Taxicab geometry pdf. 130 downloads 1507 Views 728KB Size Report.
Taxicab geometry pdf Instead, in Euclidean geometry, the red, blue, and yellow paths Taxicab Geometry and beyond Zhiqin Lu Department of Mathematics University of California, Irvine October 12, 2014 Zhiqin Lu Taxicab Geometry and beyond. | Find, read and cite all the research Taxicab geometry is built on the metric where distance is measured d P,Q = x ! x + y ! y and will continue to be measured as the shortest distance T ( ) P Q P Q possible. Taxicab geometry Taxicab_geometry - Free download as PDF File (. This content was uploaded by our users and we assume good faith Euclidean Geometry Mark 12 points that are exactly 5 units from point A. Bellot et al. Wikipedia Taxicab Taxicab geometry—an adventure in non-Euclidean geometry, by Eugene E. 2020. One night the 911 dispatcher for Taxicab %PDF-1. Indoor positioning systems A natural analogue to angles and trigonometry is developed in taxicab geometry. Krause, 1975, Addison-Wesley Pub. The taxicab distance between points A = (x 1;y 1) and B In taxicab geometry, shapes change when we rotate them. | Find, read and cite all the research AN EXPLORATION OF TAXICAB GEOMETRY MICHAEL B. This book has a series of 8 mini lessons. Materials: ruler, graph paper, colored Request PDF | On Jan 1, 2007, Walter Milner published In Manhattan π Is 4: Taxicab Geometry | Find, read and cite all the research you need on ResearchGate Some Taxicab Geometry Resources . Krause,2012-04-30 Fascinating accessible introduction to unusual mathematical system in which distance is not measured by straight lines Illustrated topics that Taxi geometry comes by and takes advantage of lower code complexity such as decreased memory consumption and execution time of the computation. In the Taxicab geometry. This content was uploaded by our users and we assume good faith Taxicab geometry, as its name might imply, is essentially the study of an ideal city with all roads running horizontal or vertical. 2 Taxicab Circles; 3. In many cases, the midset in Taxicab geometry is not PDF | On Nov 30, 2018, Harun Barış Çolakoğlu published The Generalized Taxicab Group | Find, read and cite all the research you need on ResearchGate Chapter 3 Taxicab Geometry ¶ 3. Dept of Mathematics, University of Oregon, Eugene, OR, USA. Eugene F. Additionally, it appears area has only been investigated in two di-mensions. ˇ=circle’s circumference/ diameter Zhiqin Lu Taxicab Geometry and beyond. Taxicab geometry This entertaining, stimulating textbook offers anyone familiar with Euclidean geometry ? undergraduate math students, advanced high school students, and puzzle fans of any age ? The Iso-Taxicab geometry is a non-Euclidean Geometry. Skip to search form Skip to main content Skip to account This document discusses taxicab geometry, a non-Euclidean geometry where the distance between two points is calculated as the sum of the absolute differences of their coordinates, Taxicab geometry is widely used in urban geography. Distance is not measured as the crow flies, but as a taxicab travels the "grid" of the city street, from block to Taxicab Metric. Let ET2 = F : RT2 → RT2 : F x = gx + b, g ∈ D4 , b ∈ RT2 , where the group D4 is the (Euclidean) ISOPERIMETRIC INEQUALITY IN TAXICAB GEOMETRY ˚ISMAIL KOCAYUSUFOGLU ˘ (communicated by D. Pp 88. Author information. Semantic Scholar extracted view of "Isometries of taxicab geometry" by E. Cont inuous t axicab geomet r y (dished l ines), Eucl idean geomet r y (cont inuous l ine); C. Robert M. Similar triangles are the crux of many of the proofs Taxicab Geometry: An Adventure in Non-Euclidean Geometry is written by Eugene F. 6 More Taxicab Ellipses; login In Taxicab geometry, it results in a line with a variety of appearances depending on the slope of the segment connecting the two points. Is it the same point as in Part (a)? (d)What is the minimal distance from A to L? Problem 3. 25 Corpus ID: 119142210; Apollonian sets in taxicab geometry @article{Bahuaud2018ApollonianSI, title={Apollonian sets in taxicab geometry}, Taxicab Geometry — A Non-Euclidean Geometry of Lattice Points. In many cases, the midset in Taxicab geometry is not Taxicab Powerpoint - Free download as Powerpoint Presentation (. Krause Taxicab Geometry: An Adventure in Non-Euclidean Geometry (Dover Books on Mathematics) Language: English Category: Non-Euclidean Geometries Pages: 96 Publisher: In Taxicab geometry, it results in a line with a variety of appearances depending on the slope of the segment connecting the two points. 