3.0 EXPLORATION [OF NUMBER AND SERIES]

The power of number and series to model not only the appearance of things and events but also to predict their behaviour raises the question as to whether number is an invention, or a discovery by man of some aspect of the deep structure of the cosmos.

Notes on the reduced Fibonacci Series - alternate figures in bold

3.1 Digits [and reduced numbers] placed in a triangular array continue to maintain their role and function.

3.1.1 Pascals Triangle with numbers reduced demonstrates the patterns of the triads 1, 4 and 7, 2, 5 and 8 and 3, 6 and 9.

3.1.2 Fibonacci Series reduced in a triangular array - 24th line [bottom] holds full repeat of 24 digits. The 2 central triangles reflect around the central horizontal line. The opposing digits of the 2 lower triangles reflect and add up to 9. Many vertical lines have significant symmetries.

3.1.3 Albarn Series reduced in a triangular array - 24th line [bottom] holds full repeat of 24 digits. The 2 central triangles reflect around the central horizontal line. The opposing digits of the 2 lower triangles reflect and add up to 9. Many vertical lines have significant symmetries..

3.2 Maths and Magic Squares - Magic squares demonstrate many interesting patterns hence their fascination for all manner of experimental philosophers and mathematicians through the ages.

3.2.1 Magic Squares - This page after work by Keith Critchlow in Islamic Pattern, Schocken Books New York 1976.

3.2.2 Magic Squares cont..

3.2.3 Further patterns in Magic Squares - After reducing all numbers to single digits, the digits become coloured dominos. Groups of dominos are added together, the total reduced, and the pattern revealed.

3.2.4 Durers Magic Square - analysed to reveal the patterns within its structure.

3.2.5 Durers Magic Square - analysed to reveal the patterns within its structure.

3.2.6 Patterns in simple squares revealed when the products[where vertical and horizontal co-ordinates meet] of procedures [i.e. multiplication etc.] are reduced.

3.2.7 The Ancient Multiplication Square [used in its reduced form in mandalas, temple plans etc. by the Vedic People of North India around 500 B.C.] - Re-examined here using colour coded digits to further reveal the qualitative [as well as quantitative] aspect of number.

3.2.8 The Ancient Multiplication Square - cont.

3.2.10 Staged Analysis of The Multiplication Square [Part Two - the digits as dominos] .

3.2.11 Staged Analysis of The Multiplication Square[Part Three - each four adjacent dominos is added together and reduced to form a larger domino].

3.2.12 Staged Analysis of The Multiplication Square [Part Four - each four adjacent dominos is added together and reduced to form an even larger domino].

3.2.13 Four Domino Multiplication Squares.

3.2.14 The Four Multiplication Squares [3.2.13] with transparent, additive layers.

3.2.15 The Four Multiplication Squares [3.2.13] with opaque, additive layers.

3.2.16 A single Square using superimposed line dominos.

3.2.17 A single Square using superimposed line dominos as 3.2.16 but reversed.

3.2.18 Another version of 3.2.15.

3.3 The Countgame - Is where each digit moves its own number of squares within the line. If there is no room it starts the next line and so on... This procedure highlights the particularities of even, odd and most interestingly prime numbers which form flags[from 1 - 10]

3.3.1 The Countgame begins.

3.3.2 Countgame continued... 11 and 12.

3.3.3 Countgame continued further ... 13,14 and 15.

3.3.4 Countgame analysis of flag 13 [prime].

3.3.5 Countgame - Evens as dominos.

3.3.6 Countgame - Odds including 3x flags.

3.3.7 Countgame - Flag of 17 [prime].

3.3.8 Countgame - Flag 17 after each four adjacent dominos is added together and reduced to form a larger domino.

3.3.9 Countgame - Flag 17 after each four adjacent dominos of 3.3.8 is added together and reduced to form larger dominos.

3.3.10 Countgame - Flag 17 after each four adjacent dominos of 3.3.9 is added together and reduced to form even larger dominos.

3.3.11 Countgame - Flag 17 after each four adjacent dominos of 3.3.9 is added together and reduced to form one large domino 7.

3.3.12 Summary of analysis of Prime Flag 17.

3.4 Series [after reduction].

3.4.1 More complex series such as the Fibonacci, the Lucas and their derivatives such as the Albarn Series also demonstrate, [after reduction], similar and perhaps even moreremarkable characteristics .

