“Recreational mathematics” is what Martin Gardner called what he did. I worshipped those collections of his columns as a kid, and I have them all now. As far as I can tell, it’s just a marketing term, trying not to scare off people who would really be interested.
It includes combinatorics, geometry, computer science, logic, game theory, probability, and just about any other branch of mathematics. And he didn’t shy away from tough calculations or rigorous proofs, either. I remember deep philosophy, discussions of the Axiom of Choice, and paradoxes.
I think all mathematics is recreational.
3D printed Catalan solids would actually be a nice addition to my collection of maths related objects, especially since only a few are Zome constructible, only one is Polydron constructible, and less than half are readily available as dice(I own some tetrakiscubic d24s, some d30s, some deltoidal d60s, and some d120s, and rhombic d12s, deltoidal d24s, and d48s can be found on Amazon, but that still leaves triakis tetrahedral d12s, 2 types of d24, and 3 types of d60 unaccounted).
My metaphorical kingdom for a good, commandline tool that allows for easily generating stls or vector graphics of mathematical curves and surfaces and a plotter and 3D printer reliable enough I could print at home without having to worry about troubleshooting that requires sight to do safely… Sadly, but all I’ve found that’s remotely accessible for blind generation of graphics is coding .svg files manually, and the svg markup has no calculation capabilities, so doing so requires doing all of the calculations of where elements should be positioned manually… ANd sadly, even ignoring the GUI and rendering stuff, I don’t know the first thing about programming a graphing calculator, much less one with my dream feature set(among other things, I’d want 2d graphs to allow cartesian and polar graphs to be drawn superimposed, and for 3D, I’d not only want to mix cartesian, cylindrical, and spherical coordinates, but to have r, phi, and theta for all three axises not just the zaxis(e.g. rx = 1 is an infinite cylinder of radius 1 centered on teh xaxis, while rz = 2 is a cylinder of raidus 2 centered on the zaxis) r and the ability to take boolean operations on multiple graphs(such as the part of z=cos(rz) bounded by rz = sin(7thetaz or the interesection of the aforementioned cylinders).
ANd for what it’s worth, I’m a yank, but maths is one of several Britishisms I’ve picked up(along with referring to the Atlantic as the Pond and Americans as yanks… and spelling it aluminium… also have my speech synth set to British English).
I’ve heard a lot about Gardner over the years, but I confess I’ve never really dug into his body of work… Should at least see if there are any hits for him on BARD, though I kind of doubt it, maths texts don’t get turned into audio books all that often from my experience.
I’m away from home and don’t have time to read through and respond to everything, but I can answer this one.
You can’t make joints smooth and con cavity the same simultaneously, since just fixing the joints determines everything when you have rigid pieces and are making the first derivatives fit.
The two tangent lines to the endpoints of your segment have slope + or  4, and intersect at an angle of 2arctan(4). Since that’s not a rational multiple of 360, copies of them with matching tangent lines wouldn’t fit nicely at all.
But if you instead use the region from x=.5 to .5, the angle is 90 and you can get lovely pictures:
(These graphs have been adjusted to be centered on (0,0): )
Edit: a lot of these are pretty easy to just do in wolfram alpha. I don’t have enough phone power to finish but just type “r^2+theta^2=100” into wolfram alpha and it shows you. Looks pretty cool!
Editedit:
Your calculation of the claw areas is fine. Just replace n with k and it shows that your area for the kth part is just k (with your normalizations). But the way you made the yin yang means the other part will be nk (or something similar, based on the conventions you’re using) and k plus nk is a constant.
Now I get the meaning your nick, mathbrush…
Best regards from Italy,
dott. Piergiorgio
The problem is, WolframAlpha graphs aren’t much help if you’re blind.
Two books I’ve read recently about maths that I thought were very good, at quite different levels of maths:

Fun level: Math With Bad Drawings by Ben Orlin. Laughoutloud funny with lots of approachable discussions of numbers in everyday life. Even my colleagues who hate mathematics thought this was a great book that made them feel more knowledgeable about the likes of geometry and lotteries. (One colleague’s read the whole thing and colleague #2 has borrowed it).

A bit more advanced: How to Think Like a Mathematician by Kevin Houston. The last half is above my capability but I like the first half. It contains a section on how to study mathematics I wished I’d been taught at school, a method for tackling proofs and a nice summary of mathematical logic (the last part tracks with and considerably extends my understanding of the subject from my formal logic philosophy class). Students considering university entrance or in the early stages of a numerate degree course would benefit from this.
Yeah, I suspect a lot of my musings would be easy to find the answers to if I still had a working eye and could make use of existing software. Sadly, finding software for any kind of graphics generation that works well with a screen reader and keyboardonly input is hard, and then getting the output into something blind usable is another issue, especially when one doesn’t have lots of money to spare… There is a talking version of the TI84 available, but it costs $700 and even then, it can only really let a blind person trace a graph and try to build up a mental image, brailledisplays and embossers start in the hundreds and braille displays don’t have the resolution for graphics and not all embossers can go off the grid to do freestyle, so you kind of have to do the Braille equivalent of ASCII art, which probably wouldn’t be too bad for 2d graphs that are just lines, curves, and monochrome shaded regions of the plane, and even if I could write a script that mathematically defines a 3d object, feed it to a parser that spits out an stl, and owned my own 3d printer that I could tell to print and then forget about it, I’d have no way of previewing what I’m about to print… and as far as I know, the most advanced tactile graphics display on the market has a resolution of 40*60, does the equivalent of five shades of gray, and costs 15 grand. And there just aren’t enough blind people to drive the economies of scale needed to drive prices down if the accessible tech can’t piggy back off of mainstream tech.
