In many types of toy train and slotcar track, there is one "common" length of straight track section and curved track sections are all 1/12 or 1/8 circle of the same radius; while some allow for a few variations on curve radii, interconnecting the different sizes can be troublesome.

I would suggest
that making the curved sections be 1/8 circle and having curved-track radii and straight-track length that vary by powers of sqrt(2), it should be possible to make many types of designs fit together very easily.

Although high-end model railroaders would probably want to stick with the existing track selection (or--better yet--use flex track which is adjusted to suit their particular needs) this approach could be well suited for slotcar tracks (where the variety of turning radii would make racing more challenging) and to toy trains which use tighter turning radii than would be used on real trains or models thereof.

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Yes please. Also Legos and any other sort of construction kit. A number of construction kits actually offer 45-degree bend joints but only offer straight pieces in integer multiple sizes.

The nice thing is that scaling by sqrt(2) twice gives you a power of two. So if you have three sizes of length 1, sqrt(2), and 2, you can either use 1 and 2 for "straight" and sqrt(2) for "angled" or sqrt(2) for "straight" and 1 and 2 for "angled". (And scaling up to 2sqrt(2), 4, etc.) Of course this is obvious if you just think about rotating your whole design by 45 degrees but anyway.

If you wanted to get fancy you could mix in some sqrt(3) pieces, so you could do 30/60 angles conveniently as well.

egnor: Using sqrt(3) pieces and 30-degree segments might be an alternative to sqrt(2) pieces and 45-degree segments, but wouldn't be as thoroughly scalable since you'd need things scalable by powers of two as well as by powers of sqrt(3). By contrast, anything which is scalable by powers of sqrt(2) is inherently scalable by powers of 2, so only one 'basis vector' is needed rather than two.

No, because anything scalable by sqrt(3) is also scalable by 3, which is also acceptable. It's arguably superior, in fact; assuming you can splice segments together, it seems more efficient to have more varied lengths. And you get more angles.