden change (edges).

3. It's amazing we did so well on the KDDcup02 since we used

a Lazy classifier (sort of) and the Class attribute was

definitely not continuous.

4. SVMs are good at edge detection

- finding support vectors is like finding the glow lines

of boundary between two classes (edges of the classes)

5. A new direction for binary or non-continuous classification:

Use podium-type approaches to find "support vectors"

======

- first, note, at least so far, we don't have a killer

improvement for SVMs (we only apply SVM methods

to classify. We don't have a killer idea

to either improve the speed of SVM classification,

improve the accuracy of SVM classification or both).

- Instead of using Podiums to manage training vector voting,

try to use hawaiian metrics to find all 1-dimensional

"support rings" (Really 1-D rings are intervals so

"support intervals". Looking for the first SI around

the sample where the class changes in that dimension).

- This should give us a collection of "support intervals"

which can then be "separation curve fitted" (either

in training space or in higher dimensional feature

space -after the SVM transformation is applied).

APPENDIX:

======

Fibonacci base provides multiple representers for most numbers.

For seed=s defines s-Canonical Fibonacci representation (sCF)

======

(when s=1 we will drop the s)

s-Packed Fibonacci (sPF) will be the representation with 1-bits

======moved as far right as possible, for s=1:

Fib: 233 144 89 55 34 21 13 8 5 3 2 1 < - seed

NUM

0 0 0 0 0 0 0 0 0 0 0 0 0 < - 1 rep

1 0 0 0 0 0 0 0 0 0 0 0 1 < - 1 rep

2 0 0 0 0 0 0 0 0 0 0 1 0 < - 1 rep

3 0 0 0 0 0 0 0 0 0 0 1 1

4 0 0 0 0 0 0 0 0 0 1 0 1 < - 1 rep

5 0 0 0 0 0 0 0 0 0 1 1 0 =(x+1)base2

6 0 0 0 0 0 0 0 0 0 1 1 1 =(x+1)base2

7 0 0 0 0 0 0 0 0 1 0 1 0 < - 1 rep

8 0 0 0 0 0 0 0 0 1 0 1 1 =(x+3)base2

9 0 0 0 0 0 0 0 0 1 1 0 1 =(x+4)base2

10 0 0 0 0 0 0 0 0 1 1 1 0 =(x+4)base2

11 0 0 0 0 0 0 0 1 0 0 1 1 =(x+8)base2

12 0 0 0 0 0 0 0 1 0 1 0 1 < - 1 rep

13 0 0 0 0 0 0 0 1 0 1 1 0 =(x+9)base2

14 0 0 0 0 0 0 0 1 0 1 1 1

15 0 0 0 0 0 0 0 1 1 0 1 0

16 0 0 0 0 0 0 0 1 1 0 1 1

17 0 0 0 0 0 0 0 1 1 1 0 1

18 0 0 0 0 0 0 1 0 0 1 1 0

19 0 0 0 0 0 0 1 0 0 1 1 1

20 0 0 0 0 0 0 1 0 1 0 1 0 < - 1 rep

21 0 0 0 0 0 0 1 0 1 0 1 1

22 0 0 0 0 0 0 1 0 1 1 0 1

23 0 0 0 0 0 0 1 0 1 1 1 0

24 0 0 0 0 0 0 1 0 1 1 1 1

25 0 0 0 0 0 0 1 1 0 1 0 1

26 0 0 0 0 0 0 1 1 0 1 1 0

27 0 0 0 0 0 0 1 1 0 1 1 1

28 0 0 0 0 0 0 1 1 1 0 1 0

29 0 0 0 0 0 0 1 1 1 0 1 1

30 0 0 0 0 0 0 1 1 1 1 0 1

31 0 0 0 0 0 0 1 1 1 1 1 0

32 0 0 0 0 0 0 1 1 1 1 1 1

33 0 0 0 0 0 1 0 1 0 1 0 1 < - 1 rep

34 0 0 0 0 0 1 0 1 0 1 1 0

. . . (find the rest further down in the appendix)

