Geometric toys, puzzles, and amusements have been around for who-knows-how-many millennia but in the last few decades, a convergence of technologies and mathematical understanding and explorations has exploded the possibilities and offerings. While there are countless potential geometric toy concepts, we’ll focus (at least at the end of the post) on the permutations of the Platonic Solids. If one wants to have a huge range of options (including – but hardly limited to – the famous fave five Platonic Solids), my first recommendation is Zometool – the virtues of this versatile construction system this site extolls within numerous posts. There are also 5 pages of posts with geometric toys already mentioned on the GeometryCode website, including RedHen Educational Books and Toys, Fractiles, Astro-logix, and more.
All Five Polyhedra – Octahedron outermost: Nested Platonic Solids model using Zometool; note the golden ratio division of octahedral edges defining icosahedron verticesBefore we look at Platonic Solid-inspired toys, the Hoberman Sphere is a fun toy that deserves mention; it collapses down to a dense “solid” at a fraction of the expanded circumference:
… and a related (and very fun to toss up and down and watch colors change) Hoberman Switch Pitch Ball – check out the video to see how this works.
There are countless ways to “build” a cube and the SHASHIBO Shape Shifting Box transforms into over 70 shapes, emerging from … and returning to … a cube.
Platonic (Solid) Toys and Puzzles
The most ubiquitous of the 5 Platonic solids is the (hexahedron) cube and when one subdivides its six square sides into multiple (tiled, tesselated) sub-faces interesting topological possibilities arise!
The Rubik’s Cube was invented a half-century ago, on May 19, 1974. (I just missed it by a month! Happy summer solstice 2024, BTW!) When (about that same time frame) I was an undergraduate studying Electrical Engineering (Circuits and Signal Processing) at UC Davis (California) I recall two class assignments involving the Rubik’s Cube. The first (for a computer programming class, I believe using Algol) was to solve the puzzle using any arbitrary orientation of the faces; fortunately, we were given the algorithm (not Al Gore rhythm) as implementing it was plenty challenging enough! The second class assignment (in a newly launched class titled “Computer Graphics”) was to make a 3d virtual model of the cube, and print out images like the one below, along with various assigned rotations.
The configurations of a 2×2×2 Rubik’s cube reachable using only half twists form a Nauru graph. Wolfram (2022) analyzed of the 2×2×2 Rubik’s cube via a multiway graph, the first few steps of which are illustrated above.Here are numerous 2×2, 3×3, 4×4, 5×5 + 6×6 + 7×7 toy variations of the famous (and mathematically analyzed Rubik’s Cube (dividing each square face into 4, 9, 16, 25, 36 or 49 smaller squares) that should keep spinners and fidgeters busy for quite a while. Here’s one with 486 (9 x 9 x 6) small rectangular or square facets:
“Hungarian design teacher and serious puzzler Erno Rubik assembled his first cube puzzle in 1974 and called it the Magic Cube. After a toy agent pitched the puzzle to Ideal Toy & Novelty Company, it renamed the puzzle Rubik’s Cube and began putting it in stores in 1980.” – from the Strong National Museum of Play
It’s interesting that the product description used just about every sort of related word EXCEPT dodecahedron!
There are many variations of these available online, including some inexpensive options.
Here is an octahedron puzzle …
… and an example of a tetrahedron puzzle …
While there aren’t any moving parts in these icosahedra, rotating the entire shape by hand provides a mesmerizing and aesthetic demonstration of refractive color mixing:
Do-it-yourselfers with some mirrored glass (and abundant care) might try making their own Kaliedoscapes, a name coined by fellow geometer, Sara Frucht who gifted me a small 3-triangular-mirror version decades ago which I have since given away before our most recent downsizing move. What’s quite fun is to see the “virtual polyhedra” that appear when the triangles are slightly truncated, or when an object such as a small rubber ball is dropped into the corner. Here’s a cool video that I’ve shared before with more examples.
Of course, Plato would remind us – while we spin, rotate, fidget, and morph these myriad manifestations – to look for the essence behind the ephemeral, the source beyond the symbol, the reality that transcends the reflected form.
“As a boy I always wanted to know how it looked on the inside. So I would very much like to make one you can walk into. I know all kids would love it. I know people in China that have the skills to help me do it.”
