Is this Duplo train track under too much tension?

Any piece able to freely rotate is considered the same structure. For example, for 2 1X2 legos the arrangement count is 2: top connected to bottom with both nubs, top connected to bottom with one nub because if you analyze legos you will find that such an arrangement can freely rotate over 270 degrees, and left vs right nubs result in the same structure when taking rotational symmetry into account.

For the problem I assume an 'ideal' lego with 0 manufacturing tolerance, no illegal building techniques are allowed.

Is there a name for the above combinatorics question? Is it well-posed?

Is there a closed-form solution? If not is there a generator program?

I should say that with a high enough N any generator would be very complex - imagine how degrees of rotational freedom give rise to the possibility of further structures hidden from other rotational orientations.

Straight sections are mostly ignorable since you can always add them in pairs on opposite sides of the loop if they are parallel. (although there are some interesting triangle-shapes that can be made that break that pattern)

My friend even went so far as to code up a solver for it which mostly worked and generated some interesting layouts. We never got around to adding switches into it.

It eventually led us to the math behind necklace problems because it was often hard to tell if 2 track layouts were identical: https://en.wikipedia.org/wiki/Necklace_problem

Happy to elaborate further if it is valuable to the discourse!

https://www.cailliau.org/Alphabetical/L/Lego/Duplo/Train/Rai...

I owned both "new-type" and "old-type" (black) Duplo rails as a kid. I remember that even as a 4-year-old, I was annoyed with the old-type black rails and greatly preferred the new ones.

View each track piece as a 2d vector. Add up the vectors. In a zero-tension setup, the sum is (0, 0).

As a metric for tension, assume any mismatch in position is evenly distributed. Model this as the average of all the vectors. (Thus the same displacement is more meaningful when we have fewer pieces.)

___

That's the full idea. It might seem that it is ignoring rotation, because it doesn't explicitly mention rotations, but they are included because the effective vector that a track piece provides is both a current direction as well as the displacement contributed by that piece. If we wrote some code to model this, a cursor would consist of a direction (an angle) along with an (x, y) position.

___

Some related math concepts:

* The exterior angles of a polygon sum to 360. So we could have another measure which is how far we are away from 360.

* Not useful in this case, but this also reminds me of winding numbers from complex analysis, which is a way to locally walk along a curve to understand which side is the "inside" or how many times a curve goes around a given point.

But (for me), the same is no longer true for stackoverflow. I used to participate on it both as an asker and an answerer. But something happened. It felt like it was a takeover by ever pedantic moderators. Now I participate there only rarely.

So when under severe misalignment, one side of the key would be pushed with extra lateral pressure and may deform or break.

However, this sort of severe tension is likely to be in effect while attempting to link/lock the last joint. It's likely to be done by the child when the parent is not there to supervise the feasibility of such forced link. The parent will be alerted when it's either too late or when it succeeded and there's no need to fix it.

Thus, if it were to snap a key neck, then it's just meant to be... No drama. The second key is still there to maintain the joint. Though caution, if no lesson is drawn, such section would become even weaker link!

If it somehow coerced into a loop, then Yay! here comes the locomo. If the train cars don't tip over the forced link gaps or warped sections, then the ride goes on. Otherwise, a tuneup/rebuild is due.

That would be my puzzle solution to:

1. Assign each piece type it's end offset and next piece connection angle

2. Start at 0,0 coordinate and iterate through pieces, advancing last piece position

3. Check the offset between the start and end pieces

And the result would look like images in the answer.

Updating the track to minimize offset is harder, though.

https://lamington.wordpress.com/2011/12/02/laying-train-trac...

Except for the simplest of tracks, I often wonder if the misalignment of a complex track is not stressing the pieces. Of course, instead of asking in stackexchange I dismiss the thought and just play -- er, my daughter plays -- with the train.

Don't the track pieces fit together loosely?

And aren't most Lego/Duplo pieces made of such hard and rigid plastic that they don't effectively bend at all?

So while it's still an interesting math problem about angles and lengths, I'm not sure the premise of "tension" is correct here.

"I would first check for track flatness"

This thread is a great example of how engineering is often NOT a solution to problems, classic "hammer and nail" territory here. And how engineers often ofterthink things unnecessarily ;)

[1] https://projecteuler.net/problem=208

Märklin C-track is sectional too, but has rather tight tolerances for assembly. There is no flex C-track. A computational method for solving parts lists and connection plans would be fantastic.

Currently people do it "by hand (virtually)" with a variety of apps, but that is labor-intensive.

Where is the combinatorial algebra?

Does this indicate any sort of predilection for math or for engineering ? Or is it just a usual sort of kid behavior ?

However the question is false in its initial assumption, i think: if theres too much tension anywhere in the string that joint will separate. These pieces are designed to do that.

Perhaps a better way of stating it would've involved the gaps between sections where there might be too much space and lead to derailment.

Probably need to define "too much tension". Is a bit of tension that enables you to build the thing you want and couldn't otherwise, a good or a bad thing? (e.g. maybe I want a spiral)

I'd have thought if overly tensioned, once tolerances were overcome, the track would develop a camber. Maybe build on a perfectly flat, frictionless surface and then if your track isn't perfectly level you know there's tension.

I suppose if the track is big enough then you would be able to insert a "wrong" piece without necessarily using up all the slack, so the pieces would still be somewhat loose. But in that case there would be no mechanical concern to worry about.

Actually I suspect that it suffices to check one piece. If any piece is in tension then they all will be, assuming friction with the floor is not too large. Unless you have intersections in the track, then you have to check each loop separately, or maybe you could just check the switch pieces. Might be an interesting math problem there to minimize the number of pieces to check in complex tracks.

I'll also point out that bending Lego pieces isn't always bad: https://youtube.com/@BrickBending

The asker severely underestimates the amount of force it takes to break a Duplo piece.

> Not really an answer to the question as posted — but I think the premise needs some good parenting advice: let your kids break bricks. They are pretty darn durable anyway and fabulously cheap to replace. So when they break one they will begin to learn about over-stressing materials through their own experiences

Duplo, while expensive, is a consumable, if you look at it through this old man's eyes.

Now that that's out of the way, I love all the answers here.