DOCSLIB.ORG

131 865- EDRS PRICE Technclogy,Liedhanical

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
131 865- EDRS PRICE Technclogy,Liedhanical ..DOCUMENT RASH 131 865- JC 760 420 TITLE- Training Program for Teachers of Technical Mathematics in Two-Year Cutricnla. INSTITUTION Queensborough Community Coll., Bayside, N Y. SPONS AGENCY New York State Education Dept., Albany. D v.of Occupational Education Instruction. PUB DATE Jul 76 GRANT VEA-78-2-132 NOTE 180p. EDRS PRICE MP-$0.83 HC-S10.03 Plus Postage. DESCRIPTORS *College Mathematics; Community Colleges; Curriculum Development; Engineering Technology; *Junior Colleges Mathematics Curriculum; *Mathematics Instruction; *Physics Instruction; Practical ,Mathematics Technical Education; *Technical Mathematics Technology ABSTRACT -This handbook 1 is designed'to.aS ist teachers of techhical'mathematics in:developing practically-oriented-curricula -for their students. The upderlying.assumption is that, while technology students are nOt a breed apartr their: needs-and orientation are_to the concrete, rather than the abstract. It describes the:nature, scope,and dontett,of curricula.. in Electrical, TeChnClogy,liedhanical Technology4 Design DraftingTechnology, and: Technical Physics', ..with-particularreterence to:the mathematical skills which are_important for the students,both.incollege.andot -tle job. SaMple mathematical problemsr-derivationSeand.theories tei be stressed it..each of_these.durricula-are presented, as- ate additional materials from the physics an4 mathematics. areas. A frame ok-reference-is.provided through diScussiotS ofthe\careers,for.which-. technology students-are heing- trained'. There.iS alsO &section deioted to the development of reading and studY.skills and:to general. claSSroom_Aatagement. (Author/DC),'- *************************** ******* ****** ** **************** Documents acquired.by ERIC include many info mal unpublished * materials not available. from other sources. ERIC makeseiery effart *ito obtain the best copy,available. Nevertheless, items of marginal * reproducibility are often'encountered and this affects the gUality of the microfiche and hardcopy reproductions ERIC makes available via the ERIC Document Reproduction Service (EDRS). EDES is not responsible for the\guality of the original document. Reproductions supplied by EDRS are the best that can be made from the original. ******************************************************************** U.S DEPARTMENT OF HEALTH, EDUCATION E WELFARE NATIONAL IN1TITUTE OF EDUCATION THIS DOCUMENT HAS BEEN REPRO- DUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGIN- ATINO IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRE- SENT OFFICIAL NATIONAL INSTITUTE OF EDUCATION POSITrON OR POLICY 'Tr ining PTO tain. for.Teachcrs Of echria MathetnaVet in.Twalfeaid Curricula INeefisbOratigirCfomMunity -College- THE CITY UNIVERSITY OF NEW YORK BAYSIDE1 NEW YORK. TRAINING PROGRAM FOR TEACHERS OF TECHNICAL MATHEMATICS IN TWO-YEAR CURRICULA A HANDBOOK PREPARED EOR TIlE _VFORMATIONOF. PARTICIPANTS IN THE SEMINAR PROGRAM HELD AT QUEENSBOROUGH COMIIUNITY COLLEGE OE THE CITY UNIVERsrry OF NEW YORK fUNE 7 - 18. 1976 Partially Pinch York Statc Editcatian Depart? I/EA Grant No. 76-2-132 Published by -enshorough Community College Bayside, New York - WIN 1976 Contents Page Introduction ..1 Development of the Technology Curriculum 2 ..7 Electrical Technology . .. .. .... .. .. .. Introduction to Electrical Technology . ... ... .....10 Electrical Circuit Analysis . 16 Electronics 33 ..36 Transient Circuit Analysis .... ... .. .. Development ofLaplace Transformation. Mechanical and Design Drafting Technologies 61 . .5621 Mechanical Technology ..... .... ..62 Design Drafting Technology . .. ... .... .. ...... Mathematics for Mechanical Technology and DesignDrafting Technology .63 Applied Mechanics_ 64 ..79 Strength of Materials. ... Fluid Mechanics 87 Thermodynamics . ...... .97 Machine DFsign 103 Introduction to Numerical Con rol 106 Design Drafting 112 Statics and Strength of Mate als 3 Surveying and Layout 119 Technical Drawing 121 MechanIsms 124 istrative Uses of Mathemazics Not Covered inOther Areas 128 Technical Physics 156 . 6 Probliems in Proportionality. ... ..... ... .11558 Graphs Trigonometry and Angles . 