A 1-pass r-to-l alg which produces Binary PF from Binary CF

using table lookup:

- Forevery 1000 0000 0000 convert to 0101 0101 0110

- Forevery 100 0000 0000 convert to 010 1010 1011

- Forevery 10 0000 0000 convert to 01 0101 0110

- Forevery 1 0000 0000 convert to 0 1010 1011

- Forevery 1000 0000 convert to 0101 0110

- Forevery 100 0000 convert to 010 1011

- Forevery 10 0000 convert to 01 0110

- Forevery 1 0000 convert to 0 1011

- Forevery 1000 convert to 0110

- Forevery 100 convert to 011

- Forevery 1000 0000 0000 convert to 0101010101 10

- Forevery 100 0000 0000 convert to 0101010101 1

- Forevery 10 0000 0000 convert to 01010101 10

- Forevery 1 0000 0000 convert to 01010101 1

- Forevery 1000 0000 convert to 010101 10

- Forevery 100 0000 convert to 010101 1

- Forevery 10 0000 convert to 0101 10

- Forevery 1 0000 convert to 0101 1

- Forevery 1000 convert to 01 10

- Forevery 100 convert to 01 1

- In fact it would probably be efficient code to form the 256

entry table of all conversions.

- What we see from above is that BPF converted segments are

always alternating except for the last 1 or two digits.

Closed form formula for CFn:

Roots of E = r^2 = r + 1 are (1+-SRQ(5))/2

phi = (SQR(5)+1)/2 = -root2(E) ~= .618034...

Phi = 1 + phi = 1/phi = (SQR(5)+1)/2 = root1(E) ~= 1.618034...

CFn= ( (1+SQRT(5)/2)^n - (1-SQRT(5)/2)^n ) ( Phi^n - (-phi)^n)

Lim(n->inf)CFn/CFn+1 = Phi = ~1.61803... = gm (golden mean)

CFn alternates above and below gm.

Closed form formula for sCFn?

Closed form formula for the binary CFn representation?

Closed form formula for the binary sCFn representation?

Closed form formula for the binary PFn representation?

Closed form formula for the binary sPFn representation?

A rectangle with aspect ratio = gm has the nice recursive

(fractal?) property: Removing a maximal square leaves a

recatangle with aspect ratio = gm

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Given any Fibonacci sequence, {bn, bn-1,..., b1, b0}

where b0 is the seed, b1=b0+0, bn=bn-1+bn-2 for n>1

the canonical CF-b0 representation of a positive integer, x, is

QnQn-1...Q1Q0, where the Qi's are generated recursvely by:

Ri=x initially, for i = n, n-1, ... , 1, 0

IF bi <= Ri THEN Qi=1 and Ri-1 = Ri - bi ELSE Qi=0.

____Qi___

bi | Ri

Qi*bi

-----

Ri-1

This is taking out the maximum each time (left to right)

======

6. What about taking out the minimum (right to left?

Ri=x initially, for i = 0, 1, 2, ... , n

IF bi <= Ri THEN Qi=1 and Ri+1 = Ri - bi ELSE Qi=0.

Fibonacci base sequence: ...233 144 89 55 34 21 13 8 5 3 2 1 1

( ni = n(i+1) + n(i+2) )

For byte data:

Fibonacci seed

(use twice?)

Fib:233 144 89 55 34 21 13 8 5 3 2 1 1

Pos: 11 10 9 8 7 6 5 4 3 2 1 0 0

NUM:

