Symmetric colorings of polypolyhedra

Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami models with one module per skeleton edge. Consider a coloring in which each edge of a given component receives a different color, and where the coloring (up to global color permutation) is invariant under the polypolyhedron's symmetry group. On the Five Intersecting Tetrahedra, the edges of each color form visual bands on the model, and correspond to matchings on the dodecahedron graph. We count the number of such colorings and give three proofs. For each of the non-polygon-component polypolyhedra, there is a corresponding matching coloring, and we count the number of these matching colorings. For some of the non-polygon-component polypolyhedra, there is a corresponding visual-band coloring, and we count the number of these band colorings

Counting mountain-valley assignments for flat folds

We develop a combinatorial model of paperfolding for the purposes of enumeration. A planar embedding of a graph is called a {\em crease pattern} if it represents the crease lines needed to fold a piece of paper into something. A {\em flat fold} is a crease pattern which lies flat when folded, i.e. can be pressed in a book without crumpling. Given a crease pattern $C=(V,E)$, a {\em mountain-valley (MV) assignment} is a function $f:E\rightarrow \{$M,V$\}$ which indicates which crease lines are convex and which are concave, respectively. A MV assignment is {\em valid} if it doesn't force the paper to self-intersect when folded. We examine the problem of counting the number of valid MV assignments for a given crease pattern. In particular we develop recursive functions that count the number of valid MV assignments for {\em flat vertex folds}, crease patterns with only one vertex in the interior of the paper. We also provide examples, especially those of Justin, that illustrate the difficulty of the general multivertex case

Using chronic kidney disease trigger tools for safety and learning: a qualitative evaluation in East London primary care

Background An innovative programme to improve identification and management of chronic kidney disease (CKD) in primary care was implemented across three clinical commissioning groups (CCGs) in 2016. This included a falling estimated glomerular filtration rate (eGFR) trigger tool built from data within the electronic health record (EHR). This patient safety tool notifies GP practices when falling eGFR values are identified. By alerting clinicians to patients with possible CKD progression the tool invites clinical review, the option for specialist advice, and written reflection on management. Aim To identify practitioner perceptions of trigger tool use and value from interview data, and compare these with the written reflections on clinical management recorded within the tools. Method Eight semi-structured interviews with 6 GPs, 1 pharmacist and 1 practice manager were recorded and transcribed. Thematic analysis of the interview transcripts was undertaken using framework analysis. The free-text reflective comments recorded in the trigger tools of 1,921 cases were organised by referral category ‘yes’ and ‘no’, with each category stratified by age into ‘younger’ and ‘older’ cases. Subsequently the themes arising from the interviews were compared with the descriptive analysis of the reflective comments. Findings Three themes emerged from interviews: Getting started, Patient safety and Practitioner and Practice learning. Well organised practices found the tool was readily embedded into workflow and expressed greater motivation for using it. The trigger tool was seen to contribute to patient safety, and as a tool for learning about CKD management, both individually and as a practice. Reflective comments from 1,921 trigger tools were examined, these supported the theme of patient safety from the interviews. However the free text data, stratified by age, challenged the expectation that younger cases would have higher referral rates, driven by a higher level of risk for CKD progression. Conclusion Building electronic trigger tools from the EHR can identify patients with a falling eGFR prompting review of the eGFR trajectory and management plan. Interview and reflective data illustrated that practice use of the trigger tool supported the patient safety agenda and in addition encouraged team and individual learning about CKD management

Numerical tools to validate stationary points of SO(8)-gauged N=8 D=4 supergravity

Until recently, the preferred strategy to identify stationary points in the scalar potential of SO(8)-gauged N=8 supergravity in D=4 has been to consider truncations of the potential to sub-manifolds of E_{7(+7)}/SU(8) that are invariant under some postulated residual gauge group G of SO(8). As powerful alternative strategies have been shown to exist that allow one to go far beyond what this method can achieve -- and in particular have produced numerous solutions that break the SO(8) gauge group to no continuous residual symmetry -- independent verification of results becomes a problem due to both the complexity of the scalar potential and the large number of new solutions. This article introduces a conceptually simple self-contained piece of computer code that allows independent numerical validation of claims on the locations of newly discovered stationary points.Comment: 9 pages, program code can be obtained by downloading paper's source from arxiv; new version contains code cleanup and extensions (scalar mass matrix code

Flat origami is Turing Complete

Flat origami refers to the folding of flat, zero-curvature paper such that the finished object lies in a plane. Mathematically, flat origami consists of a continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ along with a layer ordering $\lambda_f:P\times P\to \{-1,1\}$ that tracks which points of $P$ are above/below others when folded. The set of crease lines that a flat origami makes (i.e., the set on which the mapping $f$ is non-differentiable) is called its \textit{crease pattern}. Flat origami mappings and their layer orderings can possess surprisingly intricate structure. For instance, determining whether or not a given straight-line planar graph drawn on $P$ is the crease pattern for some flat origami has been shown to be an NP-complete problem, and this result from 1996 led to numerous explorations in computational aspects of flat origami. In this paper we prove that flat origami, when viewed as a computational device, is Turing complete. We do this by showing that flat origami crease patterns with \textit{optional creases} (creases that might be folded or remain unfolded depending on constraints imposed by other creases or inputs) can be constructed to simulate Rule 110, a one-dimensional cellular automaton that was proven to be Turing complete by Matthew Cook in 2004