Watching the news a couple mornings ago, an impossible triangle appeared on the screen. Hopefully some readers might be able to turn some first ideas a colleague and I had into a great applied geometry lesson. What follows are some teacher thoughts. My colleagues and I hope to cultivate classes where students become curious enough to raise some of these questions themselves.
At first glance, the labeling seems off. In Euclidean geometry, the Triangle Inequality says the sum of the lengths of any two sides of a triangle must exceed the length of the third side. Unfortunately, the shorter two sides sum to 34 miles, so the longest side of 40 miles seems physically impossible. Someone must have made a typo. Right?
But to dismiss this as a simple typo would be to miss out on some spectacular mathematical conversations. I’m also a big fan of taking problems or situations with prima facie flaws and trying to recover either the problem or some aspects of it (see two of previous posts here and here).
WHAT DOES APPROXIMATELY MEAN?
Without confirming any actual map distances, I first was drawn to the vagueness of the approximated side lengths. Was it possible that this triangle was actually correct under some level of round-off adjustment? Hopefully, students would try to determine the degree of rounding the graphic creator used. Two sides are rounded to a multiple of 10, but the left side appears rounded to a nearest integer with two significant digits. Assuming the image creator was consistent (is that reasonable?), that last side suggests the sides were rounded to the nearest integer. That means the largest the left side could be would be 14.5 miles and the bottom side 20.5 miles. Unfortunately, that means the third side can be no longer than 14.5+20.5=35 miles. Still not enough to justify the 40 miles, but this does open one possible save.
But what if all three sides were measured to the nearest 10 instead of my assumed ones place? In this case the sides would be approximately 10, 20, and 40. Again, this looks bad at first, but a 10 could have been rounded from a 14.9, a 20 from a 24.9, making the third side a possible 14.9+24.9=39.8, completely justifying a third side of 40. This wasn’t the given labeling, but it would have potentially saved the graphic’s legitimacy.
Is there another way the triangle might be correct? Rarely do pre-collegiate geometry classes explore anything beyond Euclidean geometry. One of my colleagues, Steve, proposed spherical geometry:
Does the fact that the earth is round play a part in these seemingly wrong values (it turns out “not really”… Although it’s not immediately clear, the only way to violate the triangle inequality in spherical geometry is to connect point the long way around the earth. And based on my admittedly poor geographical knowledge, I’m pretty sure that’s not the case here!)
Perhaps students eventually realize that the distances involved are especially small relative to the Earth’s surface, so they might conclude that the Euclidean geometry approximation in the graphic is likely fine.
Then again, why is the image drawn “as the crow flies”? The difficult mountainous terrain in upstate New York make surface distances much longer than air distances between the same points. Steve asked,
in the context of this problem (known location of escaped prisoners), why is the shortest distance between these points being shown? Wouldn’t the walking/driving distance by paths be more relevant? (Unless the prisoners had access to a gyrocopter…)
The value of a Euclidean triangle drawn over mountainous terrain has become questionable, at best.
FROM PERIMETER TO AREA
I suspect the triangle awkwardly tried to show the distances the escapees might have traveled. Potentially interesting, but when searching for a missing person in the mountains–the police and news focus at the time of the graphic–you don’t walk the perimeter of the suspected zone, you have to explore the area inside.
A day later, I saw the search area around Malone, NY shown as a perfect circle. (I wish I had grabbed that image, too.). Around the same time, the news reported that the search area was 22 square miles.
- Was the authorities’ 22 measure an approximation of a circle’s area, a polygon based on surface roads, or some other shape?
- Going back to the idea of a spherical triangle, Steve hoped students would ask if they could “compute that from just knowing the side lengths? Is there a spherical Herons Formula?”
- If the search area was a more complicated shape, could you determine its area through some sort of decomposition into simpler shapes? Would spherical geometry change how you approach that question? Steve wondered if any students would ask, “Could we compute that from just knowing the side lengths? Is there a spherical Herons Formula?“
- At one point near the end of the search, I hear there were about 1400 police officers in the immediate vicinity searching for the escapee. If you were directing the search for a prison escapee or a lost hiker, how would you deploy those officers? How long would it take them to explore the entire search zone? How would the shape of the potential search zone affect your deployment plan?
- If you spread out the searchers in an area, what is the probability that an escapee or missing person could avoid detection? How would you compute such a probability?
- Ultimately, I propose that Euclidean or spherical approximations seriously underestimated the actual surface area? The dense mountainous terrain significantly complicated this search. Could students extrapolate a given search area shape to different terrains? How would the number of necessary searchers change with different terrains?
- I think there are some lovely openings to fractal measures of surface roughness in the questions in the last bullet point.
Ultimately, we hope students would ask
- What caused the graphic’s errors? Based on analyses above and some Google mapping, we think “a liberal interpretation of the “approximately” label on each leg might actually be the culprit.” What do the triangle inequality violations suggest about round-off errors or the use of significant digits?
- The map appeared to be another iteration of a map used a few days earlier. Is it possible that compounded rounding errors were partially to blame?
- Surely the image’s designer new the triangle was an oversimplification of the reality. Assuming so, why was this graphic used anyway? Does it have any news value? Could you design a more meaningful infographic?
Many thanks to Steve Earth for his multiple comments and thoughts that helped fill out this post.