2 Pi R and Making Nantucket Baskets

2 π (r+1) - 2 π r = 2 π ?

My lovely wife, Etsuko, was recently excited to tell me about something she had recently figured out about making Nantucket Baskets and how she discovered it using basic math.

First the plug: Etsuko makes museum quality Nantucket Baskets. Go to the Nantucket Lightship Basket Museum on Nantucket Island and you can usually see one of her more recent works.
The Photo above is a Nantucket Lightship Basket made by Etsuko Yashiro. We also own a business  called GrayMist Studio and Shop, where we teach classes on making Nantucket baskets.

Next: to give you reference of the etymology of Nantucket Basket parts; unlike other forms of basketry, Nantucket Baskets were developed by the coopers (barrel makers) on the old whaling ships. This is why the parts of Nantucket baskets are named differently than traditional basket parts, and are similar to the parts of a barrel.

The top of Nantucket Baskets have two wooden rims: an outer rim, and an inner rim; sandwiched between the rims are the sides of the basket, called "staves." Generally speaking, staves are made of cane, and are 1 mm thick. Traditionally, rims are made by using a tape measure and some trial and error (depending on the thickness of your tape measure). Always looking to refine a process (part of being Japanese), Etsuko decided to use math to determine the lengths of the rims for round baskets.

In this photo you can see the inner rim and the outer rim with the staves sandwiched between. This is before the lashing is wound, and the seam between the rims are hidden. Photo - http://alltuckerdout.com/atobasket/make5.jpg

This is where her interesting personal discovery comes in. She says to me "I figured out that regardless of the size of the the basket - from the smallest round basket, to the largest; the difference in length between the inner rim and the outer rim is ALWAYS a 'hair' over 6 mm."

I thought this was counter-intuitive. I told her I would have thought that the difference in length between the inner rim and the outer rim would be proportional to the size of the basket. In other words, the longer the length of the rims, the greater the difference between the two. She said she always thought so too, but anecdotally noticed over the years that no matter the size of the round basket, the difference always looked about the same.

She went on to explain this in terms of basic math. I'll spare you the details, but I'll give you the overview. If you remember, the circumference of a circle is determined by:

          2 π r (r being the radius of the circle)

This would give us the length of the inner rim.

To calculate the length of the outer rim you would just add 1 to the radius or:

         2 π (r+1)

She showed me on paper, that the difference between 2 π (r+1) and 2 π r is 2 π.

Or 2 x π x (r+1) - 2 x π x r = 2 π

Otherwise known in millimeters as 6.28 mm - or "a hair over 6 mm." Since the value of the radius cancels each other out in terms of the difference, this is always true regardless of the value of the radius.

Isn't this cool? Maybe I didn't explain this well, or maybe you're saying "DUH....everyone knows that."

Etsuko and I thought it was cool.