159 Units, Constants, and Conversions 163 Learning Strategies Study Skills 166 Preface This handbook was written and compiled by the staff of the Training Program for Teachers of Technical Mathematics in Two-Year Curricula held at Queensborough Community College Bayside, New York in June 1976, as pai-t of the materials for the participants' use. .The entire program was partially funded by a grant from the New York State Department of Education (Vocational Education Division) for the pe.iod July 1975-June-- 1976,- We wish to acknowledge the work of Professor Joseph B. Aidab, electrical techriology, Professor Anthony Behr, mathematics, Professor Juliana Corn (program coordinator), mathetnaties, and ProfesSor Sheldon I. KATI, mechanical technology; and of the consultants, Mr. Amos Ben-Hut, student personnel services, Dr. Elie Lowy, physics, and Professor BarbariPennipede, basic educational skills. We also wish to thank Ms. Carol Zim, program secretary, for her diligence in deciphering the incomprehensible symbols of mathematics and science so that' the manuscript could be typed. Thanks also go to Ms. Katherine Rangos, Director of College Publications, for her unfailing optimism, ingenuity and competence in producing this book under trying conditions. Mr. Joseph Felician, manager.of printing services, and his fine staff have our gratitude. iii .introduction This handbook is compiled, not as a definitive guide to success in 'teaching technical stu- dents, but as a series of questions and suggestions, which, combined with the experiences and insights of the reader, will point the way to improved teaching, both in content and teaching methodology. Suggestions as to course content and usable applications can at hest consist of a large body of material containing more information than can be absorbed into-any particular course, th reby leaving each user theoption Of selecting those topics that apply to specific prograinsj The underlying assumption is that, while technology students are not a breed apart, thkir needs and orientation are to the concrete, ratherthan the abstract. While there are differ rices_between technology students andother students, the authors believe that in today's multi-media world, many techniques of impartinginformation must be developed; i.e., the classical "go home and read the book" days aregnhe forever. in order to help the reader build a viable teaching plan based upon these assumptions, the authors have attempted to provide a frame of reference by discussing the careers for which technology students are being trained. The nature, scope, and content of the tech- nical- curricula will be examined, with particular reference to the mathematical skills which are important for the students, both in college and on the job. The staff of the training pro. gram have compiled a list of concrete applications of mathematicS to technology, some drawn from actual textbooks currently in use in technical courses. Additional materials from the physics and mathematics areas are ineluded. A section is devoted to the develop- ment of reading and study skills, as well as gene,ml classroom management techniques. Development of the Teehnolowy Curriculum In the period from 1950 to 1960, engineering colleges shifted from a practical (laboratory- oriented) currieulum to a science (theory-oriented) one. Thus the engineers graduated from these programs gravitazed to industrial positions demanding a research and, developrrient background. This la: a void in the engineering and scientific spectrum in the more practical area. The scientist and the engineer were at one end of this spectrum, and the craftsman was at the other end. There existed, then, a need for practical technical personnel in the area between these two ends. not surprising that the demands of industry engendered the advent of a new type of / program for the academic training of these hewly required technical personnel. The prolif- eratiOn of associate degree prograrns in engineeringJechnOlogy starting in 1960 was the inevitable result. The`kinds of institutions offering educational programs in engineering technology va They belong primarily in one of the following classifications: (a) Monotechnical institutesSingle purpose institutions offering, engineer ng tech- nology. (b) Polytechnical InstitutionsInstitutes with a variety of progras related to business, health, or public service as Well as to engineering. ) Comprehensive Community