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 1 0

2 0 0 0 0 0 0 0 0 0 0 1 0 0

3 0 0 0 0 0 0 0 0 0 1 0 0 0

4 0 0 0 0 0 0 0 0 0 1 0 1 0

5 0 0 0 0 0 0 0 0 1 0 0 0 0

6 0 0 0 0 0 0 0 0 1 0 0 1

7 0 0 0 0 0 0 0 0 1 0 1 0

8 0 0 0 0 0 0 0 1 0 0 0 0

9 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 0 0 0 0 0 1 0 0 1 0

11 0 0 0 0 0 0 0 1 0 1 0 0

12 0 0 0 0 0 0 0 1 0 1 0 1

13 0 0 0 0 0 0 1 0 0 0 0 0

14 0 0 0 0 0 0 1 0 0 0 0 1

15 0 0 0 0 0 0 1 0 0 0 1 0

16 0 0 0 0 0 0 1 0 0 1 0 0

17 0 0 0 0 0 0 1 0 0 1 0 1

18 0 0 0 0 0 0 1 0 1 0 0 0

19 0 0 0 0 0 0 1 0 1 0 0 1

20 0 0 0 0 0 0 1 0 1 0 1 0

21 0 0 0 0 0 1 0 0 0 0 0 0

22 0 0 0 0 0 1 0 0 0 0 0 1

23 0 0 0 0 0 1 0 0 0 0 1 0

24 0 0 0 0 0 1 0 0 0 1 0 0

25 0 0 0 0 0 1 0 0 0 1 0 1

26 0 0 0 0 0 1 0 0 1 0 0 0

27 0 0 0 0 0 1 0 0 1 0 0 1

28 0 0 0 0 0 1 0 0 1 0 1 0

29 0 0 0 0 0 1 0 1 0 0 0 0

30 0 0 0 0 0 1 0 1 0 0 0 1

31 0 0 0 0 0 1 0 1 0 0 1 0

32 0 0 0 0 0 1 0 1 0 1 0 0

33 0 0 0 0 0 1 0 1 0 1 0 1

34 0 0 0 0 1 0 0 0 0 0 0 0

35 0 0 0 0 1 0 0 0 0 0 0 1

36 0 0 0 0 1 0 0 0 0 0 1 0

37 0 0 0 0 1 0 0 0 0 1 0 0

38 0 0 0 0 1 0 0 0 0 1 0 1

39 0 0 0 0 1 0 0 0 1 0 0 0

40 0 0 0 0 1 0 0 0 1 0 0 1

41 0 0 0 0 1 0 0 0 1 0 1 0

42 0 0 0 0 1 0 0 1 0 0 0 0

43 0 0 0 0 1 0 0 1 0 0 0 1

44 0 0 0 0 1 0 0 1 0 0 1 0

45 0 0 0 0 1 0 0 1 0 1 0 0

46 0 0 0 0 1 0 0 1 0 1 0 1

47 0 0 0 0 1 0 1 0 0 0 0 0

48 0 0 0 0 1 0 1 0 0 0 0 1

49 0 0 0 0 1 0 1 0 0 0 1 0

50 0 0 0 0 1 0 1 0 0 1 0 0

51 0 0 0 0 1 0 1 0 0 1 0 1

52 0 0 0 0 1 0 1 0 1 0 0 0

53 0 0 0 0 1 0 1 0 1 0 0 1

54 0 0 0 0 1 0 1 0 1 0 1 0

55 0 0 0 1 0 0 0 0 0 0 0 0

56 0 0 0 1 0 0 0 0 0 0 0 1

57 0 0 0 1 0 0 0 0 0 0 1 0

58 0 0 0 1 0 0 0 0 0 1 0 0

59 0 0 0 1 0 0 0 0 0 1 0 1

60 0 0 0 1 0 0 0 0 1 0 0 0

61 0 0 0 1 0 0 0 0 1 0 0 1

62 0 0 0 1 0 0 0 0 1 0 1 0

63 0 0 0 1 0 0 0 1 0 0 0 0

64 0 0 0 1 0 0 0 1 0 0 0 1

65 0 0 0 1 0 0 0 1 0 0 1 0

66 0 0 0 1 0 0 0 1 0 1 0 0