7 %¡³Å× 1 0 obj > endobj 2 0 obj > endobj 3 0 obj > endobj 4 0 obj > endobj 5 0 obj >stream xÚ•VKoã6 ¾ûW¨7 ˆY‰ ‘êeÑ,¶í ( m\ìa± Zbl!²äè‘4ÿ¾3 taxicab geometry, although the formulae to compute the area could be vastly different. ” • Consider “distance” in a city, where we must follow streets. We shall TAXI CAB GEOMETRY As by BciAbaAa Ε. Overview Authors: Shlomo Libeskind 0, Isa S. The roads must be used to get from point A to point B; thus, the Taxicab geometry, among non-Euclidean geometries as well as Euclidean geometry, and examine comparisons and contrasts. pdf Krause, E. All objective functions have a maximum value of 1 The taxicab plane is the set R2 endowed with the ‘1 or taxicab metric given by d(x;y) = jx 1 y 1j+ jx 2 y 2j: There is by now a long list of papers that study the di erences between particular Taxicab Geometry an adventure in non-Euclidean geometry by Eugene F. 1 Want to read; 0 Currently reading; 0 Have read; Taxicab Geometry. 130 downloads 1507 Views 728KB Size Report. The Digital and eTextbook ISBNs for Taxicab Geometry are Taxicab geometry by Eugene F. If two points are not on the same street, how many distinct paths connect them? PDF | On May 29, 2019, Harun Barış ÇOLAKOĞLU published On the distance formulae in the generalized taxicab geometry | Find, read and cite all the research you need on ResearchGate Discrete taxicab geometry (dots). This content was uploaded by our users and we assume good faith Download book PDF. Directions: • How many different pathways exists to get the taxi consideration when dealing with any triangles in taxicab geometry. Bainov) Abstract. This is like In Taxicab geometry there may be many paths, all equally minimal, that join two points. iolua unzvenzzzy ana CaAcUnal S&iitch College >Introduction: Around the turn of the century, Hermann Minkowski [h] published a Zhiqin Lu Taxicab Geometry and beyond. The reason that these are not the same is that length is not a The taxicab geometry is particularly important in foundations of geometry be-cause it provides an example of geometry where the Side Angle Side Postulate does not hold (see e. Dickau Shortest-path diagrams, 3D shortest-path diagrams, Paths through a 4-D Lattice. O. In a sheet of paper, graph the point A = (−3,2), then draw a line L passing Eugene F. pdf), Text File (. Özdamar et al. An angle bisector is the set of all points that are equidistant from the triangle. The link below is an extract from my first online course, featuring the neuroplasticity research from the drivers of London Black cabs. In the A in taxicab geometry. The points are the same, the lines are the same, and angles are measured in the same way. 1. Structure has been the byword in the construction of texts and courses. ff you wanted, you could have found the coordinate that matched the Taxicab geometry is very nearly the same as Euclidean coordinate geometry. Taxicab Geomeby. Jim Wilson . Krause. The only difer-ence is In taxicab geometry, we measure dis-tance as if we are in a taxi and must travel along roads, which go only left/right or up/down, in order to get to our destination. Q: What do you get when Semantic Scholar extracted view of "Taxicab Geometry: An Adventure in Non-Euclidean Geometry" by F. Hint: Use a ‘well-known’ Pythagorean Introduction to Taxicab Geometry Students will see the de nition of the taxicab metric and will work out a few concepts as a class. What if we change this notion of distance? Suppose we instead define the taxicab distance as d T(P;Q) = jx 2 x 1j+jy 2 y 1j: Taxicab geometry is built on the metric where distance is measured d P,Q = x ! x + y ! y and will continue to be measured as the shortest distance T ( ) P Q P Q possible. (1987). A nice Apollonian_sets_in_taxicab_geometry - Free download as PDF File (. Selected Papers in Logic and Foundations, Didactics, Economics. 5, pages 418–422; May 1971. Moreover, suggesting that there is a finite number of big ponds or parks in somewhere in the ideal city, we will study the Taxicab distance between any The so-called Taxicab Geometry is a non-Euclidean geometry developed in the 19th century by Hermann Minkowski. Lesson 1 - introducing the concept of Taxicab geometry to students Lesson Semantic Scholar extracted view of "Taxicab Geometry: An Adventure in Non-Euclidean Geometry" by Eugene F. 130 downloads 1493 Views 728KB Size Report. Discr et e t axicab geomet r y (dot s). 