# SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Daniel DOBRE, Ionel SIMION SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING Abstract: This article discusses the relation between the concept of symmetry and its applications in engineering. "Symmetry" is interpreted in a broad sense as repeated, coplanar shape fragments. An analysis of symmetry, which justifies its applications in engineering is given and discussed. After a brief explication of group theory and symmetry types, we show that there are industrial workpieces where symmetry is omnipresent. The analysis of symmetry can also be utilized for future research concerns the combining symmetry information with other functional characteristic of digital 3D design. Key words: symmetry, space group, automorphism, screw symmetry, regular polyhedron. 1. INTRODUCTION Symmetry is a vast subject, important in art and nature, based on mathematics. There are a wide variety of applications of the principle of symmetry in art, in the inorganic and organic nature. The notion of symmetry is equivalent to the harmony of proportions, defining the idea that man has tried, over time, cover and create order, beauty and perfection. It is seen in various forms: bilateral symmetry, translation symmetry, rotational symmetry, ornamental symmetry, crystallography symmetry etc. An object or structure is symmetrical if it looks the same after a specific of change is applied to it. The object or structure can be material, such as living organisms including humans and other animals, crystal, regular polyhedron, pavement tiles, or it can be an abstract structure such as a mathematical equation. The nature of the change can be similarly diverse, ranging from such simple operations as moving across a regularly patterned tile floor, to complex transformations of equations. This paper describes symmetry from two perspectives. The first is that of mathematics, in which symmetries are defined and categorized precisely. The second perspective looks at the application of symmetry concepts to engineering [2] [3]. displacement as composition, forms an infinite order commutative group, the continuous spatial displacement group. The group of similarities leaves the shape of a figure unchanged. The size of a figure is invariant with respect to the group of congruencies [1]. In mathematics, there is a major difference between discrete and continuous groups. Examples of discrete groups are the finite rotation groups of polygons and crystals. In one dimension, ornaments of stripes are classified by seven groups, which are systematically produced by periodic translations in one direction and reflections transverse to the longitudinal axis of translations (fig. 1). 2.2 Geometric symmetry In geometry, symmetric properties of figures and bodies indicate invariance with respect to automorphisms like rotations, translations and reflections. 2. MATHEMATICAL CONCEPT OF SYMMETRY An object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa) [1]. 2.1 Group theory In modern times, symmetries are defined mathematically by group theory. The symmetry of a set (e.g. points, numbers and functions) is defined by the group of automorphism that leaves unchanged the structure of the set (e.g. proportional relations in Euclidean space, arithmetical rules of numbers). The set of all displacement transformations in space, including the identity transformation, with consecutive Fig. 1 Symmetries in one dimension. (1-translational symmetry; 2-reflection symmetry; 3 7-composite symmetries) JUNE 2009 NUMBER 5 JIDEG 21

2 In two dimensions, there is an axis of symmetry. The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that the shape was to be folded in half over the axis, the two halves would be identical: the two halves are each other s mirror image. Thus a square has four axes of symmetry because there are four different ways to fold it (fig. 2). A reflection in [P] is the application of space itself, S: A A 1, which leads arbitrary point A in reflection of A 1 vs. [P]. A reflection point reflects a point V to the opposite side of the point, as illustrated in fig. 5. All three reflection symmetries associate exactly one reflected point A 1 with each point A on the surface [4]. The axial symmetry associates each point A with an entire circle of points A 1, which includes A. The center of the circle intersects with the symmetry axis. The plane of the circle is orthogonal to this axis (fig. 6). Fig. 2 Symmetry lines of polygons. In R 3 there exist three basic symmetry features: a point, an axis and a plane. Each of these features establishes equivalence classes of multiple surface points. We also distinguish reflection, axial and spherical symmetries. In detail, reflection symmetry associated each point A on the object surface to another surface point A 1 on the opposite side of the object. For example, a reflection line reflects points across a line. Fig. 3 illustrates such a reflection line, which reflects a point D to a reflection point D 1 on the opposite side of the line. Fig. 4 Symmetry of pyramid given the symmetry plane. Fig. 5 Symmetry of pyramid given the symmetry point. Fig. 6 Axial symmetry. 22 Fig. 3 Symmetry of prism given the symmetry axis. Plane reflection symmetry reflects points to the opposite side of the symmetry plane. This is illustrated in fig. 4, which shows an object with an associated symmetry plane [4]. JUNE 2009 NUMBER 5 JIDEG The rotation transformation acts on a 3-dimensional system by rotating the whole system through a given angle about a given axis, called the rotation axis. A 3-dimensional system might be symmetric under rotations by any angle about one or more axes or only by