3.4.2 Primacy of 147 in the series of digits 1 - 9.

3.4.3 Triangular number series [reduced].

3.4.4 Square number series [reduced].

3.4.5 Pentagonal number series [reduced].

3.4.6 Summary analysis of three series[3.4.3/4/5].

3.4.7 Digits as Series [called Indigs by Buckminster Fuller].

3.4.8 Series [reduced] generated by nuclear packing [after Buckminster Fuller].

3.4.9 Comparison of [reduced] series generated by natural digits and nuclear packed tetrahedron [after Buckminster Fuller].

3.4.10 Further comparisonsl.

3.4.11 A Series [the M Series] generated by multiplying the previous two digits then reducing the result.

3.4.12 The M Series unpacked.

3.4.13 Series or progressions demonstrate the relationships and character of the digits.

3.4.14 Simply generated Series compared.

3.4.15 Set of Series reveal the behaviour patterns of digits - To 1,2 and 3 are added 27[9], 24[6], 21[3], 18[9], 13[4], 11[2].

3.4.16 Set of Series reveal the behaviour patterns of digits - To 1,2 and 3 are added 27[9], 24[6], 21[3], 18[9], 13[4], 11[2].

3.4.17 Set of Series reveal the behaviour patterns of digits - To 1,2 and 3 are added 27[9], 24[6], 21[3], 18[9], 13[4], 11[2].

3.4.18 Summary of preceding set of series.

3.4.19 Fibonacci, Lucas and Albarn Series introduced - More complex series such as the Fibonacci, the Lucas and their derivatives such as the Albarn Series also demonstrate, after reduction, similar remarkable characteristics.

3.5 Fibonacci and Albarn Series

3.5.1 The Fibonacci Series begins.

3.5.2 Fibonacci Series explored.

3.5.3 Prevalence of each digit in the Fibonacci Series.

3.5.4 The reduced Fibonacci Series and its reduced Lucas Series relatives.

3.5.5 P.S. on Fibonacci.

3.5.6 Albarn Series minus Triads.

3.5.7 The Roles of the Triads in the reduced Fibonacci and Albarn Series.

3.5.8 The Series in domino form [Digitos].

3.5.9 The reduced Fibonacci and Albarn Series compared in domino form [Digitos].

3.5.10 The two major Series in columnar form for comparison.

3.5.11 The Albarn Series and its Sources.

3.5.12 The Fibonacci Series in bar form as a score for sound [see 7.0 Harmonics] .

3.5.13 How the Albarn Series is generated.

3.6 The Triads of Digits

3.6.1 Some examples of the Triads in action.

3.6.2 Digital relationships part one - links.

3.6.3 Digital relationships part two - on enneagrams.

3.6.4 Digital relationships part three - prime factors.

3.6.5 Digital relationships part four - more.

3.6.6 Digital relationships part five - on nonagons.

3.6.7 Digital relationships part six - more on nonagons.

3.6.8 Digital relationships part seven - magic squares and nonagons.

3.6.9 Digital relationships part eight - multiplication square and nonagons.

3.6.10 Digital relationships part nine - generating magic squares.

3.6.11 Digital relationships part nine - generating magic squares2.

3.6.12 Digital relationships part ten - generating magic squares3.

3.6.13 1 - 9 to the power of n - a pattern emerges.

3.6.14 The Primacy of 1,4 and 7.

3.6.15 Square Series and 147.

3.6.16 Triangular Series and 369.

3.6.17 Pentagonal Series and 258.

3.6.18 The Triads in the Series.

3.6.19 Postscript on Series.

3.7 Nonagons and Enneads - Some examples of the power of simple diagrams to reveal new relationships and to model collections of ideas

3.7.1 The set of number series that are generated from 1-9 around a nonagon.

3.7.2 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.3 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.4 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.5 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.6 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.7 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.8 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.9 The set of symmetrical Enneagrams as generated by the number list in 3.7.1.

3.7.10 The Enneagram numbers are turned into a bar chart for sound generation .

3.7.11 The Enneagram numbers are turned into a bar chart for sound generation .

3.7.12 Multiplication Tables [reduced] on a nonagon.

3.7.13 Colour cube and the nonagon.

3.7.14 The Ancient Enneagram.

3.7.15 An Ennead as applied from In Search of the Miraculous by P.D.Ouspensky [on the teaching of Gurdjieff].

2004 Copyright - Keith Albarn. All Rights reserved.

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