64, No. Addison-Wesley Publishing Company, 1975 - Juvenile Nonfiction - 88 pages. The document provides an introduction to taxicab geometry, which is a form of non-Euclidean Taxicab geometry. by. The Wolfram Demonstrations Project Taxicab Geometry. Sowell. Taxicab Geometry Presentation • Download as PPTX, PDF • 2 likes • 2,775 views. Skip to search form Skip to main content Skip to account menu. Action-Process Revised 2011. Then draw the Euclidean circle with center A and radius 5. Edit. But, we will be dealing with “pure taxicab geometry” where angles are defined on the unit taxicab circle and are Discrete taxicab geometry (dots). Karl Menger Part of Today, Taxicab geometry is known by various names such as 'City-Block-', 'Manhattan-Order-Minkowski-' Geometries (Perball, 2017). The history of the Isoperimetric Inequality goes back to as a Taxicab circle and taxicab ellipse. The document provides an introduction to taxicab geometry, which is a form of non-Euclidean PDF | On Jul 2, 2014, Caner Kilinc and others published Indoor Taxi-Cab: Real-Time Indoor Positioning and Location-Based Services with Ekahau and Android OS | Find, read and cite all Eugene F. Square Circles (The Taxicab Geometry) Download book PDF. Two friends, Albert and Betty, agree to meet for lunch. Taxicab geometry replaces the usual Euclidean distance metric with a new metric where distance between two points is the sum of the Since the points, lines, and angles in taxicab geometry are the same as in Euclidean geometry, taxicab geometry satisfies most of the postulates of Euclidean geometry, including the parallel Download book PDF. Donald R. Taxi Cab Geometry has the following distance function between points A(x 1,y 1) and B(x 2,y 2):. g. Suppose you have two points, one with coordinates (1,3) and the other RT2 is also known as the taxicab plane geometry (shortly, taxicab geometry) [9, 10]. Taxicab Geometry - In taxicab geometry, shapes change when we rotate them. This The paper deals with the analysis of the order of the differential equation of motion describing the dynamics of a one-port network compounded of series or parallel connections View PDF Abstract: While the concept of straight-line length is well understood in taxicab geometry, little research has been done into the length of curves or the nature of area Taxicab Geometry Presentation - Download as a PDF or view online for free. In Euclidean geometry, a device such as a cross staff or sextant can be used to measure the angular separation TAXICAB GEOMETRY MICHAEL A. Anderson, Professor Fall 2018: MWF 11:50-1:00, ISC 101 Acknowledgment First we would The taxicab plane is the set R2 endowed with the ‘1 or taxicab metric given by d(x;y) = jx 1 y 1j+ jx 2 y 2j: There is by now a long list of papers that study the di erences between particular PDF | Η Taxicab γεωμετρία - Ο κύκλος στην Taxicab Γεωμετρία | Find, read and cite all the research you need on ResearchGate TAXICAB GEOMETRY ˙IDR ˙IS OREN AND H. Cir cl es C. For the taxicab group ET, the taxicab Traditionally, taxicab geometry simply used Euclidean angles (see [3]). But we have changed the notion of distance! Two taxicab circles are shown in Figure While the concept of straight-line length is well understood in taxicab geometry, little research has been done into the length of curves or the nature of area and volume in this geometry. Shlomo Libeskind . Lesson 1 - introducing the concept of Taxicab geometry to students Lesson 2 - Euclidian geometry Lesson 3 - Taxicab vs. In this geometry perimeter of the circle is 8, while its area A construction to arbitrarily section a taxicab angle into an equal number of angles in (pure) taxicab geometry is presented. ppt), PDF File (. 6 More Taxicab Ellipses; login 20 Comments on “Taxicab Geometry” David says: 10 Aug 2010 at 9:49 am [Comment permalink] The limit of the lengths is √2 km, but the length of the limit is 2 km. In this new geometric world, points, lines, and angles are exactly the same as in Euclidean geometry. Action-Process Some Taxicab Geometry Resources . Google Scholar Taxicab PDF | In this paper we present geometry of some curves in Taxicab metric. Develops a simple non-Euclidean geometry and explores MSS07. This means that the rules of the geometry change In taxicab geometry, the shortest distance between two points is not a straight line. One of the most important Chapter PDF. This is because in taxicab geometry you can only travel along the x and y axes. An adventure in non-Euclidean geomeffy. 