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A Well-Crafted Child’s Toy and Intriguing Object

By Paul | February 18, 2008

I no longer recall how it came into my possession - very likely I bought it myself for one of my two children (Samuel who is now fast approaching 3 years and Caitlin who is around 7 months) - but I have a flexible baby toy that seems to be no longer available. It was made by Haba but does not seem to exist on their website (or elsewhere on the web so far as I can tell). Incidentally, Haba appear to make a wide range of curious children’s toys (as well as numerous other child-related paraphernalia) and their site is well worth a look if you have a baby or young child.

As it turns out, the toy has proved far more intriguing to me (and other adult visitors) than it has to my children! It’s flexibility allows it to mutate between an equilateral triangular shape and a 3D framework (where its six component pieces form half of the 12 edges of a cube).

Purely for the intellectual and creative challenge, I set out to produce an animation of the metamorphosis from equilateral triangle to cube-frame. My eventual success surprised even me. I intend to make some further refinements but the current result is shown below. [Do not adjust your speakers. I have yet to attempt to add any kind of soundtrack to the animation as yet!]


The animation was created by doing some testing of mathematical ideas in MSWLogo (a FREE “turtle” graphics program which has a 3D mode in addition to the 2D mode more common to turtle graphics in the past). This only produced some rather pathetic wireframe images, but it provided a relatively simple working environment to test the mathematics involved. This proved an extremely helpful testing ground as it took me quite some time to model the situation accurately enough to produce a pleasing result.

These ideas were then used to create more realistic images in POVray (a FREE ray-tracing program that allows complex 3D forms to be created by amalgamating mathematical objects in various ways ). Whilst this particular animation is mathematically complex, it is incredibly simple to produce some impressive objects in POVray and animate them in all manner of ways. The interested reader is welcome to inquire further but is advised that POVray is an enjoyable but time-consuming hobby!

Multiple freeze-frame images from POVray were then compiled using Windows Movie Maker (which I presume is part of the Microsoft Windows package as I certainly didn’t purchase it specifically).

The mathematically-minded reader is invited to solve the following problem:

If the length of each of the six components is considered to be 1 unit (and for simplicity we are measuring each component from the centre of one hinge to the centre of the hinge at the other end):

  1. What is the distance from the centre of the red-blue hinge to the centre of the green-yellow hinge when the model is in the equilateral triangle formation?
  2. What is the distance from the centre of the red-blue hinge to the centre of the green-yellow hinge when the model is in the cube-frame formation?
  3. What happens to this distance as the model is flexed?

I have yet to answer the third question, but the answers to the first two questions have led me to an intriguing (but seemingly improbable) conjecture that I’ve yet to prove or disprove. I would be very pleased and impressed if anybody can confirm my results for questions 1 and 2, guess at my conjecture and help me out in any way with question 3!

Topics: Mathematics, Parenting |

One Response to “A Well-Crafted Child’s Toy and Intriguing Object”

  1. Cindy Says:
    March 31st, 2008 at 7:40 am

    Hi Paul,

    The triangle/half-cube looks really simple, doesn’t it? One of those deceptive things, in that it looks simple but really isn’t. You did well to create the animation. They’re tricky things.

    I wonder if you could actually built one of these toys yourself, using wooden pieces and rubber hinges… Would that even work? I’m not a mathematician, so there could well be a major flaw in my thinking.

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