CollegesCommunity and/or junior colleges which include in their offerings various occupational-technical prograrns as well as "university parallel" or "transfer- programs. (d) UniversitiesSenior institutions which include associate degree programs in engineer- ing technology as part of their offerings. Engineering technology iS Concerned primarily with the application of established scientific and engineering knowledge and methods. Normally, engineering technology is not concerned with the development of new\principles and methods. Technical skills such as drafting, in one instance, are characteristic of engineering technology. The associate degree graduate of an engineering technology program is called an engineering technician. The graduate of a baccalaureate program in engineering technol/./ ogy is known as an engineering technologiSt..The engineering technologist has acadeinic training which lies between that for:the engineering technician at d the
Load more

Recommended publications
  • 900.00 Modelability
    986.310 Strategic Use of Min-max Cosmic System Limits 986.311 The maximum limit set of identical facets into which any system can be divided consists of 120 similar spherical right triangles ACB whose three corners are 60 degrees at A, 90 degrees at C, and 36 degrees at B. Sixty of these right spherical triangles are positive (active), and 60 are negative (passive). (See Sec. 901.) 986.312 These 120 right spherical surface triangles are described by three different central angles of 37.37736814 degrees for arc AB, 31.71747441 degrees for arc BC, and 20.90515745 degrees for arc AC__which three central-angle arcs total exactly 90 degrees. These 120 spherical right triangles are self-patterned into producing 30 identical spherical diamond groups bounded by the same central angles and having corresponding flat-faceted diamond groups consisting of four of the 120 angularly identical (60 positive, 60 negative) triangles. Their three surface corners are 90 degrees at C, 31.71747441 degrees at B, and 58.2825256 degrees at A. (See Fig. 986.502.) 986.313 These diamonds, like all diamonds, are rhombic forms. The 30- symmetrical- diamond system is called the rhombic triacontahedron: its 30 mid- diamond faces (right- angle cross points) are approximately tangent to the unit- vector-radius sphere when the volume of the rhombic triacontahedron is exactly tetravolume-5. (See Fig. 986.314.) 986.314 I therefore asked Robert Grip and Chris Kitrick to prepare a graphic comparison of the various radii and their respective polyhedral profiles of all the symmetric polyhedra of tetravolume 5 (or close to 5) existing within the primitive cosmic hierarchy (Sec.
    [Show full text]
  • 41 Three Dimensional Shapes
    41 Three Dimensional Shapes Space figures are the first shapes children perceive in their environment. In elementary school, children learn about the most basic classes of space fig- ures such as prisms, pyramids, cylinders, cones and spheres. This section concerns the definitions of these space figures and their properties. Planes and Lines in Space As a basis for studying space figures, first consider relationships among lines and planes in space. These relationships are helpful in analyzing and defining space figures. Two planes in space are either parallel as in Figure 41.1(a) or intersect as in Figure 41.1(b). Figure 41.1 The angle formed by two intersecting planes is called the dihedral angle. It is measured by measuring an angle whose sides line in the planes and are perpendicular to the line of intersection of the planes as shown in Figure 41.2. Figure 41.2 Some examples of dihedral angles and their measures are shown in Figure 41.3. 1 Figure 41.3 Two nonintersecting lines in space are parallel if they belong to a common plane. Two nonintersecting lines that do not belong to the same plane are called skew lines. If a line does not intersect a plane then it is said to be parallel to the plane. A line is said to be perpendicular to a plane at a point A if every line in the plane through A intersects the line at a right angle. Figures illustrating these terms are shown in Figure 41.4. Figure 41.4 Polyhedra To define a polyhedron, we need the terms ”simple closed surface” and ”polygonal region.” By a simple closed surface we mean any surface with- out holes and that encloses a hollow region-its interior.
