On Borromean links and related structures

The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical `Borromean rings', in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed `n-Borromean' [Liang & Mislow (1994). J. Math. Chem. 16, 27–35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20–25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.

All for One and One for All

Chapter 23 tells the story of the Borromeo family and the origin of the motif on their coat of arms. It is a link known to mathematicians as the Borromean rings. The link is formed of three rings, which cannot be separated although no two of the rings are linked. This motif has a long history. It was known to the Vikings as the Walknot, and is inscribed on the Stora Hammars 1 picture-stone. Some of John Robinson’s abstract geometrical sculptures take the form of the Borromean rings. The mathematician Hermann Brun investigated how the structure of the Borromean rings could be extended to form other links, and these are known as Brunnian links. The emblem of the Principia Discordia—a satirical counter-culture text written in 1963—is a pentagonal Brunnian link formed of five nonagons known as the Mandala Discordia.

Self-assembly of metalla[3]catenanes, Borromean rings and ring-in-ring complexes using a simple π-donor unit

Despite extensive research and several stunning breakthroughs in the synthesis of interlocked molecular species, [3]catenanes, Borromean rings and ring-in-ring complexes are exceedingly rare and their targeted synthesis remains a formidable challenge. Herein, a series of Cp*Rh-based homogeneous and heterogeneous interlocked structures have been prepared by coordination-driven self-assembly, not only including metalla[2]catenanes and molecular Borromean rings, but also linear metalla[3]catenanes and ring-in-ring complexes. The interlocked structures are all based on bithiophenyl groups. The bithiophenyl groups effectively enhance the strength of the inter-ring interactions and play a crucial role in the formation of these interlocked structures. By taking advantage of the strong interaction between π-donor (D) and π-acceptor (A) groups, the electron-deficient methylviologen cation was introduced into a cationic metallarectangle based on bithiophenyl groups. Taking inspiration from these results, a cationic metallarectangle based on A units was threaded into a metallarectangle based on D units, leading to a heterogeneous D–A ring-in-ring structure.

"THE PIANO TEACHER" AND THE QUESTION OF PERVERSION

ABSTRACT: This article aims to demonstrate how perversion, as a subject, activates investigation. As a starting point, it explores Freud’s idea surrounding fetishism as a paradigm of itself; a necessary interval between neurosis and psychosis. This analysis is followed by a presentation of various notions developed by Lacan in his teachings. Subsequently, the article analyzes Haneke’s 2001 film “The piano teacher”, here considered as a case of substitute perversion that adds to the few documented cases of this nature. The movie contributes elements that allow to think about and further develop the notion of substitute perversion in psychosis. In it, “the teacher” demonstrates the separation between love, desire, juissance and what Lacan calls “false hole” through borromean rings. Lastly, questions are posed regarding perversion; organized by answers, opposite the two aforementioned structures.

Towards the Stepwise Assembly of Molecular Borromean Rings. A Donor-Acceptor Ring-in-Ring Complex

The assembly of molecular Borromean Rings from constitutionally independent rings in a stepwise manner depends on the preparation of robust “ring-in-ring” complexes. The π-electron rich macrocycle bis-1,5-dinaphtho[50]crown-14 (1) is shown to form a donor-acceptor ring-in-ring complex with the π-electron poor cyclophane cyclobis(paraquat-4,4'-biphenylene) (24+) in solution. In the crystal superstructure of [1⊂2]·4PF6, CH···O interactions between the polyether loops of 1 and the bipyridinium units of 24+ could disfavor the threading of dialkylammonium components of a third ring.