If you’ve played with fractions and decimals (and who
hasn’t?), then you probably know that 7 is a special number. The reciprocal of 7, 1/7, has a decimal
expansion that repeats. That’s not
special—most fractions do. However, what
makes 7 (or 1/7) more special is that the block of repeating digits is as long
as is possible (one less than the denominator6 digits, in this case). Also,
the fractions 2/7 through 6/7 have decimal expansions that repeat with the same
block (bolded below):
It turns out that 19 and 61 are also cyclic primes. Just as 7 has six numbers in its repeating block, 19 has 18 and 61 has 60. The blocks for 1/19 and 1/61 are:
 1/7 = 0.142857142857142857142857142857…
 2/7 = 0.285714285714285714285714285714…
 3/7 = 0.428571428571428571428571428571…
 4/7 = 0.571428571428571428571428571428…
 5/7 = 0.714285714285714285714285714285…
 6/7 = 0.857142857142857142857142857142…
It turns out that 19 and 61 are also cyclic primes. Just as 7 has six numbers in its repeating block, 19 has 18 and 61 has 60. The blocks for 1/19 and 1/61 are:
·
1/19 = 052631578947368421…
·
1/61 =
016393442622950819672131147540983606557377049180327868852459…
Since
I was born in 1961, I thought I’d try to do something with that. It seemed reasonable to try to fit all of the
fractions into an 18 x 60 box, those with a denominator of 19 being columns 18
rows long, and those with a denominator of 61 being rows 60 columns long. For example, the blocks for fractions m/7 can
fit into a 6 x 6 square; here are four of them:
3/7

6/7


4

8


2

5


2/7

2

8

5

7

1

4

5

1


4/7

5

7

1

4

2

8

1

2

Alas,
for 19 and 61, it’s not quite that simple.
There are only 18 different decimal expansions for fractions of the form
m/19, not 60, and there are 60 decimal expansions for fractions of the form
m/61, not just 18. Also, the actual
numbers don’t generally line up, like they do for m/7. But, I was able to find one case in which 21
blocks of repeating decimals fit, and another in which 24 blocks fit. Here’s the upper left corner of the 21block
box:
14/19

11/19

2/19


7

5

1


37/61

6

0

6

5

5

7

3

7

7

0

49/61

8

0

3

2

7

8

6

8

8

5

8

9

2


1/61

0

1

6

3

9

3

4

4

2

6

This
shows that the decimal expansion of 14/19 begins 0.73684, and that of 37/61
begins 0.6065573770. Note that both
share the 3 that is in the second row, seventh column. One trick in setting up this box was ensuring
that, whenever a row fraction and a column fraction share the same cell, that
numeral is correct in both expansions.
Another
challenge was how to present this an in aesthetically interesting fashion. I came up with a solution, but will leave it
to you to decide how interesting it is.
Rather than just writing the numerals, I used modified rose curves to
represent them. In the 21block case,
they are all shown using this alphabet, which shows 0, 1, 2, 4, and 8:
In
the 24block case, I tweaked the design a bit.
Also, I decided to render those numbers in the intersection cells (like
3, above, which is shared by 14/19 and 37/61) in negative. Here are the symbols for 3, 5, 6, 7, and 9:
Now
that you know what the symbols mean, you can read the image and there’s no need
for labels, right? So, without further
ado, I present “19 61 a” and “19 61 b.”