    [Show full text]
  • The Fragmentation of Being and the Path Beyond the Void 1555 Copyright 1994 Kent D
    DRAFT Fragment 34 AUTOPOIETIC STRUCTURE In this essay, a hypothetical structure of the autopoietic system will be explored. This structure has been alluded to throughout the foregoing metaphysical development of the importance of the autopoietic concept. But it is good to have a purely structural model to refer to in order to develop the metaphysical concepts at a more concrete level of the general architecture of autopoietic systems. This development will depart from the structure given by Plato, and will attempt to give a model that is clearer and more internally coherent, developed from first principles. This is not to say that all autopoietic systems necessarily have this form, but that it is closer to the archetypal form and more clearly of an axiomatic simplicity and purity. It is posited that autopoietic systems may take an unknown variety of forms, and that we are merely giving one which is closest to the threshold of minimality. Much of the contents of this essay owes its insights to work cited earlier on software development methodologies and software development work process. This concrete discipline lent itself to developing these ideas in The Fragmentation of Being and The Path Beyond The Void 1555 Copyright 1994 Kent D. Palmer. All rights reserved. Not for distribution. AUTOPOIETIC STRUCTURE unexpected ways. Some steps toward the position stated here may be found in my series on Software Engineering Foundations and the article on The Future Of Software Process. In the latter article, I attempted to articulate what an autopoietic software process might look like. In the course of developing those ideas and realizing their connection to the work on software design methods, the following approach to defining the autopoietic system arose.
    [Show full text]
  • &Doliruqld 6Flhqfh &Hqwhu
    &DOLIRUQLD6FLHQFH&HQWHU &$/,)251,$67$7(6&,(1&()$,5 2001 PROJECT SUMMARY Your Name (List all student names if multiple authors.) Science Fair Use Only Eric A. Ford Project Title (Limit: 120 characters. Those beyond 120 will be ignored. See pg. 9) J1111 Fair Dice? Division X Junior (6-8) Senior (9-12) Preferred Category (See page 5 for descriptions.) 11 - Mathematics & Software Abstract (Include Objective, Methods, Results, Conclusion. See samples on page 14.) Use no attachments. Only text inside these boxes will be used for category assignment or given to your judges. My objective was to test my hypothesis that the regular polyhedral dice I was testing (a tetrahedron, an octahedron, a dodecahedron, an icosahedron, and a rhombic triacontahedron) would be fair dice and that a non-isohedral pentahedral solid, which I predicted would be more likely to land on its larger faces more frequently than its smaller ones, would not be fair. I determined if each die were fair by conducting a set number of throws (50 times the number of sides of the die) under the same conditions, which I recorded on separate tally sheets, using different colors for each set of ten times the number of sides of the die. I was then able to determine what values were reached for each increment and enter those data into a spreadsheet that I had set up to calculate the chi-square values. Next, I calculated the confidence percentages and confidence intervals for each set of rolls. I believed that if the chi-square values reached a confidence level of 95% for a given die, it could be considered fair.
    [Show full text]
  • Pentahedral Volume, Chaos, and Quantum Gravity
    Pentahedral volume, chaos, and quantum gravity Hal M. Haggard1 1Centre de Physique Th´eoriquede Luminy, Case 907, F-13288 Marseille, EU∗ We show that chaotic classical dynamics associated to the volume of discrete grains of space leads to quantal spectra that are gapped between zero and nonzero volume. This strengthens the connection between spectral discreteness in the quantum geometry of gravity and tame ultraviolet behavior. We complete a detailed analysis of the geometry of a pentahedron, providing new insights into the volume operator and evidence of classical chaos in the dynamics it generates. These results reveal an unexplored realm of application for chaos in quantum gravity. A remarkable outgrowth of quantum gravity has been The area spectrum has long been known to be gapped. the discovery that convex polyhedra can be endowed with However, in the limit of large area eigenvalues, doubts a dynamical phase space structure [1]. In [2] this struc- have been raised as to whether there is a volume gap [6]. ture was utilized to perform a Bohr-Sommerfeld quanti- The detailed study of pentahedral geometry here pro- zation of the volume of a tetrahedron, yielding a novel vides an explanation for and alternative to these results. route to spatial discreteness and new insights into the We focus on H = V as a tool to gain insight into spectral properties of discrete grains of space. the spectrum of volume eigenvalues. The spatial volume Many approaches to quantum gravity rely on dis- also appears as a term in the Hamiltonian constraint of cretization of space or spacetime. This allows one to general relativity, the multiplier of the cosmological con- control, and limit, the number of degrees of freedom of stant; whether chaotic volume dynamics also plays an the gravitational field being studied [3].
