The plastic ratio is given by
![\begin{align*} \rho &= \sqrt[3]{\frac{9 + \sqrt{69}}{18}} + \sqrt[3]{\frac{9 - \sqrt{69}}{18}} \\ &= 1.324717957244746025\ldots \end{align*}](https://www.johndcook.com/plastic_constant2.svg)
The Dutch architect Dom Hans van der Laan gave the number this name in 1928. He used “plastic” as an allusion to a 3D construction of the number, analogous to the 2D construction of the golden ratio.
Here’s a plastic rectangle, a rectangle whose sides have the proportions of the plastic ratio.
@Gregoresate commented on Twitter that this ratio is aesthetically “disharmonious” and “bereft of meaning.” He could have said it looks plastic.
van der Laan claimed the plastic number was how humans mentally sorted proportions, but looking at how he applied it to architecture it’s striking how disharmonious it appears. There’s arrangement but no spiritual alignment between scales. Nesting bereft of meaning, no whole. https://t.co/y6H36nBwdr pic.twitter.com/oS5or5pYki
— Gregoresate (@Gregoresate) May 27, 2021
I doubt there was any negative connotation to the word plastic in 1928. The primary meaning at probably had to do with deformability, not with synthetic materials. To the extent that the word was associated with new materials, it would have had a positive connotation at the time. It’s interesting that the use of the plastic ratio has been criticized for being plastic in the contemporary sense of being inauthentic.
Perhaps Gregoresate is a frustrated Pythagorean — and do they get frustrated, boy howdy. All I can think of to say about the plastic rectangle is that it is a tiny bit squarer than 4:3, which was formerly for many years the default aspect ratio for television and therefore for the first few generations of computer screens. Somehow I think the Pythagoreanness of 4:3 is perceived subliminally, whereas I’m pretty sure that x^^3 – x – 1 is not.
The frustrations of Pythagoreans are particularly connected with the mathematics of music, a topic that I know interests you. I have written a set of three papers on this, which can be found by searching my name on academia.edu . The second of the three papers, Tertium Quid, throws an interesting light on the prevailing standards of intradisciplinary debate in the 14th, 15th, and 16th Centuries. Those guys knew how to have fun….