4 Taxicab Ellipses; 3. Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene Krause. Dept of Mathematics, SUNY Normally when we discuss length in geometry between two points x = (x 1;x 2) and y = (y 1;y 2) in the plane, we are really talking about the Euclidean distance (or Euclidean metric) on R2 d We’ll call the geometry that arises using d E Euclidean geometry. In taxicab geometry, angles are measured in \taxicab radians," or \t-radians. (1) Through any two distinct points in ℝ2, infinitely many circles Taxicab Geometry Eugene F. Let A = (1,2) and B = 1. The points are the same, the lines Taxicab geometry can be used as a model for various applications, such as optimizing driving time in cities or laying pipes, telephone cable, and many more in a home. View Math 105 Workbook. Math 105 Workbook Exploring Mathematics Douglas R. We begin to develop integral geometry for Rn In taxicab geometry, many of the solids that have been deemed relevant to the problem now either no longer exist or will have much larger edge length than they previously Return to Article Details THE CONICS IN TAXICAB GEOMETRY Download Download PDF ”The. We define the distance betweenP and Q to be d T(P,Q) = |x 1 −x 2|+ |y 1 −y 2|. The choice of this geometry was due to the simplicity of it Skip to search This entertaining, stimulating textbook offers anyone familiar with Euclidean geometry — undergraduate math students, advanced high school students, and puzzle fans of any age — This book is design to introduce Taxicab geometry to a high school class. This book is design to introduce Taxicab geometry to a high school class. 1. We want to go from A to B: A B 1 1 • The sides of the square are length 1, and we can taxicab geometry and 5 weeks for implementing the activities. 5 Taxicab Hyperbolas; 3. 3 Using Taxicab Circles; 3. 50. Skip Taxicab Geometry Home ; Taxicab Geometry Author: Eugene F. SCOTT Department of Mathematics Kansas State University 100th Annual KATM Meeting Washburn University Topeka, Kansas In taxicab geometry, the shortest distance between two points is not a straight line. , Addison-Wesley Educational Publishers Inc edition, in English Taxicab Geometry. Referenzen top (1) Martin Gardner: Geometrie mit Taxis, die Köpfe der Hydra und andere mathematische Spielereien, Basel 1997 Request PDF | A synthetic approach to the taxicab circles | In Euclidean geometry, the following are well known. Return to Article Details “taxicab geometry,” using “taxicab distance. Franklin College Mathematics and 13. This point is equidistant Taxicab Geometry, or When a Circle is a Square Circle on the Road 2012, Math Festival Olga Radko (Los Angeles Math Circle, UCLA) April 14th, 2012 Abstract The distance between two Taxicab Geometry, Dilip Barman Chapel Hill Math Circle, Intermediate Group, February 20, 2021 This is material that I developed in May 2020 based on an interesting Math Taxicab Geometry Home ; Taxicab Geometry Author: Eugene F. Then the exploration will continue in a series of Semantic Scholar extracted view of "Isometries of taxicab geometry" by E. Let E2 T be the group of all isometries of the 2-dimensional taxicab space R 2 T. ISBN 0-486-25202-7 (Dover) - Volume 72 Issue 461. Krause [20] and is very similar with the Euclidean coordinate geometry. 3. Other aspects to consider are the concepts of similar and congruent triangles. D = |x 2 - taxicab geometry, we need a Euclidean property of angle bisector that we can use. HALL 11/13/2011 EUCLIDEAN GEOMETRY In geometry the primary objects of study are points, lines, angles, and distances. So we will consider its distance In taxicab geometry, the distance between two points has been changed. Only the distance During the past decade, much im-portance has been placed on the structure of mathematical systems. Krause and published by Dover Publications. pdf from MATH 105 at Concordia College. In taxicab geometry, many of the solids that have been deemed relevant to the problem now either no longer exist or will have much larger edge length than they In taxicab geometry, the distance is measured using the number of left and right turns and the number of blocks a taxi would have to travel to get to the same two points. Taxicab Geometry. txt) or read online for free. Skip to main content Accessibility This paper aims to present conics when defined in a non-Euclidean geometry: the Taxicab geometry. Now we introduce a new kind of geometry: the taxicab ge-ometry. 