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Most stable shape- triangle

Introduction

Some mathematical secrets hide inside this children's playground. Why do they make use of a triangular chain to attach the swing to the support? Is it only for decoration? How is it related to Mathematics?

Description

There are lots of facilities in a children’s playground, like swings or even a hammock. These are all based on principles of geometry and mechanics.





 

And if you look carefully at these structures, you will probably find that most are triangularly shaped. Their physical structure may not be triangular, but some of the components are. (See the above pictures)
 
In considering the structure of buildings or devices, we would like to think about the following:
1.      Cost – How we can use less material while still ensuring that the structure is strong and stable?
2.      Strength – The structure must not only support its own weight, but also withstand as much external force as it can. For instance, the swings above have to support the weight of the children using them.
According to Newton's second law of motion, F =ma
Where F is net force, m is the mass and a refers to acceleration.
To ensure the swing does not fall off, the net force in a vertical direction should be zero.
 
To achieve the above objectives, triangular structures may help.
 
Why?
1.      Because the triangle does not easily deform and is able to balance the stretching and compressive forces inside the structure.
2.      For economic reasons: since the triangle obviously has only 3 sides, it requires little material to make a support, thus minimizing the costs.

Overall, a triangle is the simplest geometrc figure that will not change shape when the lengths of the sides are fixed. In comparison, both the angles and the lengths of a four-sided figure must be fixed for it to retain its shape.
 

Regarding point one, this diagram shows how a triangulated structure withstands forces.


 

Actually, The Mathematical Bridge between two parts of Queens' College, Cambridge also demonstrates a triangulated structure. It is made up of a series of timber tangents joined together to make it rigid.


In architecture, this technique is called tangent and radical trussing.
By definition, a truss is a structure consisting of triangular unit(s) constructed with straight members whose ends are connected at joints referred to as nodes.

A tetrahedron-shaped base in a flying fox is a simple space truss. It contains six members joined at 4 points.

More examples in our city:

Bamboo sticks constructed in a tetrahedron shape to fix the tree in position, preventing it from falling over due to the wind.


 

The most common form of transport in Cambridge – bicycles – also demonstrates this property. The cycle frame is made of two triangles to withstand your weight.
This is an example of a simple truss.
Due to its stability, the "triangular" structure is ubiquitous in our city.
To achieve stability, the structure must satisfy this mathematical equation:
(where m is the total number of members, j is the total number of joints and r is the number of reactions, e.g. r=3 in a 2-D structure.) 
Demonstration:
In order to test how strong a triangular structure is, we are going to do an experiment.

First of all, we need to accordion fold an A4 sheet of paper.

Use two books as supporters and put the paper on top of them.
Now, we put a load on it.

Surprisingly, the A4 paper is able to support two books which are far heavier than its own weight.

 Some buildings are built with more than the minimum number of truss members required. Therefore, they do not not collapse easily even if some members are damaged. 

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