67 0 0 0 1 0 0 0 1 0 1 0 1

68 0 0 0 1 0 0 1 0 0 0 0 0

69 0 0 0 1 0 0 1 0 0 0 0 1

70 0 0 0 1 0 0 1 0 0 0 1 0

71 0 0 0 1 0 0 1 0 0 1 0 0

72 0 0 0 1 0 0 1 0 0 1 0 1

73 0 0 0 1 0 0 1 0 1 0 0 0

74 0 0 0 1 0 0 1 0 1 0 0 1

75 0 0 0 1 0 0 1 0 1 0 1 0

76 0 0 0 1 0 1 0 0 0 0 0 0

77 0 0 0 1 0 1 0 0 0 0 0 1

78 0 0 0 1 0 1 0 0 0 0 1 0

79 0 0 0 1 0 1 0 0 0 1 0 0

80 0 0 0 1 0 1 0 0 0 1 0 1

81 0 0 0 1 0 1 0 0 1 0 0 0

82 0 0 0 1 0 1 0 0 1 0 0 1

83 0 0 0 1 0 1 0 0 1 0 1 0

84 0 0 0 1 0 1 0 1 0 0 0 0

85 0 0 0 1 0 1 0 1 0 0 0 1

86 0 0 0 1 0 1 0 1 0 0 1 0

87 0 0 0 1 0 1 0 1 0 1 0 0

88 0 0 0 1 0 1 0 1 0 1 0 1

89 0 0 1 0 0 0 0 0 0 0 0 0

90 0 0 1 0 0 0 0 0 0 0 0 1

91 0 0 1 0 0 0 0 0 0 0 1 0

92 0 0 1 0 0 0 0 0 0 1 0 0

93 0 0 1 0 0 0 0 0 0 1 0 1

94 0 0 1 0 0 0 0 0 1 0 0 0

95 0 0 1 0 0 0 0 0 1 0 0 1

96 0 0 1 0 0 0 0 0 1 0 1 0

97 0 0 1 0 0 0 0 1 0 0 0 0

98 0 0 1 0 0 0 0 1 0 0 0 1

99 0 0 1 0 0 0 0 1 0 0 1 0

100 0 0 1 0 0 0 0 1 0 1 0 0

101 0 0 1 0 0 0 0 1 0 1 0 1

102 0 0 1 0 0 0 1 0 0 0 0 0

103 0 0 1 0 0 0 1 0 0 0 0 1

104 0 0 1 0 0 0 1 0 0 0 1 0

105 0 0 1 0 0 0 1 0 0 1 0 0

106 0 0 1 0 0 0 1 0 0 1 0 1

107 0 0 1 0 0 0 1 0 1 0 0 0

108 0 0 1 0 0 0 1 0 1 0 0 1

109 0 0 1 0 0 0 1 0 1 0 1 0

110 0 0 1 0 0 1 0 0 0 0 0 0

111 0 0 1 0 0 1 0 0 0 0 0 1

112 0 0 1 0 0 1 0 0 0 0 1 0

113 0 0 1 0 0 1 0 0 0 1 0 0

114 0 0 1 0 0 1 0 0 0 1 0 1

115 0 0 1 0 0 1 0 0 1 0 0 0

116 0 0 1 0 0 1 0 0 1 0 0 1

117 0 0 1 0 0 1 0 0 1 0 1 0

118 0 0 1 0 0 1 0 1 0 0 0 0

119 0 0 1 0 0 1 0 1 0 0 0 1

120 0 0 1 0 0 1 0 1 0 0 1 0

121 0 0 1 0 0 1 0 1 0 1 0 0

122 0 0 1 0 0 1 0 1 0 1 0 1

123 0 0 1 0 1 0 0 0 0 0 0 0

124 0 0 1 0 1 0 0 0 0 0 0 1

125 0 0 1 0 1 0 0 0 0 0 1 0

126 0 0 1 0 1 0 0 0 0 1 0 0

127 0 0 1 0 1 0 0 0 0 1 0 1

128 0 0 1 0 1 0 0 0 1 0 0 0

129 0 0 1 0 1 0 0 0 1 0 0 1

130 0 0 1 0 1 0 0 0 1 0 1 0

131 0 0 1 0 1 0 0 1 0 0 0 0

132 0 0 1 0 1 0 0 1 0 0 0 1

133 0 0 1 0 1 0 0 1 0 0 1 0

