A New Look at the So-Called Trammel of Archimedes

The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity

Volumes of solids swept tangentially around cylinders

In earlier work ([1]-[5]) the authors used the method of sweeping tangents to calculate area and arclength related to certain planar regions. This paper extends the method to determine volumes of solids. Specifically, take a region S in the upper half of the xy plane and allow the plane to sweep tangentially around a general cylinder with the x axis lying on the cylinder. The solid swept by S is called a solid tangent sweep. Its solid tangent cluster is the solid swept by S when the cylinder shrinks to the x axis. Theorem 1: The volume of the solid tangent sweep does not depend on the profile of the cylinder, so it is equal to the volume of the solid tangent cluster. The proof uses Mamikon's sweeping-tangent theorem: The area of a tangent sweep to a plane curve is equal to the area of its tangent cluster, together with a classical slicing principle: Two solids have equal volumes if their horizontal cross sections taken at any height have equal areas. Interesting families of tangentially swept solids of equal volume are constructed by varying the cylinder. For most families in this paper the solid tangent cluster is a classical solid of revolution whose volume is equal to that of each member of the family. We treat forty different examples including familiar solids such as pseudosphere, ellipsoid, paraboloid, hyperboloid, persoids, catenoid, and cardioid and strophoid of revolution, all of whose volumes are obtained with the extended method of sweeping tangents. Part II treats sweeping around more general surfaces

Arithmetic based fractals associated with Pascal's triangle

Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a "carry rule" for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomialcoefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr . We go on to describe Bqr as the limit of an iterated function system of "partial similarities", and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space

Volumes of solids swept tangentially around general surfaces

In Part I (Forum Geom., 15 (2015) 13-44) the authors introduced solid tangent sweeps and solid tangent clusters produced by sweeping a planar region S tangentially around cylinders. This paper extends Part I by sweeping S not only along cylinders but also around more general surfaces, cones for example. Interesting families of tangentially swept solids of equal height and equal volume are constructed by varying the cylinder or the planar shape S. For most families in this paper the solid tangent cluster is a classical solid whose volume is equal to that of each member of the family. We treat many examples including familiar quadric solids such as ellipsoids, paraboloids, and hyperboloids, as well as examples obtained by puncturing one type of quadric solid by another, all of whose volumes are obtained with the extended method of sweeping tangents. Surprising properties of their centroids are also derived

When Two Wrongs Made A Right: A Classroom Scenario of Critial Thinking as Problem Solving

Educators from kindergarten through college often stress the importance of teaching critical thinking within all academic content areas (Foundation for Critical Thinking, 2007, 2013). As indicated by the position statements of the National Council of Teachers of Mathematics, high quality mathematics education before the first grade should use curriculum and teaching practices that strengthen children’s problem-solving and reasoning processes as well as representing, communicating, and connecting mathematical ideas” The joint position statement of NAEYC and the National Council of” (NAEYC & NCTM [2002] 2010, 3). Through the educational and academic institutions critical thinking is identified as an important outcome for achieving the higher orders of learning upon successful completion of a course, a promotion, or a degree (Humphreys, 2013; Jenkins & Cutchens, 2011). Although there are numerous definitions of critical thinking, the authors have selected the definition by Scriven & Paul, 2008 as “the intellectually disciplined process of actively and skillfully conceptualizing, applying, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection reasoning, or communication as a guide to belief and action” (Scriven & Paul, 2008). Instructors should teach problem solving within the context of mathematics instruction and engage students in critical thinking by thoughtful questions with discussion of alternative results. Teaching preschool children to problem solve and engage in critical thinking in the context of mathematics instruction requires a series of thoughtful and informed decisions

Peroral Cholangioscopy in the Diagnosis and Treatment of Biliary Strictures

Objective: To determine the role of and indications for peroral cholangioscopy using the SpyGlass system in the differential diagnosis of biliary tract lesions and in case of biliary strictures, based on the literature data analysis and our own experience.Materials and methods: Peroral cholangioscopy is mainly used for the differential diagnosis of biliary tract lesions. During peroral cholangioscopy, we carefully consider gross signs of damage to the bile duct mucosa: abnormal capillary vascular pattern, granulation tissue and other types of proliferation, palpatory characteristics of the wall. Morphology should also be verified during image-guided intraductal forceps biopsy.Results: Peroral endoscopy of the biliary tract significantly increases the effectiveness of differential diagnosis between various types of biliary strictures.Direct examination of the bile duct mucosa with optical forceps biopsy and morphological verification increases the sensitivity and specificity of stricture type determination up to 83.3%–96% and 90.9%–99%, respectively. The diagnostic value of peroral cholangioscopy in the diagnosis of malignant and benign biliary tract lesions exceeds the effectiveness of endoscopic retrograde cholangiopancreatography, even with fluoroscopy-guided verification of ducts.Conclusions: Peroral cholangioscopy with its enormous potential plays an important role in management of patients with various diseases of the bilio-pancreatoduodenal area, including biliary strictures. We formulated key indications for peroral cholangioscopy based on the literature data analysis and our own experience with this technique in patients with bile duct pathology, including nondifferentiated biliary strictures

Hadron Energy Reconstruction for the ATLAS Calorimetry in the Framework of the Non-parametrical Method

This paper discusses hadron energy reconstruction for the ATLAS barrel prototype combined calorimeter (consisting of a lead-liquid argon electromagnetic part and an iron-scintillator hadronic part) in the framework of the non-parametrical method. The non-parametrical method utilizes only the known $e/h$ ratios and the electron calibration constants and does not require the determination of any parameters by a minimization technique. Thus, this technique lends itself to an easy use in a first level trigger. The reconstructed mean values of the hadron energies are within $\pm 1$ of the true values and the fractional energy resolution is $[(58\pm3)/\sqrt{E}+(2.5\pm0.3)$. The value of the $e/h$ ratio obtained for the electromagnetic compartment of the combined calorimeter is $1.74\pm0.04$ and agrees with the prediction that $e/h > 1.7$ for this electromagnetic calorimeter. Results of a study of the longitudinal hadronic shower development are also presented. The data have been taken in the H8 beam line of the CERN SPS using pions of energies from 10 to 300 GeV.Comment: 33 pages, 13 figures, Will be published in NIM

The Effective Use of Online Diagnostic Tools to help Identify Appropriate Learning Modules as a Form of Professional Development for Mathematics Teachers

Most professional development occurs in a time that is inconvenient for teachers. Learn ways to improve teachers\u27 mathematics knowledge and enable them to provide better instructions for students with the help of an online diagnostic assessment. Participants of this session will learn how district mathematics teachers used an online diagnostic model to understand and connect specific key ideas regarding mathematics operations and applications; the potential for introducing interdisciplinary mathematics scenarios to enhance student learning; and, understand and be able to teach mathematics applications with confidence