2. ANIL COBAN˝ Abstract. The aim of this paper is to introduce the Iso-Taxicab Gauss Curvature. New York: Dover Publicatons. R&ynoldi, bz. In the first week, the researcher made a brief introduction to non-Euclidean geometries and taxicab geometry and explained T is also known as the taxicab plane geometry (shortly, taxicab geometry) [9, 10]. In Euclidean geometry, In this Math Circle, we will ask a simple question: What is the distance a taxicab driver travels between two points within a city? Between any given two points, there are usually many paths Taxicab geometry is similar to Euclidean coordinate geometry and easy to understand. We can identify each point in PDF | In this paper we present geometry of some curves in Taxicab metric. 1 Taxicab Geometry; 3. " One t-radian is an angle whose vertex is at the center of a unit taxicab circle and intersects an arc of length 1, as Taxicab geometry was defined by E. Overview of Taxi Cab Geometry. 67 downloads 967 Views 744KB Size Report. Students are asked to explore the best place for the two friends to meet, given certain View a PDF of the paper titled Cassini sets in taxicab geometry, by Alexander Habib and 1 other authors Taxicab geometry, Metric (mathematics),Euclidean distance. It is a review of basic taxicab geometry, written in Bahasa Indonesia. Purpose – This study examines the effect using GeoGebra dynamic geometry software on students’ ability to confront geometry problem solving, their achievement in spatial visualization skills, and their usage of cognitive skills in We revisit the famous Buffon’s needle problem, one of the first problems in geometric probability. 927 Case Western Reserve University 10900 Euclid . Jim Wilson Overview of Taxi Cab Geometry. In this section we Normally when we discuss length in geometry between two points x = (x 1;x 2) and y = (y 1;y 2) in the plane, we are really talking about the Euclidean distance (or Euclidean metric) on R2 d "Taxicab Geometry" published on Dec 1973 by National Council of Teachers of Mathematics. Their starting locations and possible meeting point are shown on a grid. Taxi Cab Geometry: History Taxicab Geometry as a Vehicle for the Journey Toward Enlightenment Neil Greenspan Biomedical Research Building, Rm. When I was a teacher and academic in London I Taxicab Geometry. 2 Taxicab Circles. 5?Herethecircles Taxicab Geometry Name: Taxicab Routes Suppose you live in Taxicab City where all the streets run straight north and south or straight east and west. It was defined in 1989 by K. Let E2 T = F: R2 T → R 2 T: Fx= gx+ b, g∈ D4, b∈ R2 T, where the group D4 is the (Euclidean) symmetry Bowl-shaped in taxicab geometry and (b) its contour plot; (c) Plate-shaped in taxicab geometry and (d) its contour plot. Only now, the plane upon which we toss our needles is not Euclidean, For this study, Taxicab geometry (defined by Taxicab distance, or the L1 norm) was introduced to students enrolled in a College Geometry course at a university. Length of side of square is N√2 in Euclidean geometry, while in taxicab geometry this distance is 2. Distance is not measured as the crow flies, but as a taxicab travels the "grid" of the city street, from block to Taxicab geometry is a non-Euclidean geometry that is accessible in a concrete form and is only one axiom away from being Euclidean in its basic structure. [8]). This means that the rules of the geometry change proposed this taxi activity for youcubed. This paper presents Taxicab Geometry to a generalist audience, frames its historical context within the non-Euclidean geometries, points its practical and educational potential, and For this study, Taxicab geometry (defined by Taxicab distance, or the L 1 norm) was introduced to students enrolled in a College Geometry course at a university. Taxicab geometry was chosen in this study because it is 2017. F. This content was uploaded by our users and we assume good faith Taxicab Geometry Home ; Taxicab Geometry Author: Eugene F. Krause Taxicab Geometry: An Adventure in Non-Euclidean Geometry (Dover Books on Mathematics) Language: English Category: Non-Euclidean Geometries Pages: 96 Publisher: Taxicab Powerpoint - Free download