134 0 0 1 0 1 0 0 1 0 1 0 0

135 0 0 1 0 1 0 0 1 0 1 0 1

136 0 0 1 0 1 0 1 0 0 0 0 0

137 0 0 1 0 1 0 1 0 0 0 0 1

138 0 0 1 0 1 0 1 0 0 0 1 0

139 0 0 1 0 1 0 1 0 0 1 0 0

140 0 0 1 0 1 0 1 0 0 1 0 1

141 0 0 1 0 1 0 1 0 1 0 0 0

142 0 0 1 0 1 0 1 0 1 0 0 1

143 0 0 1 0 1 0 1 0 1 0 1 0

144 0 1 0 0 0 0 0 0 0 0 0 0

145 0 1 0 0 0 0 0 0 0 0 0 1

146 0 1 0 0 0 0 0 0 0 0 1 0

147 0 1 0 0 0 0 0 0 0 1 0 0

148 0 1 0 0 0 0 0 0 0 1 0 1

149 0 1 0 0 0 0 0 0 1 0 0 0

150 0 1 0 0 0 0 0 0 1 0 0 1

151 0 1 0 0 0 0 0 0 1 0 1 0

152 0 1 0 0 0 0 0 1 0 0 0 0

153 0 1 0 0 0 0 0 1 0 0 0 1

154 0 1 0 0 0 0 0 1 0 0 1 0

155 0 1 0 0 0 0 0 1 0 1 0 0

156 0 1 0 0 0 0 0 1 0 1 0 1

157 0 1 0 0 0 0 1 0 0 0 0 0

158 0 1 0 0 0 0 1 0 0 0 0 1

159 0 1 0 0 0 0 1 0 0 0 1 0

160 0 1 0 0 0 0 1 0 0 1 0 0

161 0 1 0 0 0 0 1 0 0 1 0 1

162 0 1 0 0 0 0 1 0 1 0 0 0

163 0 1 0 0 0 0 1 0 1 0 0 1

164 0 1 0 0 0 0 1 0 1 0 1 0

165 0 1 0 0 0 1 0 0 0 0 0 0

166 0 1 0 0 0 1 0 0 0 0 0 1

167 0 1 0 0 0 1 0 0 0 0 1 0

168 0 1 0 0 0 1 0 0 0 1 0 0

169 0 1 0 0 0 1 0 0 0 1 0 1

170 0 1 0 0 0 1 0 0 1 0 0 0

171 0 1 0 0 0 1 0 0 1 0 0 1

172 0 1 0 0 0 1 0 0 1 0 1 0

173 0 1 0 0 0 1 0 1 0 0 0 0

174 0 1 0 0 0 1 0 1 0 0 0 1

175 0 1 0 0 0 1 0 1 0 0 1 0

176 0 1 0 0 0 1 0 1 0 1 0 0

177 0 1 0 0 0 1 0 1 0 1 0 1

178 0 1 0 0 1 0 0 0 0 0 0 0

179 0 1 0 0 1 0 0 0 0 0 0 1

180 0 1 0 0 1 0 0 0 0 0 1 0

181 0 1 0 0 1 0 0 0 0 1 0 0

182 0 1 0 0 1 0 0 0 0 1 0 1

183 0 1 0 0 1 0 0 0 1 0 0 0

184 0 1 0 0 1 0 0 0 1 0 0 1

185 0 1 0 0 1 0 0 0 1 0 1 0

186 0 1 0 0 1 0 0 1 0 0 0 0

187 0 1 0 0 1 0 0 1 0 0 0 1

188 0 1 0 0 1 0 0 1 0 0 1 0

189 0 1 0 0 1 0 0 1 0 1 0 0

190 0 1 0 0 1 0 0 1 0 1 0 1

191 0 1 0 0 1 0 1 0 0 0 0 0

192 0 1 0 0 1 0 1 0 0 0 0 1

193 0 1 0 0 1 0 1 0 0 0 1 0

194 0 1 0 0 1 0 1 0 0 1 0 0

195 0 1 0 0 1 0 1 0 0 1 0 1

196 0 1 0 0 1 0 1 0 1 0 0 0

197 0 1 0 0 1 0 1 0 1 0 0 1

198 0 1 0 0 1 0 1 0 1 0 1 0

