First, to be clear, I think there will always be an interest in "puzzle math", competitive problem solving of the sort seen in Math Olympiad, and competitive solving of popular puzzles like Rubik's Cube. There will also be serious mathematicians who may have little use for computers except as word processors. I'm talking about something a little different here. Since it's hard to define, I'll give an instance: flexagons, which I would explain as a method by which a strip of paper can be folded so it is isomorphic to an oriented binary tree and a traversal can be carried out by a series of folding and unfolding steps. I first encountered them in Madachy's Mathematical Recreations (hence my subject line) as a teen and I forget the other ones in that book. I guess magic squares would qualify as a mathematical recreation, but I have always been more excited by those that involve some craft: cutting, folding, gluing, etc.
Of course, since I did have access to a microcomputer at the time (though it was less common) one of the things I did a few years later was to code up the algorithm from the book on a computer (which is optimized for a human writing out a paper table, as I eventually realized).
So, examples aside, I have kids in their teens, and throughout their childhood I have shown them some "craft" mathematics that I find interesting (admittedly often using a computer as a time-saver, but it's not required). Here, for instance is an icosahedron and dodecahedron colored to illustrate how their rotation groups are equivalent to the group of 60 even permutations of five elements (pardon the sloppy glue work).
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It may be an overall lack of interest, since this kind of activity has always had limited appeal, but I find that I am increasingly fascinated by the process of pulling mathematics away from electronic computers where everything is just way too easy, and constructing physical analogs. My kids think this is just weird, and they may be right.
In my estimation, we ought to be living in a "golden age of non-computerized mathematical recreations" since computers themselves can be used to discover and optimize the kinds of things that can be built with crafts (e.g. use a 3D printer to make prototypes until you find one that is easy to craft by hand). There are also newer craft products like polymer clay and a lot of interest in crafts in general (stores like Michaels), though rarely aimed at mathematics. The field looks wide open.
So what I wonder is if the main historical appeal of these crafts, such as flexagons, is that they could be used as analog devices before computers existed. In this view, they are now as obsolete as slide rules, given the existence of computers--in which not only can a binary tree be constructed, but a flexagon itself can be modeled and rendered in animation.
The question is whether it's interesting to have touchable mathematics. I mean, aside from manipulatives in a Montessori preschool. Are these things of any interest to teens and adults with access to powerful electronic computers?
