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## The Center of the Problem

The e-mail and its request were quite unexpected, being sent out to members of our Mathematics Department....

From: Dean Kahn [Editor of The Bellingham Herald]
Date: Friday, January 19, 2007 3:21 PM
Subject: Query
I wonder if there's a way to calculate the geographical center of the city of Bellingham. I've talked to some city staff people and surveyors, and they are a bit befuddled, or use approaches that result in two focal points, not one.

Then it occurred to me that there might be a mathematical formula for calculating the center of an irregularly shaped object (in the case of the city limits, very irregularly shaped). If a member of the department is interested in taking this on, I'd love to hear what he or she has to say.

Question 1: How would you respond...that is, how would you find the geographic center of your town?

Question 2: You also might want to find the geographical center of each of the 50 states? Or possibly of the United States itself...but do you use 50 states or 48 states? For an interesting perspective on this, you might consider the Americasroof site or this ASK YAHOO site.

Note: The headlines in the March 23, 2007, issue of the USA Today newspaper read: "Geographic Center of USA 'Moving'." What does this mean?

Hint: Think string and hanging pictures.

Solution Commentary: The following e-mail discussion reveals solutions to the problem:

From: Millie Johnson [WWU Mathematics Professor]
Sent: Friday, January 19, 2007
To: Dean Kahn [Editor]
If you have a decent map, you could numerically estimate the centroid and get as close as you want to the center depending on how finely you cut the grid lines.

That is, you need the geographic region on a piece of flat paper and you need a straightedge and ruler. Establish a grid over the entire region. The finer the grid, the closer the estimate of the centroid (center of mass-in this case geographic center) will be.

The centroid is the point on planar region that if placed on the tip of a sharpened pencil lead, would theoretically balance the region. Since you are only considering the geography, then the assumption is that the region is homogeneous. The balancing problem uses an adjusted formula if you consider, say, density of population. Yup, there are formulas for these computations.

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From: Millie Johnson
Sent: Friday, January 19, 2007
To: Dean Kahn [Editor]
Subject: Part II
Part II and a lot less trouble:
1. Cut out a map of Bellingham on stiff material, cardboard, tagboard.
2. Hang the map by a string.
3. Draw a vertical line across the map from the point of suspension.
4. Remove the string and rotate the map a little (a few degrees), hang it by a newly attached string, and draw another vertical line. The lines drawn should intersect at approx the centroid.

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From: Dean Kahn [Editor]
Sent: Friday, January 19, 2007
To: Millie Johnson
Subject: Part II
A fellow at the USGS suggested cutting out a map of the city, gluing it to firm paper, cutting the paper to fit the map outline, then using a pen or other pointed object to find the balance point, and that, more or less, is the center of the city. Would that approximate the "string" approach?

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From: Millie Johnson
Sent: Friday, January 19, 2007
To: Dean Kahn [Editor]
Subject: RE: Part II
Same thing. The string method is usually a bit easier, since the balance point is rather tricky to find on unusual shapes.

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From: Dean Kahn [Editor]
Sent: Friday, January 19, 2007
To: Millie Johnson
Subject: Part II
With the string method, and with such an irregular shape as the city of Bham, how many string reattachments are necessary? Is just two sufficient? Also, it sounds like it doesn't matter where on the perimeter to attach the string initially. Is that correct?

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Date: Sat, 20 Jan 2007
From: Millie Johnson
To: Dean Kahn [Editor]
Subject: RE: FW: Part II
Irregularity is irrelevant. The centroid (see my first message) is the balance point. Since there is one centroid for Bham, then theoretically regardless of where you attach the string, the vertical line will cross the centroid (balance point). Since two lines intersect in a point and theoretically there should be ONE point, then two string attachments should do. However, since there will be experimental error, you could try a bunch of string attachments and sort of take an average of the intersection locations.

The first method that I sent to you (cutting up the region into a fine grid might be more accurate, but would take a lot more time.

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Additional Note: From the March 23, 2007, issue of the USA Today newspaper: "The 1959 site (i.e. geographic center) was determined, according to David Doyle, chief geodetic surveyor for the National Geodetic Survey, by using a map of the USA, pasted onto a hard surface like cardboard, and then balancing it on an object such as a pencil. Some rudimentary math was applied to allow for Hawaii and Alaska, he said."