Number Patterns, Geometry and Fractals
Here are some explanations of how some of these images where created.
The images with spiraling tile patterns, such as triangles, squares and hexagons use golden ratio proportions to create the spirals. There is also something interesting that isn't really visible, the ratio of the areas of the smaller shapes to the larger are exact proportions of 4 for triangles, 3 for sqares, and 2 for hexagons. These are also the only regular shapes that can tile continuously with out gaps or require the use of other irregular shapes.
The images with the circle patterns were created with digit sum patterns. Digit summing is the process of adding all the digits up until you get one digit.
Number 1 2 3 4 5 6 7 8 9 Digit Sum
0 9 18 27 36 45 54 63 72 81 9
1 10 19 28 37 46 55 64 73 82 1
2 11 20 29 38 47 56 65 74 83 2
3 12 21 30 39 48 57 66 75 84 3
4 13 22 31 40 49 58 67 76 85 4
5 14 23 32 41 50 59 68 77 86 5
6 15 24 33 42 51 60 69 78 87 6
7 16 25 34 43 52 61 70 79 88 7
8 17 26 35 44 53 62 71 80 89 8
9 18 27 36 45 54 63 72 81 90 9
The followin images were create using similar concepts with basic number patterns of doubling numbers, tripling numbers, quadrupling numbers, and the infinitely repeating decimal number patter of 1/7 in base 10 and base 4 as represented by the Enneagram symbol on the lower right picture below.
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Doubling Series Digit Sums:
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1, 2, 4, 8, 16 = 1+6 = 7, 32 = 3+2 = 5, 64 = 6+4 = 10 = 1+0 = 1
1, 2, 4, 8, 16 = 1+6 = 7, 14 = 1+4 = 5, 28 = 2+8 = 10 = 1+0 = 1
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1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, ...
Tripling Series Digit Sums:
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1, 3, 9, 27 = 2+9 = 9, 81 = 8+1 = 9, 243 = 2+4+3 = 9
2, 6, 18 = 1+8 = 9, 54 = 5+4 = 9, 162 = 1+6+2 = 9, 486 = 4+8+6 = 18 = 1+8 = 9
4, 12 = 1+2 = 3, 36 = 3+6 = 9
5, 15 = 1+5 = 6, 45 = 4+5 = 9
7, 21 = 2+1 = 3, 63 = 6+3 = 9
8, 24 = 2+4 = 6, 72 = 7+2 = 9
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1, 3, 9, .....
2, 6, 9, .....
4, 3, 9, .....
5, 6, 9, .....
7, 3, 9, .....
8, 6, 9, .....
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Quadrupling Series Digit Sums:
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1, 4, 16 = 1+6 = 7, 28 = 2+8 = 10 = 1+0 = 1, 4 , ....
1, 4, 7, 1, ....
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2, 8, 32 = 3+2 = 5, 56 = 5+6 = 11 = 1+1 = 2, 8, ....
2, 8, 5, 2, ....
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3, 12 = 1+2 = 3 ....
6, 24 = 2+4 = 6 ....
Quintupling Series Digit Sums:
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1, 5, 25 = 2+5 = 7, 35 = 3+5 = 8, 40 = 4+0 = 4, 20 = 2+0 = 2, 10 = 1+0 = 1, 5 , ....
1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, ....
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As you can see this pattern is the inverse
of the doubling pattern. It is the same number
pattern going in the opposite direction. 2 and 5
are also reciprical numbers, meaning:
1/2 = 0.5
and
1/5 = 0.2
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So the have the same pattern, but the
flow is in the opposite direction.
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Divisions of Seven, Shown to be Reciprocals of Four
Shown Below is the algebraic representation of the digit sum of an infinitely repeating decimal of divisions of 7.
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1/7 = 0.142857142857142857..... = X 4
2/7 = 0.285714285714285714..... = 2X 8
3/7 = 0.428571428571428571..... = 3X 3
4/7 = 0.571428571428571428..... = 4X 7
5/7 = 0.714285714285714285..... = 5X 2
6/7 = 0.857142857142857142..... = 6X 6
7/7 = 1.000000000000000000..... = 7X = 1 1
8/7 = 1.142857142857142857..... = 8X = X+1 5
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Doubling Series has to match the pattern: 1, 2, 4, 8, 7, 5, 1, ....
If X =4, then:
4*2 = 8 = 2X
8*2 = 16 = 1+6 = 7 = 4X
7*2 = 14 = 1+4 = 5 = 8X = X+1 4+1 = 5 Matches the Pattern
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If you use X = 1, 2, 3, 5, 6, 7, 8, or 9 the patterns will not match.
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The pattern above is called the Eneagram from Vedic mathematics, and the pattern around the triangle is called the web of life. It follows the infinitely repeating decimal of fractions of 7 in base 10. The triangle pattern in the middle also comes from decimal pattern in base 4, which is a 3 pointed circle.
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