3 a minimum angle of 360 /n, where n is an integer greater than 1, and multiples of it. Such a rotation axis is called an axis of full or n-fold rotational symmetry, respectively. A sphere, for example, has an infinite number of axes of full rotational symmetry all axes passing through its center. Systems having this type of symmetry are said to posses spherical symmetry. Spherical symmetries reflect each point A on the object surface to an entire sphere, whose center is the symmetry point. Figure 7 illustrates a spherical symmetry. A number of axes of full rotational symmetry are indicated. A regular tetrahedron has three axes of 2-fold rotational symmetry (through the midpoints of pairs of opposite edges) and four 3-fold rotational symmetry axes (through each vertex and the center of the opposite face). To calculate the order of the group, observe that a given vertex can be moved to one of four positions. Hence, the order of the group of direct symmetries (all rotations) is S(T) = 24. A cube has six 2-fold axes (through the midpoints of pairs of opposite edges), four 3-fold axes (through pairs of opposite vertices) and three 4-fold axes (through the centers of pairs of opposite faces. Fig. 7 Spherical symmetry. A cylinder has full rotational symmetry only about a single axis, the longitudinal axis of the cylinder. Also a cone possesses this symmetry called axial symmetry (fig. 8) The symmetry group of the cube or octahedron S(C) They both have the same number of edges, being 12. The number of faces and vertices are interchanged. Because these two solids are dual to each other they have the same symmetry group. The order of the group of direct symmetries (all rotations) is S(C) = 24. The elements are: - 3 rotations (by ± π/2 or π) about centers of 3 pairs of opposite faces; - 1 rotation (by π) about centers of 6 pairs of opposite edges; - 2 rotations (by ± 2π/3) about 4 pairs of opposite vertices (diagonals). Together with the identity this accounts for all 24 elements. Fig. 8 3-dimensional systems with axial symmetry. An infinitely long cylinder with no ends has, in addition to its axial symmetry, displacement symmetry by any interval in the direction of its axis. This combination of symmetries, displacement symmetry by any interval along an axis of axial symmetry, is called cylindrical symmetry. 2.3 Symmetry of regular polyhedrons In three dimensions, rotations about a common axis give us the cyclic groups C n. For n 3, C n is the rotational symmetry group of the pyramid built on a regular n gonal base (with axis of rotational symmetry passing through the apex and the center of the base). All the Platonic solids are symmetric about their centers (fig. 9). Fig. 9 Regular polyhedrons The symmetry group of the dodecahedron or icosahedron S(D) They both have the same number of edges, being 30. Because these two solids are dual to each other they have the same symmetry group. The order of the group of direct symmetries (all rotations) is S(D) = 60. The elements are: - 4 rotations (by multiples of 2π/5) about centers of 6 pairs of opposite faces; - 1 rotation (by π) about centers of 15 pairs of opposite edges; - 2 rotations (by ±2π/3) about 10 pairs of opposite vertices. Together with the identity this accounts for all 60 elements. We define the dual of a regular polyhedron to be another regular polyhedron, which is formed by JUNE 2009 NUMBER 5 JIDEG 23

