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They are according to the buzz on the Internet (and most stable too), despite competition from circles. Mythbusters even proclaim that "triangles are the strongest shape because any added force is evenly spread through all three sides".

Is there a way to make some precise sense of the question, and if so, how does one actually prove that triangles are the "strongest"?

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    $\begingroup$ Related en.wikipedia.org/wiki/Structural_rigidity . $\endgroup$
    – user5402
    Aug 12, 2015 at 11:39
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    $\begingroup$ It depends on the definition of "strong". $\endgroup$
    – Kartik
    Aug 12, 2015 at 14:59
  • $\begingroup$ @Kartik I am wondering if there is a (reasonable and non-trivial) definition that applies to closed convex curves, say, and makes triangle the strongest. $\endgroup$
    – Conifold
    Aug 13, 2015 at 23:41
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    $\begingroup$ @Conifold I can suggest this definition: If the sides are of fixed length and only the joints can move, then the shape which will not cange shape under application of forces is the strongest. In this definition, triangles would be strongest. But if you say that the sides are not of fixed length, then other things may be stronger, e.g. The Circle may be the strongest as it will "distribute" the force evenly and prevent itself from breaking, while the side of a triangle might break under the same force. $\endgroup$
    – Kartik
    Aug 14, 2015 at 1:47
  • $\begingroup$ @Kartik Where do you have the force applied, at some single point? I don't like the idea that only joints can move because it singles out points artificially. I'd rather imagine some uniform material that resists stretching/compression and measure the "give" when a uniformly distributed force is applied along the curve, not sure if it makes sense. $\endgroup$
    – Conifold
    Aug 16, 2015 at 2:56

3 Answers 3

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As you asked about the strength of a triangular shape then let me introduce to the triangular chain consisting of three rigid links or bars connected to each other by pin joints(allowing rotation between two joined links) .

The degree of freedom (n) of a plane chain is given by the Grasshoff's law as $$n=3(l-1)-2j-h$$ for a triangular chain we have $$l=\text{no. of links}=3$$ $$j=\text{no. of binary joints}=3$$ $$h=\text{no. of higher pairs}=0$$ Hence, we get $$n=3(3-1)-2(3)-0=6-6=0$$ The degree of freedom of the triangular chain (equivalent to plane triangular shape) has zero degree of freedom this indicates that links of the triangular chain can't move even a bit if links are strong enough even under the application of external forces.

Thus a triangular shape is the strongest one which is also called a rigid structure. It is also called a perfect frame in physical structures.

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    $\begingroup$ I suppose for smooth curves the "number of degrees" is infinite, so triangle kind of maximizes its negative among closed curves. $\endgroup$
    – Conifold
    Aug 16, 2015 at 3:00
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Here's one part of it.

As far as polygons go, a triangle is the only one that is defined by its side lengths. If you have a triangle of sides 5,6, and 7, there is only one shape it can take. The same cannot be said of other polygons. Imagine a square. It can be squished into a diamond with the same side lengths.

There is SSS congruence for triangles, but no analogous congruence for other polygons.

That's what diagonal bracing does in physical structures. Creates triangles.

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    $\begingroup$ I understand the intuitions, but is there a variational "strength" functional defined on "shapes", which can be proved to be maximized by a triangle. Like area for "shapes" of fixed perimeter is maximized by a circle. $\endgroup$
    – Conifold
    Aug 12, 2015 at 5:27
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    $\begingroup$ Not that I'm aware of. But I'm certainly no expert. Possibly an engineering group could better answer that. Another thing to be aware of: stresses on a triangle are more directed to compressions and tensions on the sides, where as on other polygons, the can be applied to a joint, with no compressional or tensile strength to support the joint. $\endgroup$ Aug 12, 2015 at 6:38
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    $\begingroup$ Technically, this is also true of circles (defined by circumference), but those can still be squashed into ellipses and they aren't particularly practical for engineering in the way triangles are. $\endgroup$
    – Kevin
    Aug 12, 2015 at 13:52
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If the links are rigid (i.e. can't be deformed) then there will be no relative motion between any two pin-jointed links of a triangle while quadrilateral & other polygons have degree of freedom $\ge 1$ hence other polygons can be made to out of shape by applying force so triangles are only rigid or the strongest shapes.

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