199 0 1 0 1 0 0 0 0 0 0 0 0

200 0 1 0 1 0 0 0 0 0 0 0 1

201 0 1 0 1 0 0 0 0 0 0 1 0

202 0 1 0 1 0 0 0 0 0 1 0 0

203 0 1 0 1 0 0 0 0 0 1 0 1

204 0 1 0 1 0 0 0 0 1 0 0 0

205 0 1 0 1 0 0 0 0 1 0 0 1

206 0 1 0 1 0 0 0 0 1 0 1 0

207 0 1 0 1 0 0 0 1 0 0 0 0

208 0 1 0 1 0 0 0 1 0 0 0 1

209 0 1 0 1 0 0 0 1 0 0 1 0

210 0 1 0 1 0 0 0 1 0 1 0 0

211 0 1 0 1 0 0 0 1 0 1 0 1

212 0 1 0 1 0 0 1 0 0 0 0 0

213 0 1 0 1 0 0 1 0 0 0 0 1

214 0 1 0 1 0 0 1 0 0 0 1 0

215 0 1 0 1 0 0 1 0 0 1 0 0

216 0 1 0 1 0 0 1 0 0 1 0 1

217 0 1 0 1 0 0 1 0 1 0 0 0

218 0 1 0 1 0 0 1 0 1 0 0 1

219 0 1 0 1 0 0 1 0 1 0 1 0

220 0 1 0 1 0 1 0 0 0 0 0 0

221 0 1 0 1 0 1 0 0 0 0 0 1

222 0 1 0 1 0 1 0 0 0 0 1 0

223 0 1 0 1 0 1 0 0 0 1 0 0

224 0 1 0 1 0 1 0 0 0 1 0 1

225 0 1 0 1 0 1 0 0 1 0 0 0

226 0 1 0 1 0 1 0 0 1 0 0 1

227 0 1 0 1 0 1 0 0 1 0 1 0

228 0 1 0 1 0 1 0 1 0 0 0 0

229 0 1 0 1 0 1 0 1 0 0 0 1

230 0 1 0 1 0 1 0 1 0 0 1 0

231 0 1 0 1 0 1 0 1 0 1 0 0

232 0 1 0 1 0 1 0 1 0 1 0 1

233 1 0 0 0 0 0 0 0 0 0 0 0

234 1 0 0 0 0 0 0 0 0 0 0 1

235 1 0 0 0 0 0 0 0 0 0 1 0

236 1 0 0 0 0 0 0 0 0 1 0 0

237 1 0 0 0 0 0 0 0 0 1 0 1

238 1 0 0 0 0 0 0 0 1 0 0 0

239 1 0 0 0 0 0 0 0 1 0 0 1

240 1 0 0 0 0 0 0 0 1 0 1 0

241 1 0 0 0 0 0 0 1 0 0 0 0

242 1 0 0 0 0 0 0 1 0 0 0 1

243 1 0 0 0 0 0 0 1 0 0 1 0

244 1 0 0 0 0 0 0 1 0 1 0 0

245 1 0 0 0 0 0 0 1 0 1 0 1

246 1 0 0 0 0 0 1 0 0 0 0 0

247 1 0 0 0 0 0 1 0 0 0 0 1

248 1 0 0 0 0 0 1 0 0 0 1 0

249 1 0 0 0 0 0 1 0 0 1 0 0

250 1 0 0 0 0 0 1 0 0 1 0 1

251 1 0 0 0 0 0 1 0 1 0 0 0

252 1 0 0 0 0 0 1 0 1 0 0 1

253 1 0 0 0 0 0 1 0 1 0 1 0

254 1 0 0 0 0 1 0 0 0 0 0 0

255 1 0 0 0 0 1 0 0 0 0 0 1

More hobbit rings, thinner and better centered ;-))))

To push the idea a little further, consider a Fibonacci starter

value of .1 rather than 1 (results in 16 bit representations and

results in more plateaus which should be even thinner ;-)))

