We have the occasional “discussion” here at the farm (ranging from will we have enough pumpkins to who isn’t doing their fair share of laundry), but our most recent conflict came over the Golden Spiral. Which is interesting on several levels: it’s only been since 2010 that the Golden Spiral (derived from the Golden Ratio) has existed in our minds as a topic of discussion; most people don’t argue about it; it is an example of mathematics in use in an ordinary household; and it distracted us from another round of “the reason no one has underwear is that the laundry pile has eaten all of it.”
A lot of the arms of the squid end in Golden Spirals by design–they look great, and they also give us an opportunity to use the concept in field trips and in math connections on our website and for when we do presentations at schools. I overlay the Golden Rectangle template on the cutting map to make it easier to lay out in the field.
I brought this section of the map out to show how to cut the spirals, and when I explained that the spiral was made up of a series of quarter circles with radii that are the sides of the squares, I showed this picture as a how-to:
Alan didn’t believe that it would work out like this. He tried to describe his reasons for disagreeing with me, but we quickly ran into a lack of vocabulary to frame the issue: he just kept saying that spiral doesn’t work like that, and I just kept saying (in all caps) that BY DEFINITION a Golden Spiral has to be drawn like this, and then we got out a compass, which showed that I was right (in the sense that the quarter-circle technique would work in the field)–but we were still confused. So, why does the Golden Spiral work like that? What if I’d made another kind of spiral–would there be a shortcut then?
Alan’s position was, in essence, that a spiral should, BY DEFINITION, have a constant rate of change, so you shouldn’t be able to put together a series of quarter circles to make it.
He’s had very little academic math but a lot of practical, real-world math. I’ve had some academic math (through a couple of semesters of calculus, a while ago) and an interest in math in general–just the interesting parts, though. Between the two of us, we came up with the ideas that yes, this rectangle overlay will help when cutting the maze, but if we’d put in another kind of spiral it wouldn’t work.
Then we had a brief but fascinating discussion about mathematical intuition and folk physics and being a veterinarian vs. being a farmer, and then we segued into Newtonian and non-Newtonian and Euclidean and non-Euclidean realms (okay, that was actually just me talking and everyone else edging out the door.) Alan and the rest of our maze cutting crew headed out in to the real world to actually get something done and I just went to the computer and googled why can you make a golden spiral using a golden rectangle, and related search terms.)
And the [short] answer is: there are different kinds of spirals (which is obvious) and the Golden Spiral is a special one that gets larger by a factor of
for every quarter turn (BY DEFINITION!) and that for other spirals, you don’t necessarily have that every quarter-turn constraint–most of them just get larger continuously, as Alan predicted.
So, it’s good I picked the Golden Spiral for use in the squid, because the rectangle template makes them “easy” to cut…
Here are some pages to explore for a better explanation of this topic
And this one is by Vi Hart via Khan Academy–if you haven’t seen her work, you should check it out–amazing!