A passing chemist told me a four-carbon ring with double bonds all the way round is impossible — why?

Worth separating two different strains that get muddled here, because your model kit felt fine for a reason.

Angle strain is what you get from forcing bond angles away from where they want to be. Cyclobutane already has this — its corners are near ninety degrees where carbon prefers about 109 — and it copes, at an energy cost.

The linearity requirement of cumulated double bonds is much more severe. You would need to bend something that wants to be 180 degrees down to 90. That is not a strained molecule, that is a different bonding arrangement entirely.

Model kits are built to be forgiving because otherwise nobody could assemble anything. The plastic joints flex, so they will happily let you build molecules that would cost hundreds of kilojoules per mole. Building it proves the connectivity is imaginable, nothing more.

It is the double bonds sitting next to each other, not the ring size — and the specific problem has a name.

When a carbon has a double bond on both sides, that arrangement is called cumulated, and cumulated double bonds force the three carbons involved into a straight line. That is not a preference, it comes out of the bonding geometry: the central carbon's two double bonds have to point in opposite directions.

Now try to build a ring out of pieces that each insist on being straight. Four straight segments cannot close into a four-membered loop; you would need the corners to bend by ninety degrees, and the geometry does not permit it.

So the impossibility is not about four carbons and not about double bonds in general. Four-carbon rings exist perfectly well — cyclobutane is ordinary. What cannot happen is every carbon carrying two double bonds at once in a small ring.

The interesting footnote is that a four-carbon ring does exist as a species — it just is not the molecule you built.

Small carbon clusters have been studied extensively, and a four-atom carbon ring is a known arrangement. It is not a square with alternating or cumulated double bonds; it is a distorted rhombus with a bonding pattern that does not map onto the lines-between-atoms picture at all.

Which is a good general lesson about model kits. The stick model is a bookkeeping device for ordinary organic molecules, and it works remarkably well there. Push it into unusual regions — small clusters, electron-deficient compounds, metals — and the sticks stop corresponding to anything, while the atoms carry on existing quite happily.

"Cumulated double bonds have to be linear" is the specific fact I was missing, and it makes the whole thing obvious in hindsight.

It also explains why the kit felt fine — those joints flex far more than real bonding does, which is exactly what makes them usable and exactly what makes them misleading here.