And maybe I didn’t explain the parabola thing well… When you join two parabolic segments so both of the joined endpoints are tangent to the same line(e.g. the joint is smooth), the segments can be on the same or opposite sides of the tanget line… if they are on opposite sides of the tangent line, for a segment that has the vertex of the parabola as its midpoint, this forms what could be called a parabolic wave with one segment forming a through and the other a crest… but if the parabloas are on the same side of the tangent line, they wrap around some center point… my intuition is that you would have to truncate pretty close to the vertex of the parabola to get a convex, closed curve repeating the process and that most would have selfintersections… Or working from the opposite end, imagine making a curvy version of a pentagram by drawing parabolas such that the symmetry axis of the pentagram coincide with the axis of the parabolas, and the parabolas are tangent to the pentagram at its edge midpoints.
On a somewhat related note, if you take four copies of y =sin(x) from x= 0 to x = pi, I’m pretty sure you get a closed, smooth, convex curve joining them endpoint to endpoint… a fact that got me wondering about what exponent for a supercircle(aka a superellipse with equal major and minor axises, x^n+y^n=r^n for n > 2) is closest to this sine curved derived shape… and of how to fill in the gaps to make a solid if you took three orthogonal copies of the curve as sort of the skeleton of a solid… and my apologies if I’m not making much sense, had I an accessible means of creating visual aids, I’d be sharing them… and as much as I love Zome Tool and Polydron, Zome is useless for anything involving curves(unless you count approximating circular arcs with decagons or dodecagons and elliptical arcs with irregular hexadecagons), and Polydron is useless for curves other than quarter circles of unit radius(unless you cound approximating circular arcs with icasagons or triacontagons, both of which are of an unruly size in Polydron).
Though, to give props to what manipulatives I have access to, I feel like I wouldn’t understand the sphenocorona and the other sporadic Johnson solids had I not built them with polydron(sadly, not all Johnson solids are polydron constructible(the base edges of the pentagonal pyramid and pentagonal cupola are too sharp for the hinge joints polydron pieces make). Also don’t think I would have realized the icosahedron can be seen as a combination of a corona and megacorona without polydron… and I have found a few polydron constructible Johnson nearmisses(on of my favorites separates a dodecahedron into two bowls made of siz pentagons, rotates one by 36degrees, and fills the gaps with pairs of triangles. The Polydron model has no strain whatsoever, the pentagons are truly regular, but the triangles are very slightly isosceles(I think the triangletriangle edges are something like 1.07 times the length of the pentagonpentagon and trianglepentagon edges).
Also, when I found out about cyclogons(basically cycloids, but the rolling shape is a polygon rather than a circle), it got me wondering wht the patterns traced by the vertices and edges of a polyhedron as it rolls around a tiling of hte plane or around another polyhedron(mainly thinking cube on a square tiling or around another cube, tetrahedron, octahedron, or icasahedron around a triangular tiling or each other, , a dodecahedron around another dodecahedron, or a truncated icasahedron around a hexagonal tiling where it’s only allowed to roll over hexhex edges… or going back to cycloids, instead of circles rolling around circles, using cones or cylinders and tracing out the path of a segment instead of a single point(such as rolling a cone around a cone, tracing the path of a segment connecting the rolling con’s apex to it’s base edge or rolling a cylinder around a cylinder and tracinga diagonal of the rolling cylinder’s longitudinal section… or tracing a meridian of a rolling sphere or doing traditional cycloids, but insted of a single point, apply a gradient to a diameter and trace a family of cycloids…
Ugh, so many ideas in my head, no good way to get them out.
Ah that’s so cool! Sorry I didn’t address your visual impairment, I only get snatches of internet and went through your post very briefly. I like your ideas and thank you for the clarification!
I used to be very strong in Mathematics at school, winning Scotlandwide Mathematical Challenge prizes in a couple of years. Just the other day I found 1980s newspaper articles about this with photos of young me and other prizewinners! I continued to study Maths for two years at university, but part of a computer science honours degree, and almost astronomy honours, but had to choose which so went with computer science. Then as I started a software engineering PhD aged 22 my progressive neurological disease started, and I had to drop out. And later retrained very parttime from scratch (3 more degrees!) as an academic historian. But I do retain a fondness for recreational maths puzzles. Lewis Carroll’s maths puzzles were a huge influence on me as a youngster.
My husband switched from astronomy to a PhD in formal theorem proving/computer algebra/software. Officially under the umbrella of computer science, but very strongly mathematical leaning. Later he developed software for NAG of Oxford (Numerical Algorithms Group Limited) who provide numerical analysis etc libraries. And then switched to space technology research, within a university computer science department. He has to do pretty high level maths things daily. I boggle. But I never got that far.
And on the subject of Maths the ClubFloyd transcript of my game Napier’s Cache has just gone back up, after being offline (not at my request) for years. It was marvellous fun for me to see how the players reacted to the game. Which can still be played/downloaded from IFDB links. The game name comes because it is based on a true life event of mathematician John “logarithms” Napier. And a true story in my personal family history. AllThingsJacq.com  Interactive Fiction  ClubFloyd  April 5, 2020  Napier's Cache by Vivienne Dunstan