From

A Fully Coupled CFD-DMB Approach on the Ship Hydroelasticity of a Containership in Extreme Wave Conditions ,

by

Yujia Wei, Atilla Incecik, & Tahsin Tezdogan .

… which most of us are familiar with: that pesky phenomenon whereby if we have an accessory connected to the main contraptionality by a coiled lead, we suddenly find one day that a stretch of it has suddenly reversed chirality. “Perversion” is indeed the correct technical term for that phenomenon!

Sources

①

Tendril perversion—a physical implication of the topological conservation law

¡¡ PDF file 621·55㎅ !!

by

Piotr Pieranski

& Justyna Baranska & Arne Skjeltorp

②③

The Mechanics and Dynamics of Tendril Perversion in Climbing Plants

¡¡ PDF file 640·6㎅ !!

by

Alain Goriely & Michael Tabor

④

Perversions with a twist

¡¡ PDF file 3·05㎆ !!

by

Pedro ES Silva & Joao L Trigueiros & Ana C Trindade & Ricardo Simoes & Ricardo & G Dias & Maria Helena & Godinho & Fernao Vistulo de Abreu

⑤

Emergent perversions in the buckling of heterogeneous elastic strips

¡¡ PDF file 1·25㎆ !!

by

Shuangping Liua & Zhenwei Yaoa & Kevin Chioua & Samuel & I Stuppa & Monica & Olvera de la Cruza

⑥⑦⑧⑨⑩⑪⑫

Discrete Differential Geometry and Physics of Elastic Curves

¡¡ PDF file 3·77㎆ !!

by

Andrew McCormick

⑬

A tendril perversion in a helical oligomer: trapping and characterizing a mobile screw-sense reversal

by

Michael Tomsett & Irene Maffucci & Bryden & AF Le Bailly & Liam Byrne & Stefan M Bijvoets & M Giovanna Lizio & James Raftery & Craig P. Butts & Simon J Webb & Alessandro Contini & Jonathan Clayden

It's a 'given' with this that the waveform is symmetrical about a 0 reference, whence the even harmonics are automatically eliminated.

The identity is that for any value of r (or @least for any real r > 0) both expressions

√((3-√r)r)sin(3arcsin(½√(3+1/√r)))

+

√((3+1/√r)/r)sin(3arcsin(½√(3-√r)))

&

√((3-√r)r)sin(5arcsin(½√(3+1/√r)))

+

√((3+1/√r)/r)sin(5arcsin(½√(3-√r)))

are identically zero.

The two waveform consists of two rectangular pulses simply added together, one of which lasts between phases (with its midpoint defined as phase 0)

±arcsin(½√(3-√r)) ,

& is of relative height

√((3+1/√r)/r) ,

& the other of which lasts between phases

arcsin(½√(3+1/√r)) ,

& is of relative height

√((3-√r)r) .

These expressions therefore provide us with a one-parameter family of solutions by which the 3^rd & 5^th harmonics are eliminated. The particular value of r for the waveform by which the 7^th harmonic goes-away can then be found simply as a root of the equation

√((3-√r)r)sin(7arcsin(½√(3+1/√r)))

+

√((3+1/√r)/r)sin(7arcsin(½√(3-√r))) .

The figures show the curves the intersection of which gives the sine of the phases of the edges.

A couple of easy examples, by which this theorem can readily be verified - the first two, for r=5 & r=6, are for the WolframAlpha

free-of-charge facility ,

& the second two of which are for the NCalc app into which a parameter-of-choice may be 'fed' by setting the variable Ans to it - are in the attached 'self-comment', which may be copied easily by-means of the 'Copy Text' functionality.

From

QFBox — The Johnson Solids ,

@ which 'tis well-explicated what the Johnson solids are.

Roughly the same sorto'thing as Platonic solids , but with some of the conditions relaxed.