Inchworm

An inchworm hangs from a single gossamer thread.

Doesn’t the inchworm have some marigolds somewhere to measure? A pity my own marigolds aren’t blooming yet, I thought. Then, thinking twice, I wondered why anyone would bother to write a song about it if inchworms and marigolds didn’t appear at the same season. Double checking my marigolds, sure enough the short ones have just started blooming.

Oh, it was Danny Kaye’s song:

4 Responses to “Inchworm”

  1. Noetica Says:

    An inchworm, bored with marigolds, Copenhagen, and the whole Hollywood-musicals shmeer, attempts to traverse an extensible piece of string of initial length one metre. Being a metric inchworm, it advances one centimetre with each “step”. After its first step, it is one centimetre along the string. Then the string increases in length by one metre (uniformly, so that our intrepid invertebrate is now two centimetres along the string). The cycle repeats: our vermicular hero takes another step of one centimetre along the string (so it is now three centimetres from the start), then string increases in length, uniformly, by one metre; and again, and so on.

    Questions:

    Q1. Does our imperishable would-be geometer moth ever make it to the far end of the string?

    Q2. If not, why not?

    Q3. If so, how many cycles will it take to get there?

  2. Noetica Says:

    Oops. Please amend:

    “then string increases in length”

    to

    “then the string increases in length”

  3. Nijma Says:

    What’s this then, a Philosopher with a math problem?

    But having defined our multi-legged friend as our “hero” puts a different complexion on it; I feel compelled to rescue it in some way, maybe make sure the journey is completed by some trick of Nijma ex machina, perhaps reaching a hand into the scene to arrange things for a more satisfactory ending, in defiance of the laws of physics. Or even to make sure the laws of physics are not suspended for the sake of the story. I mean, how often do you stretch a string and find it does not return to its previous length as soon as the ends are released? By simply stipulating that the string returns to its original length at the end of each cycle, the creature will complete its journey in something like 100 cycles, with each cycle covering 1% of the journey.

    But let’s follow the hapless creature for a few cycles to see what happens instead. In the first cycle, our hero has taken a 1 cm. step along a 1 meter string, for 1% of the journey. After the string is stretched, the creature is 2 cm. along a 2 meter string, still having completed 1% of the journey. When the string stretches at the end of the 2nd cycle, the creature goes from 3 cm. to 4.5 cm. along the string, which has stretched from 2 m to 3 m., so instead of having completed 2% of the journey, it has only completed 1.5% of the journey. In the third cycle, our creature is 5.5 cm. along the string after taking its step, and 7.33333… cm. after the stretch, while the string has stretched from 3 m. to 4 m. This time, only 1.8% of the journey is complete, instead of the 3% you could expect with an immobile string. Already a pattern is taking shape. With each successive cycle, the creature makes progress, but each time the progress is less and less. Will the journey be complete after 300 or so cycles, in the fashion of a mortgage that is paid for 3 times over by the time it’s paid off? Or will the creature’s progress successively diminish with each cycle until it reaches zero or even to the point of reversal?

    At this point we must examine the problem not as a math exercise, which no doubt could be solved quickly with the right software and the right machine to run it on, but as a philosophy exercise.

    Why does it matter whether or not our hero gets to the end of the string?

    It doesn’t.

    The creature is traversing the string out of boredom. It is an adventurer, not a mindless competitor for some artificial goal. The meaning is in the journey, not the destination. We are confused for a moment because usually this means “stopping to smell the flowers” which our hero has grown bored with.

    When I was a kid, I thought the lyrics of the song said “inchworm, inchworm, measuring the magical”, not “measuring the marigolds” and was disappointed years later to find out the true lyrics. This inchworm has just decided to live the dream, forget the marigolds, and go directly to “measuring the magical”.

  4. Noetica (n. pl.) Says:

    O Nijma, you have done well. Well enough, I mean – if I may post patronisingly. There is in fact a definite and more worldly answer, which I’m sure Ø would be able to work out. It leads to something interesting about transcendental numbers. But let’s leave that idea hanging, would-be-geometer-moth-like, for a little while.

    How engaging the images are! How richly “present” the foliage is in the first one, especially. (Where did you get these? Did you capture them yourself?)

    Now, prompted by your observations above I bring to the table a happy connexion between magic and measuring: the Sanskrit word māyā. This source, among the more flowery (nay, marigolden) effusions on the topic, says that māyā was “traditionally associated with mystery and magic as well as with measurement”. This source agrees; and so does another one, which even brings inches into play. Clearly our worm has something to show us that transcends the ever-cycling world of appearance and mere seeming, yes?

    (Ah, you don’t get this sort of thing at LH or LL.)


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