Implementation of mathematical curves and Hooke’s chain theory in architecture
Tianwei Lu
Introduction
At first, when Leonardo Da Vinci addressed the problem of the shape of a chain having two fixed ends with both ends are fixed in place, he assumed shape of such string is that of a quadratic parabola with equation 𝑦 = 𝑎𝑥 2 + 𝑏𝑥+𝑐 . In Galileo’s work, Dialogues Concerning Two New Sciences , published in 1638, the quadratic curve was applied in a real-life context, related to a physical phenomenon. Galileo assumed that the shape of a light chain when hanging under its own weight with two fixed ends on the same horizontal level creates the shape of a parabola, which is incorrect, as stated in a later note in the journal. For a chain which hangs under its own weight can only be approximated using a parabola. 1 In this case, Galileo has followed previous assumptions and thus submitted to confirmation bias.
Figure 1 Catenary (red) and parabola (blue)
In 1691, Huygens proved that the shape would not be a parabola by using similar triangles, making a breakthrough of not assuming the string to be weightless where evenly spaced weights are attached (this model will be explained in section 2.2), but using sections of the string that have weights. This adjustment allows for a less erroneous proof, which will be shown in 2.1. 2
Proof of using the modelling methods
2.1 Catenary curve
The equation is given as:
𝑦 = cosh𝑎𝑥 𝑎 In the context of a string or chain hanging under its own weight, there are three main forces that are applied on a section of said string or chain: 𝑒 𝑎𝑥 −𝑒 −𝑎𝑥 2𝑎 =
1 See Dialogues Concerning Two New Sciences Day two and Day three. 2 See Curves for the mathematically curious.
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