## Sunday, March 27, 2022

### Simple Geometry

The men's retreat I was at this weekend is on a small lake -

Let me note in passing that the last men's retreat I went to was in the early 1990s and I will not be going again -

- a runner asked how far around it was.  I was about to answer "2.7 miles" when the director answered from across the room: "It is 2.58 miles on the inside of the road and 2.74 miles on the outside." That seemed an unusual level of precision until I remembered that they have short triathlons for the youth campers every summer, swimming something around 0.3 mile, then circling the lake running once and bicycling it twice. For that, one does need some precision.  I had reached my total by pacing it of in OCD fashion in a previous year, walking counterclockwise and thus on the outside of the road.  So it seemed about right.

Because we are usually looking at a 440 yd/400m track for measured events most of the time, and moving to the outside lane is usually brief, the difference is usually disregarded except for the sprints or the first lap at most. We know that passing on the curve means running a little farther, but it is negligible.  But it adds up quickly, doesn't it?  0.16 mile is a big deal in a timed event.

james said...

Assistant Village Idiot said...

Two lanes. I would have to estimate

james said...

Widely separated lanes? Maybe my modeling is missing something, but I figure the road has three classes of local shape: straight (outer length = inner), convex (outer length >inner), and concave (outer length < inner). I'd figure the concave and convex sections would mostly cancel out, except for the overall circling of the lake. Approximate by a circle, and delta-C = 2Pi delta-R, which gives a distance between inner and outer edges of a bit over 130 feet.

Assistant Village Idiot said...

I believe you

Thos. said...

A typical two-lane road is ~24' wide. It might be wider, but in remote areas, it's more likely to be a little bit narrower.

If the road is a perfect circle, the outside diameter is 48' larger than the inside diameter, making the outside perimeter just under 151'longer than the inside, regardless of the size of the inner circle.

James is right - unless there's something very surprising about the shape of the lake, this road would have to be impossibly wide in order to have the difference in lengths reported.

Assistant Village Idiot said...

Confirmed. What's the thought here? Should I take the whole post down or leave it up as a bad example and a warning to others?

Leave it up as a reminder that math is always a disappointment

Be a good do-be and tell the camp director how wrong he is!

Assistant Village Idiot said...

I'll be back in the summer and will personally pace it off on the inside track and the outside track.

james said...

I don't know about you, but by the end of 5 miles my pace might be an inch or so different from when I started.

Assistant Village Idiot said...

Yes, I thought of that. I will be there for a week and can split it up and also try in both directions. As it will be something around 6,000 steps each time, I might borrow my wife's fitbit for that instead, so I can just listen to a podcast. Though I mind counting much less than other people. Counting was a calming OCD habit of mine for years and I have gone over 10,000 several times. I get irritated later because I feel I didn't get anything done in those hours, but at the time I don't mind it at all.