I'm no structural engineer but if triangles are the strongest geometrical shape then why aren't frame tubes triangular, maybe it comes down to cost, torsional rigidity or flex but this would seem to affect all forms of structural engineering. Maybe it's just me missing the obvious but as long as the cross bar wasn't pointy end up (oouuch) it would act towards reducing weight.

The triangular tube itself might be stronger, but I would think at the end where it is welded the round tube would be stronger.

I am a mechanical engineer.
The strongest geometrical shape is a circle. That's why tubes are usually round.

Pace make a very nice frame from rectangular section tubing and there was a cast magnesium frame (Kirk) with an I-section, but triangular sections may concentrate stress at the edges too much. Circular sections dissipate stress evenly.

A diamond bike frame is actually not ideal for resisting the torque or twisting force, but it has to be flat to permit a rider to stradle the frame. Most bikes are sufficiently strong to resist the bending, but longer tandems do have problems.

In addition to diamond frames, you get space-frame designs, cross-shaped frames, and monocoque frames.

I've just had a quick surf through google and seem to have found both Triangle and circle/dome responses. here's a link to page showing this: https://newton.dep.anl.gov/newton/ask...3/eng/ENG1.HTM .

...and they start out by mentioning 3-dimensional shapes???
The reason why you see so many triangular shapes on bridges, et al, is because straight steel beams cost less than curved beams.Some of those beams are tensile links, too. If you look at the bottoms of the longest bridge spans, you will notice tha the arch is usually of a circular profile, or ellipsoidal.
Ellipsoidal tubes are stronger in one direction than in the perpendicular direction, making construction trickier. A triangulr cross-section would likewise be stronger directionally. I have never seen triangular cross-section tubes used for any cycling purpose.

My Dedacacci Alu 7003 uses a near-triangular tube profile on the three main tubes.

I have seen several Al frames with large "triangular" tubes, all of which have a fairly decent (1/4" to 3/8"?) radius on the edges. What impresses me are some of the extrusions I see - I think Colnago has some - that are essentially square cross section with semicircular depressions in each side. Come to think, they may be trying to replicate their logo in the tube cross section.

Cheers...Gary