Triangles come in many flavours. But all triangles have one thing in common apart from having three sides : they are stable. The best way to understand this is to think of a different shape, for example a square. It can easily be transformed into a parallelogram.
This is why you see triangles all over the place in the world around you. In electricity pylons, cranes, bridges, and many houses.
Equilateral triangle, and from the middle of each side, one were to place an interior frame board, spanning from each center of the exterior sides to the next, so that now, the triangle has is segmented into four triangles. How much stronger has the triangle become? Doubled, tripled or more in strength? I ask in that I have an interest in geodesic domes and because, it is the basics of a truss system and I wondered if there is a definitive equation on this.
We are only talking about the pressure placed on a single tip of course. Your email address will not be published. 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. Hence the minimum for a 2D structure is achieved by the triangle. If three lengths can form a triangle at all, that is, if the longest is less that the sum of the other two, there is only one triangle that they can form.
You can uniquely solve for the angles, and the only way for them to change would be for at least one of the lengths to change. If you have four lengths, from which you can assemble a quadrilateral, there are infinitely many sets of angles that could work.
Just consider how a cardboard box collapses flat when the top and bottom are removed: you can see that the shape moves smoothly through various angle solution sets with no change in side lengths, starting at 90,90,90,90, and ending at ,0,,0. For simplicity, let's think about shapes made up of line segments, which can't bend or stretch in any way, and "joints" between them, which can rotate but can't pull apart.
You can imagine solid metal bars with pins put through them, for example. First, imagine a unit-length line segment in a plane. We can say that this segment has three degrees of freedom : you can't fully specify it with less than three numbers two for the position, one for the rotation.
This is a reasonable definition of "rigidity" within this model. So let's define it: a shape is rigid if it has no more than three degrees of freedom. In other words, it can be moved around as a whole, and it can be rotated as a whole, but there's nothing else that can change about it. Now imagine two unit-length line segments. There are now six degrees of freedom: each segment needs two numbers to define its position, and one for its rotation.
But if you fasten them together at one end, that takes away two degrees of freedom. The position of the second line segment is now completely determined by the position and rotation of the first one and the length of the first one, but we've said the segments can't stretch or shrink.
So now we only need four numbers: the position of the first one, the rotation of the first one, and the rotation of the second one. This turns out to be true in general: a joint between two segments, in this model, removes two degrees of freedom.
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