    [Show full text]
  • On the Shape Simulation of Aggregate and Cement Particles in a DEM System
    Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2015, Article ID 692768, 7 pages http://dx.doi.org/10.1155/2015/692768 Research Article On the Shape Simulation of Aggregate and Cement Particles in a DEM System Huan He,1,2 Piet Stroeven,2 Eric Pirard,3 and Luc Courard3 1 College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100122, China 2Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands 3GeMMe (Minerals Engineering, Materials and Environment), University of Liege,` Chemin des Chevreuils 1, 4000 Liege,` Belgium Correspondence should be addressed to Huan He; [email protected] Received 30 April 2015; Revised 6 July 2015; Accepted 7 July 2015 Academic Editor: Ana S. Guimaraes˜ Copyright © 2015 Huan He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Aggregate occupies at least three-quarters of the volume of concrete, so its impact on concrete’s properties is significant. Both size and shape of aggregate influence workability, mechanical properties, and durability of concrete. On the other hand, the shape of cement particles plays also an important role in the hydration process due to surface dissolution in the hardening process. Additionally, grain dispersion, shape, and size govern the pore percolation process that is of crucial importance for concrete durability. Discrete element modeling (DEM) is commonly employed for simulation of concrete structure. To be able to do so, the assessed grain shape should be implemented.
    [Show full text]
  • A Study of the Phase Mg2cu6ga5, Isotypic with Mg2zn11. a Route to an Icosahedral Quasicrystal Approximant Qisheng Lin the Ames Laboratory, [email protected]
    Chemistry Publications Chemistry 2003 A study of the Phase Mg2Cu6Ga5, Isotypic with Mg2Zn11. A Route to an Icosahedral Quasicrystal Approximant Qisheng Lin The Ames Laboratory, [email protected] John D. Corbett Iowa State University Follow this and additional works at: http://lib.dr.iastate.edu/chem_pubs Part of the Inorganic Chemistry Commons, and the Materials Chemistry Commons The ompc lete bibliographic information for this item can be found at http://lib.dr.iastate.edu/ chem_pubs/956. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Chemistry at Iowa State University Digital Repository. It has been accepted for inclusion in Chemistry Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Inorg. Chem. 2003, 42, 8762−8767 A Study of the Phase Mg2Cu6Ga5, Isotypic with Mg2Zn11. A Route to an Icosahedral Quasicrystal Approximant Qisheng Lin and John D. Corbett* Department of Chemistry, Iowa State UniVersity, Ames, Iowa 50011 Received June 9, 2003 The new title compound was synthesized by high-temperature means and its X-ray structure refined in the cubic space group Pm3h, Z ) 3, a ) 8.278(1) Å. The structure exhibits a 3-D framework made from a Ga14 and Mg network within which large and small cavities are occupied by centered GaCu12 icosahedral and Cu6 octahedral clusters, respectively. The clusters are well bonded within the network. Electronic structure calculations show that a pseudogap exists just above the Fermi energy, and nearly all pairwise covalent interactions remain bonding over a range of energy above that point.
