Funky Protractors for Exploring Angle Measure

In this article, we discuss funky protractor tasks, which we designed to provide opportunities for students to reason about protractors and angle measure. We address how we have implemented these tasks, as well as how students have engaged with them.

A Bound on a Convexity Measure for Point Sets

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

Risk of Engagement of Bipolar Bone Defects in Posterior Shoulder Instability

Background: The risk of re-engagement of bipolar bone defects in posterior shoulder instability has not yet been investigated. Hypothesis: Posterior glenoid defects can lead to the engagement of supposedly noncritical reverse Hill-Sachs lesions (RHSLs). Study Design: Descriptive laboratory study. Methods: In a retrospective multicenter study, 102 cases of posterior shoulder dislocations and resulting RHSLs were collected. Of these cases, all patients with available computed tomography (CT) scans, with a reduced shoulder joint, and without bony posterior glenoid rim defects or concomitant dislocated fractures of the humeral head were included. The gamma angle (measure of the critical size and localization of RHSLs) and the delta angle (measure of the degree of internal rotation necessary for engagement to occur) of the RHSLs were determined on standardized CT scans. Virtual posterior glenoid defects were created, and the effect of increasing defect size on the delta angle was determined. Results: The mean gamma angle of the 19 patients included in this study was 94.5° (range, 69.7°-124.8°). After creation of the virtual posterior glenoid defects, a mean reduction of the delta angle by 2.3° ± 0.2° (range, 1.9°-2.9°) per millimeter defect was observed. The cumulative change in the delta angle showed a highly significant correlation with the absolute and relative size of the glenoid defect ( R = 0.982, P < .001 and R = 0.974, P < .001, respectively). Conclusion: Concomitant posterior glenoid defects might lead to the engagement of noncritical RHSLs. When measuring the gamma angle to identify critical RHSLs, posterior glenoid bone loss should be accounted for.