Highway Ramps

I was sitting in traffic back April, and I began thinking about highway ramps. I was waiting to get on one after all. I noted that any intersection of 2 highways must have 8 exit ramps to accommodate all possible directions of travel. That is, approaching the intersection from any one of the 4 possible directions (ie. North, South, East, or West), you would have 2 possible ramps to take. Multiply that by each of the 4 directions, and you get 8 total ramps. Simple enough.

This got me to wondering how many ramps would be in a 3-way intersection. For our purposes, the third highway can be vertical. So, in addition to North, South, East, and West, you would also have Up and Down. How many ramps would such an intersection have? Each direction now has 4 possible exits. Now, if you are heading North, you have the option of turning East, West, Up, or Down. Muliply that by the 6 possible directions, and you now have an interchange with 24 ramps. That’s quite a lot more.

I figured that there must be an equation that you could use to calculate the number of ramps (r) for any intersection based on the number of intersecting 2 way highways (h). So, still sitting in traffic, I managed to come up with the following equation:

r=(2(h-1))2h

I later refined it to this slightly cleaner equation:

r=4h²-4h

There you have it! It works. Not bad considering it’s been a while since I’ve taken any math classes. This should come in handy the next time you need to calculate the number of ramps you’ll need to construct for an intersection of 6 highways. (120 ramps, in case you were wondering.)

Today, I made the two lovely pipe cleaner models you see above. The first is a standard cloverleaf with 4 ramps. The second is the 3 highway intersection with 24 ramps. That one was considerably more complex and harder to make. Plus, the pack of pipe cleaners I got only had 5 colors, so I had to also get a sparkly pack. It’s not immediately clear in the photo, but this intersection is essentially made up of 3 cloverleafs (one per axis). I’ve no idea how you would figure out how to model more complex intersections, but you are welcome to try!

-Sean