Triangles come in countless flavours. There are equilateral triangles (all 3 sides have equal length), scalene triangle (none that the sides have actually equal length), isosceles triangles (at the very least two sides have actually equal length), right-angled triangles, obtuse triangle (one angle is better than (90) degrees), and acute triangles (all angles are much less than (90) degrees). But all triangles have actually one thing in usual (apart from having three sides): they are stable.

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The best means to recognize this is to think of a various shape, for example a square. If you make a square from 4 metal rods v hinges at their corners, friend will find that that doesn’t stay square. The can quickly be transformed right into a parallelogram. Girlfriend don’t need to bend the sides to carry out that, it wake up just since of the hinges at the corners.

For a triangle, no matter what type, this can not happen. It’s inherently rigid. It is a really special residential property to have: all other polygons (shapes made out of right line pieces associated at the finish to form a circuit) space not rigid. This is why you check out triangles all over the place in the world roughly you. In electricity pylons, cranes, bridges, and many houses.

Three-dimensional room is a little much more permissive. Mean you type a polyhedron by hinging with each other rigid encounters at their edges. In the 19th century the French mathematician Augustin luigi Cauchy proved that all convex polyhedra are rigid. Convex method that any type of line connecting two points the are part of the polyhedron is also contained in the polyhedron. This way that the five familiar Platonic solids room rigid.

The Platonic solids

What deserve to be said about non-convex polyhedra? The diagram listed below shows one icosahedron top top the left, a convex polyhedron v twenty faces, contrasted v a solid known as a little stellated dodecahedron ~ above the right. This is not convex as, for example, the lines between two of the pointed vertices room not always contained in the polyhedron.

An icosahedron and also a tiny stellated dodecahedron

It turns out that the tiny stellated dodecahedron is rigid, however for a an extremely long time nobody was able to prove whether it is true for all non-convex polyhedra. Then, in 1977, the mathematician Robert Connelly uncovered one the isn’t. It has actually eighteen triangular faces, but later a simpler example was found by Klaus Steffen with only fourteen triangle faces and nine vertices.

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Steffen’s flexible polyhedron has actually 14 triangle faces, 21 edges and 9 vertices

You can like to make your very own flexible polyhedron, so right here is a net. Over there is also a pdf version that friend may uncover convenient because that printing.