Fib 159 98 61 37 23 14. 8.9 5.5 3.4 2.1 1.3 0.8 0.5 0.3 0.2 0.1 0.1

Pos: 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

num_

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1

2 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1

3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0

4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0

5 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0

6 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1

7 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1

8 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0

9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0

11 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0

12 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1

13 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1

14 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0

15 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

16 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0

17 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1

18 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1

19 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0

20 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0

21 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0

22 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0

23 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1

24 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1

25 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

26 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0

27 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0

28 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1

29 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1

30 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0

31 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0

32 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0

33 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0

34 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1

35 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1

36 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0

37 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0

38 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0

39 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1

40 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

41 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0

42 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0

43 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0

44 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0

45 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1

46 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1

47 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

48 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0

49 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0

50 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1

51 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1

52 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1

53 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0

54 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0

55 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0

56 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1

57 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1

58 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0

59 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0

60 0 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0

61 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1

62 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1

63 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1

64 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0

65 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0

66 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0

67 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1

68 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

69 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0

70 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0

71 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0

72 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0

73 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1

74 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1

75 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0

76 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0

77 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0

78 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1

79 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1

80 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0

81 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0

82 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0

83 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0

84 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1

85 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1

86 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0

87 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0

88 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0

89 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1

90 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1

91 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0

92 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0

93 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0

94 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0

95 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1

96 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1

97 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0

98 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0

99 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0

100 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1

101 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1

102 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0

103 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0

104 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0

105 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0

106 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1

107 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1

108 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0

109 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0

110 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0

111 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1

112 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1

113 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1

114 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0

115 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0

116 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0

117 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1

118 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1

119 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0

120 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0

121 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0

122 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1

123 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1

124 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1

125 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0

126 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0

127 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0

128 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1

129 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1

130 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0

131 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0

132 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0

133 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0

134 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1

135 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1

136 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0

137 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0

138 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0

139 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1

140 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1

141 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0

142 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0

143 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0

144 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0

145 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1

146 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1

147 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0

148 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0

149 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0

150 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1

151 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1

152 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0

153 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0

154 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0

155 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0

156 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1

157 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1

158 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0

159 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0

160 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

161 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1

162 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1

163 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0

164 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0

165 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0

166 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0

167 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1

168 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1

169 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0

170 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0

171 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0

172 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1

173 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1

174 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1

175 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0

176 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0

177 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0

178 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1

179 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1

180 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0

181 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0

182 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0

183 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

184 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1

185 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1

186 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0

187 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

188 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0

189 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1

190 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1

191 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0

192 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0

193 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0

194 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0

195 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1

196 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1

197 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0

198 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0

199 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

200 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1

201 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1

202 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0

203 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

204 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0

205 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0

206 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1

207 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1

208 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0

209 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0

210 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0

211 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1

212 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1

213 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0

214 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0

215 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0

216 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0

217 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1

218 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1

219 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0

220 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0

221 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0

222 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1

223 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1

224 1 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0

225 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0

226 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0

227 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0

228 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1

229 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1

230 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0

231 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0

232 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0

233 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1

234 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1

235 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1

236 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0

237 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0

238 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0

239 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1

240 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1

241 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0

242 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0

243 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0

244 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1

245 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1

246 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1

247 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0

248 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0

249 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0

250 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1

251 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1

252 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0

253 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0

254 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0

255 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0

"Thin-ness of plateaus" needs to be studied and quantified.

NOTES on FIBBONACCI NUMBERS and SEQUENCES:

1. Taking the fraction to be 1/B where B is any of

the standard Fibonacci numbers {1,2,3,5,8,13,21,34...}

gives a sequence of base numbers which will include 1.

