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Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the SternBrocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

A zipper folding of a polygon P given a source point x ∈ ∂P is the polyhedron generated by identifying all points in ∂P equidistant from x, measured along the perimeter of P , in essence “zipping” the boundary of the polygon. A theorem of Alexandrov shows that as long as every glued point has nonnegative curvature, then any zipper folding of a convex polygon leads to unique convex polyhedron (where a doubly covered polygon is considered a “flat” polyhedron). Alexandrov’s theorem is existential, but a more recent constructive proof by Bobenko and Izmestiev allows for the explicit construction of the polyhedron by solving a certain differential equation [2]. An implementation of the constructive algorithm has been coded by Stefan Sechelmann, which will output the folded convex polyhedron given a input triangulation of the polygon with gluing instructions. However, it is difficult to extract the creases and adjacencies from the initial polygon in their final output polyhedron. We seek a more combinatorial approach to computing this information. Previous work has also looked at determining all the combinatorially different polyhedra obtained via foldings, primarily for regular convex polygons as well as a few other shapes such as the Latin cross [3]. In this paper, we classify and compute the convex foldings of a diamond shape which are obtained via zipper foldings. Our primary goal was to seek a simpler combinatorial approach to testing for the correct set of folds, or crease pattern. As was observed by Alexandrov and noted in [1], there are a finite number of possible crease patterns. However, in our experience, verifying or discounting a crease pattern has been surprisingly difficult in more

We consider the chromatic number of a family of graphs we call box graphs, which arise from a box complex in n -space. It is straightforward to show that any box graph in the plane has an admissible coloring with three colors, and that any box graph in n -space has an admissible coloring with n + 1 colors. We show that for box graphs in n -space, if the lengths of the boxes in the corresponding box complex take on no more than two values from the set { 1 , 2 , 3 } , then the box graph is 3 -colorable, and for some graphs three colors are required. We also show that box graphs in 3-space which do not have cycles of length four (which we call "string complexes") are 3 -colorable.

A clinical consequence of symptomatic Alzheimer’s disease (AD) is impaired driving performance. However, decline in driving performance may begin in the preclinical stage of AD. We used a naturalistic driving methodology to examine differences in driving behavior over one year in a small sample of cognitively normal older adults with ( n = 10) and without ( n = 10) preclinical AD. As expected with a small sample size, there were no statistically significant differences between the two groups, but older adults with preclinical AD drove less often, were less likely to drive at night, and had fewer aggressive behaviors such as hard braking, speeding, and sudden acceleration. The sample size required to power a larger study to determine differences was calculated.

There are many unanswered questions about the evolution of the ancient ‘Silk Roads’ across Asia. This is especially the case in their mountainous stretches, where harsh terrain is seen as an impediment to travel. Considering the ecology and mobility of inner Asian mountain pastoralists, we use ‘flow accumulation’ modelling to calculate the annual routes of nomadic societies (from 750 m to 4,000 m elevation). Aggregating 500 iterations of the model reveals a high-resolution flow network that simulates how centuries of seasonal nomadic herding could shape discrete routes of connectivity across the mountains of Asia. We then compare the locations of known high-elevation Silk Road sites with the geography of these optimized herding flows, and find a significant correspondence in mountainous regions. Thus, we argue that highland Silk Road networks (from 750 m to 4,000 m) emerged slowly in relation to long-established mobility patterns of nomadic herders in the mountains of inner Asia. The authors use modelling to show that the network of trading routes known as the Silk Road emerged from hundreds of years of interactions between pastoralists as they moved their herds and flocks between higher and lower elevations in generally mountainous regions. The Silk Road refers to a network of ancient trade routes that have crossed central Asia since time immemorial. But how did it get started? Conventional models usually start by inferring the easiest paths between sites already known to be part of the network. This introduces a circular argument as it biases the results towards what is already known. Here Michael Frachetti and colleagues take a different approach to show that the network emerged from hundreds of years of interactions between pastoralists moving their livestock between higher and lower elevations in response to the seasons in this generally mountainous region. They suggest that the Silk Road network therefore materialized slowly from the long-established, local mobility patterns of nomadic herders. This finding may encourage archaeologists to seek more nuanced explanations for the evolution of ancient connectivity.

We obtain a complete classification of all simple closed geodesics on the eight convex deltahedra by solving a related graph coloring problem. Geodesic segments in the neighborhood of each deltahedron vertex produce a limited number of crossing angles with deltahedron edges. We define a coloring on the edge graph of a deltahedron based on these angles, and we show that the set of graph colorings compatible with edge-colorings of the neighborhood graphs of radius one classifies all possible simple closed geodesics on all convex deltahedra.

Background/Objectives: Road tests and driving simulators are most commonly used in research studies and clinical evaluations of older drivers. Our objective was to describe the process and associated challenges in adapting an existing, commercial, off-the-shelf (COTS), in-vehicle device for naturalistic, longitudinal research to better understand daily driving behavior in older drivers. Design: The Azuga G2 Tracking DeviceTM was installed in each participant’s vehicle, and we collected data over 5 months (speed, latitude/longitude) every 30-seconds when the vehicle was driven. Setting: The Knight Alzheimer’s Disease Research Center at Washington University School of Medicine. Participants: Five individuals enrolled in a larger, longitudinal study assessing preclinical Alzheimer disease and driving performance. Participants were aged 65+ years and had normal cognition. Measurements: Spatial components included Primary Location(s), Driving Areas, Mean Centers and Unique Destinations. Temporal components included number of trips taken during different times of the day. Behavioral components included number of hard braking, speeding and sudden acceleration events. Methods: Individual 30-second observations, each comprising one breadcrumb, and trip-level data were collected and analyzed in R and ArcGIS. Results: Primary locations were confirmed to be 100% accurate when compared to known addresses. Based on the locations of the breadcrumbs, we were able to successfully identify frequently visited locations and general travel patterns. Based on the reported time from the breadcrumbs, we could assess number of trips driven in daylight vs. night. Data on additional events while driving allowed us to compute the number of adverse driving alerts over the course of the 5-month period. Conclusions: Compared to cameras and highly instrumented vehicle in other naturalistic studies, the compact COTS device was quickly installed and transmitted high volumes of data. Driving Profiles for older adults can be created and compared month-to-month or year-to-year, allowing researchers to identify changes in driving patterns that are unavailable in controlled conditions.