    [Show full text]
  • 6.5 X 11 Threelines.P65
    Cambridge University Press 978-0-521-12821-6 - A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics Peter Hilton and Jean Pedersen Index More information Index (2, 1)-folding procedure 44 BAMA (Bay Area Mathematical Adventures) 206 (2, 1)-tape 44 Barbina, Silvia xiii (4, 4)-tape 99 Berkove and Dumont 205 1-period paper-folding 17 Betti number 231 10-gon from D2U 2-tape 50 big guess 279 11-gon 96 bold italics x 12-8-flexagon 16 Borchards, Richard 236 12-celled equatorial collapsoid 184–185 braided cube 117 12-celled polar collapsoid 179–185 braided octahedron 116 16-8-flexagon 16 braided Platonic solids 123, 125 2-period folding procedure 49 braided tetrahedron 115 2-period paper-folding 39 braiding the diagonal cube 129–130 2-symbol 99 braiding the dodecahedron 131–136 20-celled polar collapsoid 182 braiding the golden dodecahedron 129, 131 3-6-flexagon 195 braiding the icosahedron 134–138 3-period folding procedure 97 3-symbol 104 Caradonna, Monika xiii 30-celled collapsoid 183–186 Cartan, Henri 237 6-flexagons 4 Challenger space shuttle 2 6-6 flexagons 7–11 closed orientable manifold 232 8-flexagon 11–16 closed orientable surface 230 9-6 flexagon 9 coach 261 90-celled collapsoid 190 coach theorem 260–263, 264 90-celled collapsoid (more symmetric) 193 coach theorem (cyclic) 271 coach theorem (generalized) 267–271 Abel Prize 236 collapsible cube 194 accidents (in mathematics there are no) 275, 271 collapsoids 38, 175–194 Albers, Don xv collapsoids, equatorial 176 Alexanderson and Wetzel 206 collapsoids, polar 176 Alexanderson, Gerald L.
    [Show full text]
  • Metamorphosis of the Cube
    Metamorphosis of the Cube Erik Demaine Martin Demaine Anna Lubiw Joseph O'Rourke Irena Pashchenko Dept. of Computer Science, University of Waterloo Dept. of Computer Science, Smith College Waterloo, Ontario N2L 3G1, Canada Northampton, MA 01063, USA feddemaine, mldemaine, [email protected] forourke, [email protected] 1 Introduction The foldings and unfoldings shown in this video illus- trate two problems: (1) cut open and unfold a convex polyhedron to a simple planar polygon; and (2) fold and glue a simple planar polygon into a convex polyhedron. A convex polyhedron can always be unfolded, at least when we allow cuts across the faces of the poly- Figure 1: Crease patterns to fold the Latin cross into hedron. One such unfolding is the star unfolding [2]. (from left to right) a doubly covered quadrilateral, a When cuts are limited to the edges of the polyhedron, pentahedron, a tetrahedron, and an octahedron. it is not known whether every convex polyhedron has an unfolding [3], although there is software that has never failed to produce an unfolding, e.g., HyperGami [6]. enumeration takes exponential time, but we have shown The problem of folding a polygon to form a convex that there can be an exponential number of valid glu- polyhedron is partly answered by a powerful theorem ings. We are currently working on extending this algo- of Aleksandrov. The following section describes this in rithm beyond the case where whole edges of the polygon more detail. After that, we explain the examples that must be glued to other whole edges. are shown in the video, and discuss some open problems.
    [Show full text]
  • A Polyhedral Byway
    BRIDGES Mathematical Connections in Art, Music, and Science A Polyhedral Byway Paul Gailiunas 25 Hedley Terrace, Gosforth Newcastle, NE3 IDP, England [email protected] Abstract An investigation to find polyhedra having faces that include only regular polygons along with a particular concave equilateral pentagon reveals several visually interesting forms. Less well-known aspects of the geometry of some polyhedra are used to generate examples, and some unexpected relationships explored. The Inverted Dodecahedron A regular dodecahedron can be seen as a cube that has roofs constructed on each of its faces in such a way that roof faces join together in pairs to form regular pentagons (Figure 1). Turn the roofs over, so that they point into the cube, and the pairs of roof faces now overlap. The part of the cube that is not occupied by the roofs is a polyhedron, and its faces are the non-overlapping parts of the roof faces, a concave equilateral pentagon (Ounsted[I]), referred to throughout as the concave pentagon. Figure 1: An inverted dodecahedron made from a regular dodecahedron. Are there more polyhedra that have faces that are concave pentagons? Without further restrictions this question is trivial: any polyhedron with pentagonal faces can be distorted so that the pentagons become concave (although further changes may be needed). A further requirement that all other faces should be regular reduces the possibilities, and we can also exclude those with intersecting faces. This still allows an enormous range of polyhedra, and no attempt will be made to classify them all. Instead a few with particularly interesting properties will be explored in some detail.