******75 46 28. 17. 11 6.8 4.2 2.6 1.6 1 0.6 0.4 0.2 0.2 = 1/5

Pos:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

num_

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1

2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1

3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1

4 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1

5 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1

6 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1

7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

8 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

9 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1

10 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1

11 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1

12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1

13 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1

14 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1

15 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1

16 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1

17 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1

18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1

19 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1

20 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1

21 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1

22 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1

23 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1

24 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1

25 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1

26 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1

27 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1

28 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1

29 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1

30 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1

31 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1

32 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1

33 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1

34 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1

35 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1

36 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1

37 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1

38 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1

39 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1

40 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1

41 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1

42 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1

43 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1

44 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1

45 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1

46 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1

47 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1

48 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1

49 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1

50 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1

51 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1

52 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1

53 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1

54 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1

55 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1

56 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1

57 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1

58 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1

59 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1

60 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1

61 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1

62 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1

63 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1

64 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1

65 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1

66 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1

67 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1

68 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1

69 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1

70 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1

71 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1

72 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1

73 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1

74 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1

75 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1

76 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1

77 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1

78 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1

79 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1

80 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1

81 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1

82 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1

83 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1

84 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

85 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1

86 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1

87 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1

88 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1

89 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1

90 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1

91 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1

92 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1

93 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1

94 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1

95 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1

96 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1

97 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1

98 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1

99 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1

100 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1

101 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1

102 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1

103 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1

104 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1

105 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1

106 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1

107 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1

108 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1

109 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1

110 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1

111 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1

112 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1

113 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1

114 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1

115 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1

116 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1

117 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1

118 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1

119 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1

120 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1

121 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1

122 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1

123 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1

124 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1

125 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1

126 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1

127 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1

128 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1

129 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1

130 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1

131 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1

132 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1

133 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1

134 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1

135 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1

136 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1

137 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1

138 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1

139 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1

140 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1

141 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1

142 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1

143 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1

144 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1

145 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1

146 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1

147 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1

148 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 1

149 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1

150 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1

151 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1

152 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1

153 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1

154 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1

155 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1

156 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1

157 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1

158 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1

159 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1

160 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1

161 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1

162 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1

163 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1

164 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1

165 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1

166 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1

167 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1

168 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1

169 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1

170 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1

171 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1

172 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1

173 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1

174 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1

175 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1

176 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1

177 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1

178 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1

179 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1

180 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 1

181 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1

182 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1

183 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1

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2. In the canonical Fibonacci sequence,

Lim(n -> inf)Fn/Fn-1 = ~1.61803... =gm, the golden mean

(and the convergences is oscillatory above and below gm).

3. There is a closed form formula for Fn

(nth element of the canoical Fibonacci sequence):

Fn = ( (1+SQRT(5)/2)^n - (1-SQRT(5)/2)^n )

------

SQRT(5)

4. A rectangle with aspect ratio = gm has the nice recursive

(fractal?) property: Removing a maximal square leaves a

recatangle with aspect ratio = gm

______

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|______|____|___|

5. Given any Fibonacci base sequence (FBS), {bn, bn-1, ..., b1, b0}

where b0 is the seed, b1=b0+0, bn=bn-1+bn-2 for n>1

the canonical FBS(b0) representation of a positive integer, x, is

QnQn-1...Q1Q0, where the Qi's are generated recursvely by:

Ri=x initially, for i = n, n-1, ... , 1, 0

IF bi <= Ri THEN Qi=1 and Ri-1 = Ri - bi ELSE Qi=0.

____Qi___

bi | Ri

Qi*bi

-----

Ri-1

This is taking out the maximum each time (left to right)

======

6. What about taking out the minimum (right to left?

Ri=x initially, for i = 0, 1, 2, ... , n

IF bi <= Ri THEN Qi=1 and Ri+1 = Ri - bi ELSE Qi=0.

____Qi___

bi | Ri

Qi*bi

-----

Ri+1

Problems: Only an estimate of x is produce

and multiple x's can have the same representative.

That about taking a geometric sequence with base (~1.61803)^2 = gm^2 ?

************76. 46. 29. 17. 11. 6.8 4.2 2.6 1.6 1 0.6 0.3 .

Pos:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1/(gm)^2

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*************************************************************

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349541385321

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**********************************

CF

PF

21

3485321

349541385321

vvvvvvvvvvvv

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**********************************

CF PF

21 21

3485321 3485321

349541385321 349541385321

vvvvvvvvvvvv vvvvvvvvvvvv

0

1 1 1

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