Better definitions of Reap and Sow

Question

I am having tremendous difficulty understanding Reap and Sow in Mathematica. The definitions are pretty circular. When I look at the definition of Sow, I get, "specifies that e should be collected by the nearest enclosing Reap,specifies that e should be collected by the nearest enclosing Reap whose pattern matches tag, and specifies that e should be collected once for each pattern that matches a tag". Then for reap, they do the same thing by using Sow in two of the definitions. I don't fully understand either. And the examples use tags in a way I don't really think of as tags in common English or real analysis. So, what are they actually talking about?

Answer 1

Here a quick example how Reap and Sow are typically used:

Reap[
  Do[
   If[OddQ[i], Sow[i, "Odd"], Sow[i, "Even"]],
   {i, 1, 20}],
  _, Rule
  ][[2]]

{"Odd" -> {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, "Even" -> {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}}

Here _ in Reap means that arbitrary tags should be collected by this particular Reap.

Other patterns than _ are useful when we are dealing with nested instances of Reap. In the following, the inner Reap collects only tags of the form "Odd" while the outer collects "Even":

b = Reap[
    a = Reap[
       Do[
        If[OddQ[i], Sow[i, "Odd"], Sow[i, "Even"]],
        {i, 1, 20}],
       "Odd", Rule
       ][[2]],
    "Even", Rule
    ][[2]];
a
b

{"Odd" -> {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}}

{"Even" -> {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}}

Replacing Rule with any other function, say f leads to

Reap[
  Do[
   If[OddQ[i], Sow[i, "Odd"], Sow[i, "Even"]],
   {i, 1, 20}],
  _, f
  ][[2]]

{f["Odd", {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}], f["Even", {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}]}