4 connecting the centers of the faces of the original polyhedron (fig.10). Fig. 10 Dual of a regular polyhedron. 3. SCREW (HELICAL) SYMMETRY Helical symmetry is the kind of symmetry seen in everyday objects such as springs, drill bits and spiral staircases. It can be thought of as rotational symmetry along with translation along the axis of rotation (fig.11). interval h. Since there are n steps per turn, each step is essentially a wedge of angle 360 /n (fig. 11). So a rotation of the staircase about its axis by 360 /n puts each step exactly in a position either above where the step below it was. A displacement by interval h/n along its axis can return the staircase to its original appearance. Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis: - infinite helical symmetry (an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby the translation distance on that axis at which the object will appear exactly as it did before); - n-fold helical symmetry (objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle; - non-repeating helical symmetry. Fig. 12 Helical staircase. 4. SYMMETRY IN ENGINEERING 24 Fig. 11 Screw transformation. Consider a helical staircase (fig. 12). Let h denote the change of height involved in each complete turn of the staircase and n the number of steps in one complete turn. The height of each step above the one below it is then h/n. If it were infinitely long, it would have displacement symmetry along its axis with minimum displacement JUNE 2009 NUMBER 5 JIDEG Lots of metallic parts such as flexible membranes from the structure of elastic couplings, sprocket wheels, cars rims etc, used as pieces for devices, mechanisms and machines have been designed according to the principle shape follows function and the beauty of these objects increased at the same time with their functional efficiency [3]. An aspect of beauty is symmetry, which represents relative simplicity within complexity. We illustrate this in fig The membrane is formed from thin spokes by making radial indents in the central portion, connected by inner and outer diameters of the shape (fig. 13) [2]. Many logos start with a basic shape, a rectangle, a diamond or an oval, and then the graphic artist uses symmetry to create the design.

5 The Mitsubishi company logo (fig. 18) began with a diamond that was rotated 120 degrees, than another 120 degrees from that. Toyota logo has horizontal reflection symmetry across a vertical line through its center. The logo is made up of three ellipses. The two inner ellipses are 90 0 rotations of each other. Fig. 13 Spoked membrane for elastic coupling. Translation Rotation Fig. 14 Three dimensional object with translational and rotational symmetries. Fig. 17 Rims with multiple spokes (rotational symmetry). Fig. 15 3D model with axial symmetry [3]. 5. CONCLUSION Fig. 18 Corporate logos. Fig. 16 Objects with helical and rotational symmetry [5]. Symmetry is a very important concept in mathematics and can be applied in many different areas including JUNE 2009 NUMBER 5 JIDEG 25

6 equations, shapes, workpieces and aero dynamical buildings. Symmetry also plays an important role in human visual perception and aesthetics. Knowing that a shape or object has symmetry can help us solve problems involving that shape (e.g. technique for segmenting objects into parts characterized by different symmetries). We believe that for many common objects, the construction of 3D surface shape using symmetries types is necessary for practical applications. The use of 3D modelling and simulation concepts and tools can highlight the design in the machine building process REFERENCES [1] Thrun S., Wegbreit B. (2005), Shape from symmetry, Proceedings of the 10 th IEEE International Conference on Computer Vision (ICCV), pp [2] Dobre, D., (2008), Development Basics of a Product (Bazele dezvoltarii de produs), Bucharest, Romania. [3] Dobre D., Simion I., (2009), Special applications of fair surfaces representation, The 3 rd International Conference on Engineering Graphics and Design, in Acta Technica Napocensis, Series: Applied Mathematics and Mechanics, no. 52, vol. Ia, pp , Cluj-Napoca, Romania. [4] Aldea, S., (1984), Descriptive geometry. Bodies and surfaces study (Geometrie descriptivă. Studiul corpurilor şi al suprafeţelor), U.P.B., Romania. [5] Simion, I., (1998), Engineering Graphic (Grafică inginerească), Bren Publishing House, Bucharest, Romania. Authors: Eng. Daniel Dobre, Ph.D, Lecturer, University POLITEHNICA of Bucharest, Department of Descriptive Geometry and Engineering Graphics, Eng. Ionel Simion, Ph.D, M Eng, Professor, University POLITEHNICA of Bucharest, Head of Descriptive Geometry and Engineering Graphics Department, E- mail: 26 JUNE 2009 NUMBER 5 JIDEG