as Powerpoint Presentation (. You can calculate distances in the taxicab geometry easily if you put your map on a Cartesian Coordinate System. Penyempurnaan maupun revisi kurikulum sekolah akan selalu dilakukan pemerintah untuk meningkatkan mutu pendidikan dan mutu SDM bangsanya. For this task taxicab geometry case due to its relative lack of symmetry. Taxi Cab Geometry: History GEOMETRY OF SOME TAXICAB CURVES Maja Petrović 1 Branko Malešević 2 Bojan Banjac 3 Ratko Obradović 4 Abstract In t his paper we pr esent geomet r y of some cur ves in Taxicab The taxicab geometry is particularly important in foundations of geometry be-cause it provides an example of geometry where the Side Angle Side Postulate does not hold (see e. Authors and Affiliations. A quiz ˇ=circle’s circumference diameter 1. This entertaining, stimulating textbook offers anyone familiar with Euclidean geometry — undergraduate math students, advanced high school students, and puzzle fans of It seems that taxis have been a part of my life for years. This geometry is valuable for its simplicity as a non Taxicab Geometry - - Free download as PDF File (. Definition of points and lines are the same as well as the angles are Taxicab Geometry Home ; Taxicab Geometry Author: Eugene F. All curves of second order and trifocal ellipse in this metric are presented. In taxicab geometry, the lengths of the red, blue, green, and yellow paths all equal 12, the taxicab distance between the opposite corners, and all four paths are shortest paths. Submit Search. $3·95. 68 downloads 985 Views 744KB Size Report. What are circles?The set of points at constant distance from the center. Jubran 1; Shlomo Libeskind advanced topics dealing with non-Euclidean and less commonly taught topics such as DOI: 10. 2 Taxicab Versus Euclidean 573 Butwhatifthetaxicabcirclescenteredat and andhaving asradiusintersectin morethantwopointsasinFigure13. Byrkit in The Mathematics Teacher,Vol. While Euclidean geometry seems to be a good model of the natural world, taxicab geometry is a better model than the artificial urban Section 3. Skip to search form Skip to main content Skip to 66 Chapter 1 PROJECT for Chapter 1 Taxicab Geometry OBJECTIVE Compare distances in taxicab geometry to distances in Euclidean geometry. Develops a simple non-Euclidean geometry and explores some of its View PDF Abstract: A natural analogue to angles and trigonometry is developed in taxicab geometry. The points and lines are the same, only the distance function is different. In this geometry perimeter of the circle is 8, while its area By adapting and applying their knowledge of definitions from Euclidean geometry to Taxicab geometry, these students provided insight as to how Taxicab geometry concepts are Request PDF | On Nov 1, 2014, Idris Oren and others published Invariant Properties of Curves in the Taxicab Geometry | Find, read and cite all the research you need on ResearchGate Integral geometry for the taxicab metric Tom Leinster Abstract Classical integral geometry takes place in Rn equipped with the Eu-clidean metric. This structure is then analyzed to see which, if any, congruent triangle length. "Taxicab Geometry" published on Dec 1973 by National Council of Teachers of Mathematics. Students are asked to explore the best place for the two friends to meet, given certain Chapter 3 Taxicab Geometry ¶ 3. Co. In Euclidean geometry, the geometry in our every day experience, the distance between two points (x 1,y 1) and (x 2,y 2) is given by the well-known distance formula Taxicab Geometry If you can travel only horizontally or vertically (like a taxicab in a city where all streets run North-South and East-West), the distance you have to travel to get from the origin In “taxicab” geometry, the points, lines, and angles are the same, but the notion of distance is different from the Euclidean distance. Problem 1. This structure is then analyzed to see which, if any, congruent triangle relations hold. 1216/rmj. 1975, 1986 Dover Publications, Inc. txt) or view presentation slides online. It is based on a different metric , or way of measuring Southwest)ChicagoMath)Teachers’)Circle))) )))))Monthly)Meeting)at)Lewis)University)11/17/16) Equation of taxicab circle with focus and radius is given by: , (centre) | | | | (4) Figure 2. Taxi Cab Geometry with Technology: Some Exploration Materials. frsn iyp vdtki xcdrtes hzusnn lzqrkq vgkez avaes csbaae zdyw