    [Show full text]
  • A" Helium Atom" of Space: Dynamical Instability of the Isochoric
    A \Helium Atom" of Space: Dynamical Instability of the Isochoric Pentahedron C. E. Coleman-Smith1, ∗ and B. M¨uller1, y 1Department of Physics, Duke University, Durham, NC 27708-0305, USA (Dated: November 8, 2018) We present an analysis of the dynamics of the equifacial pentahedron on the Kapovich-Millson phase space under a volume preserving Hamiltonian. The classical dynamics of polyhedra under such a Hamiltonian may arise from the classical limit of the node volume operators in loop quantum gravity. The pentahedron is the simplest nontrivial polyhedron for which the dynamics may be chaotic. We consider the distribution of polyhedral configurations throughout the space and find indications that the borders between certain configurations act as separatrices. We examine the local stability of trajectories within this phase space and find that locally unstable regions domi- nate although extended stable regions are present. Canonical and microcanonical estimates of the Kolmogorov-Sinai entropy suggest that the pentahedron is a strongly chaotic system. The pres- ence of chaos is further suggested by calculations of intermediate time Lyapunov exponents which saturate to non zero values. I. INTRODUCTION It has long been known that black holes act as thermodynamic systems whose entropy is proportional to the area of their horizon [1] and a temperature that is inversely proportional to their mass [2]. This has raised the question how quickly and how perfectly a black hole thermalizes and thereby effectively destroys any injected information. Sekino and Susskind have argued that black holes are \fast scramblers" [3, 4], i. e. systems that render information practically unretrievable at the maximal possible rate.
    [Show full text]
  • New Geometric Constraint Solving Formulation: Application to the 3D Pentahedron
    New Geometric Constraint Solving Formulation: Application to the 3D Pentahedron Hichem Barki1, Jean-Marc Cane2, Dominique Michelucci2, and Sebti Foufou1;2 1 CSE Dep., College of Engineering, Qatar University, PO BOX 2713, Doha, Qatar 2 LE2I, UMR CNRS 6306, University of Burgundy, 21000 Dijon, France {hbarki,sfoufou}@qu.edu.qa {jean-marc.cane,dominique.michelucci}@u-bourgogne.fr Abstract. Geometric Constraint Solving Problems (GCSP) are nowa- days routinely investigated in geometric modeling. The 3D Pentahedron problem is a GCSP defined by the lengths of its edges and the planarity of its quadrilateral faces, yielding to an under-constrained system of twelve equations in eighteen unknowns. In this work, we focus on solving the 3D Pentahedron problem in a more robust and efficient way, through a new formulation that reduces the underlying algebraic formulation to a well-constrained system of three equations in three unknowns, and avoids at the same time the use of placement rules that resolve the under- constrained original formulation. We show that geometric constraints can be specified in many ways and that some formulations are much better than others, because they are much smaller and they avoid spurious de- generate solutions. Several experimentations showing a considerable per- formance enhancement (×42) are reported in this paper to consolidate our theoretical findings. Keywords: Geometric Constraint Solving Problems, Parametrization, 3D Pentahedron 1 Introduction GCSPs have retained much of the researchers attention since several decades [6, 5, 13]. This attention may be justified by the advances in computing systems, in terms of both hardware capabilities and software facilities, which translated into a growing need for new CAD/CAM techniques and opened new perspectives for the implementation of researchers ideas.
    [Show full text]