### Geometrical symmetry and the fine structure of regular polyhedra

Geometrical symmetry and the fine structure of regular polyhedra Bill Casselman Department of Mathematics University of B.C. cass@math.ubc.ca We shall be concerned with geometrical figures with a high

### Shape Dictionary YR to Y6

Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use

### Begin recognition in EYFS Age related expectation at Y1 (secure use of language)

For more information - http://www.mathsisfun.com/geometry Begin recognition in EYFS Age related expectation at Y1 (secure use of language) shape, flat, curved, straight, round, hollow, solid, vertexvertices

### Geometry Vocabulary Booklet

Geometry Vocabulary Booklet Geometry Vocabulary Word Everyday Expression Example Acute An angle less than 90 degrees. Adjacent Lying next to each other. Array Numbers, letter or shapes arranged in a rectangular

### Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

### Engineering Geometry

Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the right-hand rule.

### PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT

UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

### Activity Set 4. Trainer Guide

Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES

### CK-12 Geometry: Exploring Solids

CK-12 Geometry: Exploring Solids Learning Objectives Identify different types of solids and their parts. Use Euler s formula to solve problems. Draw and identify different views of solids. Draw and identify

### Session 9 Solids. congruent regular polygon vertex. cross section edge face net Platonic solid polyhedron

Key Terms for This Session Session 9 Solids Previously Introduced congruent regular polygon vertex New in This Session cross section edge face net Platonic solid polyhedron Introduction In this session,

### ENGINEERING DRAWING. UNIT I - Part A

ENGINEERING DRAWING UNIT I - Part A 1. Solid Geometry is the study of graphic representation of solids of --------- dimensions on plane surfaces of ------- dimensions. 2. In the orthographic projection,

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### ME 111: Engineering Drawing

ME 111: Engineering Drawing Lecture # 14 (10/10/2011) Development of Surfaces http://www.iitg.ernet.in/arindam.dey/me111.htm http://www.iitg.ernet.in/rkbc/me111.htm http://shilloi.iitg.ernet.in/~psr/ Indian

### 1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

### Framework for developing schemes of work for the geometry curriculum for ages 14-16

Framework for developing schemes of work for the geometry curriculum for ages 14-16 CURRICULUM GRADES G - F GRADES E - D GRADES C - B GRADES A A* INVESTIGATION CONTEXT Distinguish Know and use angle, Construct

### Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in

### SYMMETRY M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier

Mathematics Revision Guides Symmetry Page 1 of 12 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier SYMMETRY Version: 2.2 Date: 20-11-2013 Mathematics Revision Guides Symmetry Page

Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

### Chapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold

Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional

### TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry

TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry ASLE Expectation: Focus Objective: Level: Strand: AR04MGE040408 R.4.G.8 Analyze characteristics and properties

### BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

### Grade 7/8 Math Circles Winter D Geometry

1 University of Waterloo Faculty of Mathematics Grade 7/8 Math Circles Winter 2013 3D Geometry Introductory Problem Mary s mom bought a box of 60 cookies for Mary to bring to school. Mary decides to bring

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

### Grade 7/8 Math Circles Winter D Geometry

1 University of Waterloo Faculty of Mathematics Grade 7/8 Math Circles Winter 2013 3D Geometry Introductory Problem Mary s mom bought a box of 60 cookies for Mary to bring to school. Mary decides to bring

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. esolutions Manual - Powered by Cognero Page 1 1. A figure

### SHAPE, SPACE AND MEASURES

SHPE, SPCE ND MESURES Pupils should be taught to: Understand and use the language and notation associated with reflections, translations and rotations s outcomes, Year 7 pupils should, for example: Use,

### Construction and Properties of the Icosahedron

Course Project (Introduction to Reflection Groups) Construction and Properties of the Icosahedron Shreejit Bandyopadhyay April 19, 2013 Abstract The icosahedron is one of the most important platonic solids

### Dodecahedron Faces = 12 pentagonals with three meeting at each vertex Vertices = 20 Edges = 30

1 APTER Introduction: oncept of Symmetry, Symmetry Elements and Symmetry Point Groups 1.1 SMMETR Symmetry is one idea by which man through the ages has tried to understand and create order, periodicity,

### Name Date Period. 3D Geometry Project

Name 3D Geometry Project Part I: Exploring Three-Dimensional Shapes In the first part of this WebQuest, you will be exploring what three-dimensional (3D) objects are, how to classify them, and several

### Hyperbolic Islamic Patterns A Beginning

Hyperbolic Islamic Patterns A Beginning Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-2496, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### UNIT PLAN. Big Idea/Theme: Shapes can be analyzed, described, and related to our physical world.

