Why is rigidity associated with triangles

The idea is that no matter what shape we start out with, any shape you draw in the plane at all will look the same after a rigid transformation is applied, except that its location and situation might be different from what it used to be. It might also be its mirror image, if you have allowed transformations to flip the orientation of the plane.

This would change a circle at the origin into an ellipse. If you are interested in geometry that is not founded upon distance, then you can adopt some geometry axioms that assume notions of congruence of segments and angles. Rigid transformations of the plane would then be ones which do not disturb segment congruence nor angle congruence.

Now, there is a notion of rigidity in reality that has more to do with its resistance to changing shape. This is called structural rigidity. This is really not the same animal as rigidity of shape in geometry, although it's obviously related. In the diagram you supplied, the point seems to be that we are assuming the segments do not change length, but that the joints are on hinges.

You are able to apply physical forces to both, and see how they behave. One would classify the triangle as a rigid shape because none of its edgelengths or angles would change because of the intrinsic shape of the object. The square on the other hand isn't geometrically limited to having 90 degree angles when you apply pressure to it, so it can change into a rhombus rather quickly. Moreover, you could easily imagine building an object with length changing segments and rigid hinges, so that a square could be pulled into a rectangle, but not pushed over into a rhombus.

I don't think that object would be considered "rigid" either. I don't mean to say that the physics notion is totally disjoint from mathematics: for sure physical rigidity can be analyzed with mathematics.

It's just that the wiki page on geometry was not where you wanted be if you're interested in structural rigidity. I believe that rigidity is used in a relatively intuitive way here. However, if I wanted to ignore any pretense of geometric purity and just try to define the concept in a way that works, I would do it like this: it seems like one needs to have a notion of a ''bottom'' edge, which we consider to be fixed.

The others should be free to move, and in particular we want the preserved properties to be perimeter and area. Here, the concept of geodesic distances is difficult to formalize but easy to understand; it is simply the shortest distance between two points if one is restricted to traveling along the shape's edges. The Euclidean distance is a distance is just the length of the line in the plane connecting two points. These lead to frames as rigid as a triangle.

If more than these many links are provided, parts of designed frame start moving as a mechanism with extra degrees of freedom. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more.

What is the precise definition of a rigid shape? Ask Question. Asked 8 years, 2 months ago. Active 2 years, 10 months ago. Viewed 19k times. For instance, I've seen the following image: I understand intuitively why the triangle is rigid and quadrilateral is non-rigid.

Glorfindel 3, 10 10 gold badges 22 22 silver badges 36 36 bronze badges. My understanding of what your definition is isn't that great, any chance you could expand in an answer? In particular, for an isometry, is distance necessary to be preserved only along edges?

All other shapes can be deformed with a simple push if the shape is hinged at the corners for example, a rectangle can be pushed over into a parallelogram. But not the trusty triangle, which explains its ubiquitous use in construction, from pylons to bracing. Triangles are also special because they are the simplest polygon — a common approach to a tricky geometrical problem, such as analysing a complex surface, is to approximate it by a mesh of triangles.

This approach is also used in the real world to achieve some of the exotic shapes we now see in modern architecture, such as the curved shape of 30 St Mary's Axe, aka the Gherkin, or the canopy over the courtyard in the British Museum.

The method of triangulation is also vital in building our virtual world. The CGI characters we see in film and on TV are usually approximated by an incredibly fine mesh of triangles, as this makes it easier to digitally store and manipulate them. The strength of triangles also extends to the three dimensional world. A pyramid comprised of four triangles is the three dimensional analog of the triangle in the two dimensional world.

Any three dimensional object that can be reduced to a collection of triangles by adding triangular gussets is similarly rigid. The thing I find really cool about triangles is that the mathematics of triangles is wrapped up in a very neat package called trigonometry. In contrast, most mathematical disciplines seem to have no obvious beginning or end. Perhaps Differential Equasions would serve as an example??? There is also something mysterious about triangles that extends beyond the world of mathematics and engineering.

Our government, for example, is made up of three branches, the Executive, the Legislative and the Judicial. Triangles, then, are special not only the physical world but, seemingly, in the intellectual world as well.