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dynamic - Manipulate in Manipulate


Question 63982 (Altering values in Manipulate with dynamically generated controls) got a very interesting answer from @belisarius, in which he used a Manipulate within a Manipulate to achieve the goal. I am puzzled by his solution and will demonstrate my embarrassment by a somewhat simplified example.


In Mathematica there are actually two independent sets of variables: those that are used by the frontend and those that are used by the kernel. All interactivity in Mathematica, such as Manipulate, Button, ... is based on interaction with the user and therefore belongs completely to the domain of the frontend. But we use kernel commands with kernel variables to construct these user interfaces. Fortunately, in Mathematica it is practically never necessary to bother about the difference between frontend and kernel variables, the frontend and the kernel are very well integrated.


Let us start with a simple example, based on the question mentioned above. The button displays the current value of the two variables x and s.


Row[{Manipulate[x, {{x, s}, -1, 1}, Button[Dynamic[{x, s}], s = RandomReal[{-1, 1}]]],
  Spacer[30], Dynamic[{x, s}]}]


In the second argument of Manipulate, the s is the initial value for x. When we move the slider, we see that in the Manipulate expression the value of x is continuously updated but not outside this expression; the variable s does not change at all. That is as expected. Moreover, when we look at the input form of the Manipulate expression, we see that the current value of x is stored in the second argument of Manipulate at the position of s. That is well documentated. There is no connection at all between s and x. When we press the button, the value of both the kernel s and the frontend s changes, but is has no effect on x.


Also, when we inspect the box structure of the displayed Manipulate expression with the toggle Ctrl-E, we see a.o. the variable \$CellContext`x$$. That is, I think, the indication of a frontend variable that has no connection with a kernel variable at all. Its value is displayed on the button, but not outside the Manipulate expression. Moreover, there is a variable $CellContext`s, which, I think, is a frontend variable that is automatically linked to the kernel variable s. So we see its value both on the button and in the Dynamic output.


The following command is a simplified form of the construction of @belisarius. We wrap the Manipulate in another Manipulate:


Row[{Manipulate[Manipulate[x, {{x, s}, -1, 1}, Button[Dynamic[{x, s}], s = RandomReal[{-1, 1}]]]],
   Spacer[30], Dynamic[{x, s}]}]

When we move the slider for x, we see as before the updating on the button, but not in the Dynamic output. When we click on the button for s, of course the value of s changes, but now the value of x is set to the value of s. That means that the effect of the outer Manipulate is that by clicking on the button the inner Manipulate is completely re-evaluated, including the initialization of x to s. It might be simple to understand, but I fail to see an explanation for that.


Another effect is that in the input form we no longer see the values of x and s, and that in the box form the variable \$CellContext`x$$ is no longer is visible. Instead we have the frontend variable \$CellContext`x that is now NOT coupled to the kernel variable x.


Any hint that helps me in better understanding what is going on here is highly appreciated.




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