We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutation group and the original ribbon group of Ellis-Monaghan and Moffatt to take (partial) twists and duals, or twuals, of ribbon graphs. A ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twuals. This extends the original ribbon group action, which only used the canonical identification of edges, to the more natural setting of self-twuality up to isomorphism. We then show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twuality properties of these bouquets propagate through their orbits. Thus, we can determine (partial) twualities via these one vertex graphs, for which checking isomorphism reduces simply to checking dihedral group symmetries. Finally, we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges, in contrast with the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin. We also show how the automorphism group of a ribbon graph yields self-dual, -petrial or –trial graphs in its orbit, and produce an infinite family of self-trial graphs that do not arise as covers or parallel connections of regular maps, thus answering a question of Jones and Poulton.

The purpose of the study was to analyse whether prolonged and regular physical training in children leads to changes in myocardial systolic deformation and rotational mechanics. For that purpose, cardiac MRI feature tracking was performed retrospectively in 35 pre-adolescent male soccer players and 20 matched controls. There were no changes in global strain, but left ventricular twist and apical rotation were greater in soccer players, which adds to the features of paediatric athlete’s heart.

How does DNA, the molecule containing genetic information, change its three-dimensionalshape during the complex cellular processes of replication, recombination and repair? Thisis one of the core questions in molecular biology which cannot be answered without helpfrom mathematical modeling. Basic concepts of topology and geometry can be introduced inundergraduate teaching to help students understand counterintuitive complex structuraltransformations that occur in every living cell. Topoisomerases, a fascinating class ofenzymes involved in replication, recombination and repair, catalyze a change in DNAtopology through a series of highly coordinated mechanistic steps. Undergraduate biologyand mathematics students can visualize and explore the principles of topoisomerase actionby using easily available materials such as Velcro, ribbons, telephone cords, zippers andtubing. These simple toys can be used as powerful teaching tools to engage students inhands-on exploration with the goal of learning about both the mathematics and the biologyof DNA structure.

Premature fusion of cranial sutures underlies the clinical condition of ‘craniosynostosis’, a common human disorder that occurs in both nonsyndromic and syndromic forms. The subgroup of syndromic craniosynostoses usually associates limb abnormalities and facial dysmorphism to skull distortion. Over the past decade, some of the genes causing these phenotypes have been identified. Among these, the gene encoding FGFR2, one of four members of the fibroblast growth factor receptor (FGFR) family, has been shown to account for several severe conditions including Apert, Pfeiffer, Crouzon, Beare–Stevenson and Jackson–Weiss syndromes. Two other FGFRs, FGFR1 and FGFR3, also account for craniosynostoses of variable severity [Pfeiffer, Crouzon with acanthosis nigricans (a pre-malignant skin disorder), and Muenke syndromes]. By contrast, Saethre–Chotzen syndrome and craniosynostosis (Boston-type) arise from mutations in the Twist and muscle segment homeobox 2 (MSX2) transcription factors, respectively. Whereas most FGFR mutations are likely to cause ligand-independent activation of the receptor, leading to an upregulation of signaling pathways, mutations in the basic helix–loop–helix (bHLH) transcription factor Twist appear to induce loss of protein function. This review will summarise and discuss some of the cellular and molecular mechanisms involved in normal and abnormal craniofacial development, focusing on the possible interactions between the different factors controlling membranous ossification.

The wobble GoU pairs have been implicated in several biological processes where RNA molecules play a key role. We review the geometrical and conformational properties of wobble GoU pairs on the basis of available crystal structures of RNAs at high resolution. The similarities with the wobble A+oC pairs and UoU pairs are illustrated, while the differences with the recently discovered bifurcated G[bull ]U pairs are contrasted.

Buckminster Fuller has coined the name tetrahelix for a column of regular tetrahedra, each sharing two faces with neighbours, one 'below' and one 'above' [A. H. Boerdijk, Philips Research Reports 7 (1952), p. 309]. Such a column could well be employed in architecture, because it is both strong and attractive. The (n — 1)-dimensional analogue is based on a skew polygon such that every n consecutive vertices belong to a regular simplex. The generalized twist which shifts this polygon one step along itself is found to have the characteristic equation

(λ - 1)2{(n - 1)λn-2 + 2(n - 2)λn-3 + 3(n - 3)λn-4 + . . . + (n - 2)2λ + (n - 1)} = 0,

which can be derived from tan nθ = n tan θ by setting λ = exp (2θi).