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Two mathematical aspects of the centuries-old Japanese sashiko stitching form hitomezashi are discussed: the encoding of designs using words from a binary alphabet, and duality. Traditional hitomezashi designs are analysed using these two ideas. Self-dual hitomezashi designs related to Fibonacci snowflakes, which we term Pell persimmon polyomino patterns, are proposed. Both these designs and the binary words used to generate them appear to be new to their respective literatures., Comment: 25 pages; 21 figures

D-forms have in the past been created from inflexible materials, or considered as abstract mathematical objects. This paper describes a number of realisations of D-forms, and the related pita-forms, in textiles. Examples are given in which the created 3D space is purposefully employed and others in which ornamentation of the constituent surfaces is the highlighted feature. In particular, a set of biscornu has been fashioned to provide a 3D sampler of the axial point group $D_{4d}$ and its subgroups, using hitomezashi., Comment: 11 pages, 19 figures

Through the hands-on creation of two sashiko pieces of work - a counted thread kogin bookmark and a single running stitched hitomezashi sampler - participants will explore not only the living cultural history of this traditional Japanese needlework but will also experience the mathematics of sashiko in a tangible form, and will take away with them items of simple beauty., Comment: 8 pages, 8 figures, workshop paper intended for Bridges Math Art conference 2020

We describe the construction of a new family of developable rollers based on the Platonic solids. In this way kinetic sculptures may be realised, with the Platonic solids quite literally in their heart. We also describe the strong way in which the Platonicons circumscribe the Platonic solids., Comment: 4 pages, 5 figures, submitted for Bridges Helsinki 2020; revised version after reviewing

This paper introduces a new family of solids, which we call \textit{polycons}, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable rollers such as the oloid and particular polysphericons. The paper concludes with a discussion of the polycons as stationary and kinetic works of art., Comment: 14 pages, 10 figures, submitted to Journal of Mathematics and the Arts

Sphericons and D-forms are 3D objects created and described by artists, which have separately received attention in the mathematical literature in the last 15 or so years. The attempt to classify a seamed, crocheted form geometrically led to the observation, which appears not to have been previously made explicit, that these objects are related. General results concerning $(N,k)$-icons and seam-, D- and pita-forms are given. Instructions to crochet D-forms are given in the Appendix., Comment: 11 pages, 12 figures; submitted to Journal of Mathematics and the Arts; Revised version uploaded June 15, 2016 Revised version of Jan 6, 2017 contains 17 figures and 14 pages; Further revision (as accepted by Journal) uploaded March 2017

We consider the integrable minimal models ${\cal M}(m,m';t)$, corresponding to the $\varphi_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit\,} $m, m'\to\infty$, $m/m'\to p/p'$ where $p, p'$ are coprime and the limit is taken through coprime values of $m,m'$. We view these off-critical minimal models ${\cal M}(m,m';t)$ as the continuum scaling limit of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. Applying Corner Transfer Matrices to the Forrester-Baxter RSOS models in Regime III, we argue that taking first the thermodynamic limit and second the {\it logarithmic limit\,} yields off-critical logarithmic minimal models ${\cal LM}(p,p';t)$ corresponding to the $\varphi_{1,3}$ perturbation of the critical logarithmic minimal models ${\cal LM}(p,p')$. Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ${\cal LM}(p,p')$. We also calculate the logarithmic limit of certain off-critical observables ${\cal O}_{r,s}$ related to One Point Functions and show that the associated critical exponents $\beta_{r,s}=(2-\alpha)\,\Delta_{r,s}^{p,p'}$ produce all conformal dimensions $\Delta_{r,s}^{p,p'}<{(p'-p)(9p-p')\over 4pp'}$ in the infinitely extended Kac table. The corresponding Kac labels $(r,s)$ satisfy $(p s-p' r)^2< 8p(p'-p)$. The exponent $2-\alpha ={p'\over 2(p'-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension $\Delta_t={1-\alpha\over 2-\alpha}={2p-p'\over p'}=\Delta_{1,3}^{p,p'}$ for the perturbing field $t$. As befits a non-unitary theory, some observables ${\cal O}_{r,s}$ diverge at criticality., Comment: 18 pages, 5 figures; version 3 contains amplifications and minor typographical corrections

The clustering coefficient, path length and average vertex degree of two urban train line networks have been calculated. The results are compared with theoretical predictions for appropriate random bipartite graphs. They have also been compared with one another to investigate the effect of architecture on the small-world properties., Comment: 6 pages, prepared in RevTeX

The leading excitations of the dilute $A_L$ model in regime 2 are considered using analytic arguments. The model can be identified with the integrable $\phi_{1,2}$ perturbation of the unitary minimal series $M_{L,L+1}$. It is demonstrated that the excitation spectrum of the transfer matrix satisfies the same functional equations in terms of elliptic functions as the exact S-matrices of the $\phi_{1,2}$ perturbation do in terms of trigonometric functions. In particular, the bootstrap equation corresponding to a self-fusing process is recovered. For the special cases $L=3,4,6$ corresponding to the Ising model in a magnetic field, and the leading thermal perturbations of the tricritical Ising and three-state Potts model, as well as for the unrestricted model, $L=\infty$, we relate the structure of the Bethe roots to the Lie algebras $E_{8,7,6}$ and $D_4$ using Coxeter geometry. In these cases Coxeter geometry also allows for a single formula in generic Lie algebraic terms describing all four cases. For general $L$ we calculate the spectral gaps associated with the leading excitation which allows us to compute universal amplitude ratios characteristic of the universality class. The ratios are of field theoretic importance as they enter the bulk vacuum expectation value of the energy momentum tensor associated with the corresponding integrable quantum field theories., Comment: 32 pages (tcilatex)

The universal amplitude ratio $R_{\xi}$ for the ($q\leqslant 4$)-state Potts model in two dimensions is determined by using results for the dilute A model in regime 1. The nature of the relationship between the Potts model and the dilute A model, both related to $\phi_{2,1}$ perturbed conformal field theory, is discussed., Comment: 11 pages, LaTeX2e, submitted to J Stat Phys

The universal amplitude ratio $\tilde{R}_{\xi}$ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4., Comment: 5 pages, LaTeX, submitted to J. Phys. A. One paragraph rewritten to correct errors

Some universal amplitude ratios appropriate to the $\phi_{2,1}$ peturbation of the c=7/10 minimal field theory, the subleading magnetic perturbation of the tricritical Ising model, are explicitly demonstrated in the dilute A$_3$ model, in regime 1., Comment: 8 pages, LaTeX using iop macros

On the basis of the result obtained by applying Baxter's exact perturbative approach to the dilute A_3 model to give the E_8 mass spectrum, the dilute A_L inversion relation was used to predict the eigenspectra in the L=4 and L=6 cases (corresponding to E_7 and E_6 respectively). In calculating the next-to-leading term in the correlation lengths, or equivalently masses, the inversion relation condition gives a surprisingly simple result in all three cases, and for all masses., Comment: 9 pages, LaTeX

This article presents an informed reflection on the evolution of teacher-to-learner feedback provided on written assignments in first-year university mathematics subjects. The feedback provided addresses not only mathematical accuracy and skills, but also the development of graduate attributes, such as discipline-appropriate written communication. Effective and efficient practices that have been collectively refined and enhanced, for more than a decade, are described and examined. This model for formative assessment in mathematics subjects is critiqued in the light of the scholarly literature on feedback and assessment.