Repeating Decimal Decimals

Copyright 2000 by Paul Niquette. All rights reserved.

Any rational solver -- pun intended -- knows that the ratio of any pair of integers produces a pattern of decimal digits that will repeat endlessly. Thus does the finite have the power to generate the infinite.
 
1/2 = 
0.500000000000...

2/3 = 
0.666666660000...
1/3 = 
0.333333333333...

2/5 =
0.400000000000...
1/4 = 
0.250000000000...

2/7 =
0.285714285714...
1/5 = 
0.200000000000...

2/9 =
0.222222222222...
1/6 = 
0.166666666666...

3/4 =
0.750000000000...
1/7 =
0.142857142857...

3/5 =
0.600000000000...
1/8 =
0.125000000000...

3/7 =
0.425671428571...
1/9 =
0.111111111111...

3/8 = 
0,375000000000...

As shown above, the repeating pattern ranges from utterly simple (000..., 111..., 222..., 333...) to the not so simple (142857142857...).
 

Here is a rather complex pattern:
0.645732645732645732...
What ratio of integers produced it?

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