A Couple Construction Toys

I got a couple construction toys and thought I'd look at some things that can be built with them, focusing on loops and polyhedral structures.

Setup

I saw both of these manipulatives recently on a trip to Office Depot but ordered them online to get a cheaper deal. There are some size differences between different sets. I opted for smaller scale pieces, since they were cheaper and I could get more of them for modeling. For young kids, I would opt for a larger scale. They don't seem to have standard names, so I'll simply refer to them by their order of rotational symmetry. Here are some options for each:

$8$-fold symmetry

Office Depot kit (500 pieces for $15)
VIAHART Brain Flakes (500 pieces for $17)
Snowflakes (200 large pieces for $9.44, 200 small pieces for $5.84)

$6$-fold symmetry

Office Depot kit (200 pieces for $22.50)
Iselyn Building Toy (500 pieces for $22)
STEM Building Blocks (200 pieces for $6.29)

While the pieces look pretty different, they link in the same way: rotate one piece $180^\circ$ relative to the other and join them at their slots/gaps.

Challenges

Here are some ideas for activity arcs for both of the building materials. You'll want to use one or the other, since they don't play nicely with one another.

Loops
- What sorts of closed loops can you make?
- What's the smallest number of pieces that can form a closed loop?
- For a number like $6$ or $8$, can you find all of the different ways to make a loop that requires that number of pieces?
Balls
- What sorts of balls can you make?
- What's the smallest ball you can make?
- Are there any polyhedra that can guide you in designing a ball?
Tubes
- What sorts of tubes can you make?
- Can you find a pattern that allows you to build a tube as long as you like?
- Can you make a tube that bends around into a torus?
Towers
- What sorts of towers can you make?
- What's the tallest tower you can build that stands on its own?
- How high can you build using as few pieces as possible? (For example, what if you limit yourself to 100 pieces?)

For a low-floor activity, sometimes I like to run sessions where I give students a model that they have to interrogate and then build themselves, using it as a reference. A lot of the balls below would be great for that.

Notes

While I was expecting the $6$-fold symmetry pieces to be the most interesting since the $8$-fold symmetry pieces are more superficially rectilinear, minor imperfections in the pieces can lead to structures you should be able to build coming out slightly wonky. It's also the case that if you try and use a large proportion of the six slots on a piece, some of the attached pieces will be on the verge of popping back out. These two elements lead to the pieces fighting you as you build a little more than they should. This might have had more to do with my going with a cheap option than the construction set, generally. On the other hand, the $8$-fold symmetry set had no such issues. The disks also had more give, allowing me to fudge some arrangements, whereas a similar arrangement with the $6$-fold symmetry set would have seen pieces popping apart.

The easiest loops to make are the ones where pieces are essentially vertical or horizontal. With the $8$-fold symmetry pieces, here are a few:

Since the slots are spaced $45^\circ$ apart, the regular polygons are the square and octagon.

Here are a few loops with the $6$-fold symmetry pieces:

Since the gaps are spaced $60^\circ$ apart, the regular polygons are the triangle and hexagon.
For many of the above loops, the designs can be "hinged" to create variant loops:

Many can also be elongated by adding more pieces along their edges.
The more interesting loops for the $8$-fold symmetry set move away from this alternating rectilinear pattern. A useful one that requires six pieces is rectilinear in a different way:

This family of regular polygons requires fudging at the extremes, with the hexagon being the most stable:

I've found this rhombus particularly useful for building structures:

This triangle is also easy to build off of:

This pentagon has not yet found a use:
If you think about the loops as providing faces or other structures in a polyhedron, this provides a nice heuristic for building balls.

Below is a structure related to $6$-piece loop above. If we think about the purple pieces as faces and the blue pieces as edges, then this design is a cube, as shown on the left. On the right, we see a way to alternate cubes. This can be used to make a tube, cubes of larger scales, or an arbitrarily large lattice.

In the structure below, each color is part of the rectilinear octagon loop mentioned above. You can also see the triangular loop with sides of three different colors. If we focus on these triangles, this shape is an octahedron.

The structure below features only the rhombus loop as its faces. This might be my favorite of the structures I've made with the $8$-fold symmetry set.

The structure below is an elongation of the above structure, gyrating the two halves and inserting square loops in between.

The structure below is based on chaining rhombus loops around the equator with acute angles north-south, leaving the poles open.

This structure instead chains rhombus loops around the equator with acute angles east-west, leaving the poles open.

The structure below chains the non-rectilinear hexagon loop around the equator, leaving the poles open.
Here is a cube based on an $8$-piece square-shaped loop.

The model below is based on three hexagonal half-loops. (The obvious extension is to have three full hexagonal loops, but they end up popping out of the two nexus pieces.)

For the $6$-fold symmetry pieces, most of the balls I've found make use of the hexagonal ring. Here is the prettiest, where the hexagons serve as the four faces of a tetrahedron:

Here is a similar model in which we can think of the six hexagons as positioned as the eight faces of an octahedron. Because the gaps are so large and square-shaped, another natural interpretation is a cuboctahedron:

Below are some variations on a hexagonal prism or cylinder.