There are 5 ways to classify divisors in the divisor plot:
1. Column
When considered as a member of a column x, a divisor is simply one element of the set Dx. This
is the classification which began this whole exploration.
2. Row
When considered as a member of a row Ry, a divisor is simply one of an infinite set of integers,
which all have the same value
- equal to its y coordinate. We have considered divisors along a row as representing a periodic signal over time,
such as a
blinking light or a beat in a
musical polyrhythm. Differences in periods among rows are based on integers, and every
possible integer period occurs in R.
Seeing divisors as members of rows is one way to visualize the Sieve or Eratosthenes.
3. Zero Modulo Ray
When considered as a member of a zero modulo ray, a divisor is one of an infinite, ordinal set of integers, arranged along
one of the zero modulo rays Zn.
4. Square Root Parabola
When considered as a member of a square root parabola, a divisor is one of an ordinal set of integers from 0 to n, occurring
on one of the square root parabolas Pn.
5. Product Curve
When considered as a member of a product curve, a divisor is one of an infinite, ordinal set of integers, occurring
on one of the product curves Cn.
Relations Between these Classifications
Below we see that divisors occur where Z intersects with R, and
also that divisors occur where P intersects with R.
Not surprisingly, divisors lie at the intersections of Z and P.
Note also that divisors occur where C intersects with R.
Again, not surprisingly, divisors lie at the intersections of Z and C.
And also where C intersects with P.
Below we see all of these classifications overlaid. Note that all 5 sets of curves
(D, R, Z, P, and C) intersect at
divisors locations.
Given these classifications, we now have three ways to define Dx (besides in terms of divisors):
(1) Dx = the set of y values where x intersects P
(2) Dx = the set of y values where x intersects Z
(3) Dx = the set of y values where x intersects C
Observe that for any n, Zn and Rn are associated.
The picture below shows that they each provide one of the divisors of a conjugate pair, whose
product equals their x coordinate value.
Two examples are illustrated:
x = 14, and x = 24.
And as we have seen earlier, each of these pairs of divisors are members of a square root parabola:
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