UNIT PLAN Grade Level: 2 Unit #: 5 Unit Name: Geometry Big Idea/Theme: Shapes can be analyzed, described, and related to our physical world. Culminating Assessment: Use real world objects from home, create

### CLASSIFYING FINITE SUBGROUPS OF SO 3

CLASSIFYING FINITE SUBGROUPS OF SO 3 HANNAH MARK Abstract. The goal of this paper is to prove that all finite subgroups of SO 3 are isomorphic to either a cyclic group, a dihedral group, or the rotational

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

### ascending order decimal denominator descending order Numbers listed from largest to smallest equivalent fraction greater than or equal to SOL 7.

SOL 7.1 ascending order Numbers listed in order from smallest to largest decimal The numbers in the base 10 number system, having one or more places to the right of a decimal point denominator The bottom

### RIT scores between 191 and 200

Measures of Academic Progress for Mathematics RIT scores between 191 and 200 Number Sense and Operations Whole Numbers Solve simple addition word problems Find and extend patterns Demonstrate the associative,

### Geometry Performance Level Descriptors

Geometry Performance Level Descriptors Limited A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Geometry. A student at this level has an emerging

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### A FAMILY OF BUTTERFLY PATTERNS INSPIRED BY ESCHER

15th INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS 1 5 AUGUST, 2012, MONTREAL, CANADA A FAMILY OF BUTTERFLY PATTERNS INSPIRED BY ESCHER Douglas DUNHAM ABSTRACT: M.C. Escher is noted for his repeating

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### 10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres. 10.4 Day 1 Warm-up

10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres 10.4 Day 1 Warm-up 1. Which identifies the figure? A rectangular pyramid B rectangular prism C cube D square pyramid 3. A polyhedron

### A Teacher Resource Package for the Primary Grades

A Teacher Resource Package for the Primary Grades Notes for Teachers: These are intended to be a starting point to help your class apply what they have learned in the Geometry and Spatial Sense strand

### Most classrooms are built in the shape of a rectangular prism. You will probably find yourself inside a polyhedron at school!

3 D OBJECTS Properties of 3 D Objects A 3 Dimensional object (3 D) is a solid object that has 3 dimensions, i.e. length, width and height. They take up space. For example, a box has three dimensions, i.e.

### Date: Period: Symmetry

Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into

### SOLIDS, NETS, AND CROSS SECTIONS

SOLIDS, NETS, AND CROSS SECTIONS Polyhedra In this section, we will examine various three-dimensional figures, known as solids. We begin with a discussion of polyhedra. Polyhedron A polyhedron is a three-dimensional

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### 13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

### MATHEMATICS Grade 6 Standard: Number, Number Sense and Operations

Standard: Number, Number Sense and Operations Number and Number C. Develop meaning for percents including percents greater than 1. Describe what it means to find a specific percent of a number, Systems

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

### Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) August-September (12 days) Unit #1 : Geometry

Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Common Core and Research from the CCSS Progression Documents Geometry Students learn to analyze and relate categories

### Space and Shape (Geometry)

Space and Shape (Geometry) General Curriculum Outcomes E: Students will demonstrate spatial sense and apply geometric concepts, properties, and Revised 2011 Outcomes KSCO: By the end of grade 9, students

### Junior Math Circles March 10, D Geometry II

1 University of Waterloo Faculty of Mathematics Junior Math Circles March 10, 2010 3D Geometry II Centre for Education in Mathematics and Computing Opening Problem Three tennis ball are packed in a cylinder.

### Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

### MATHEMATICS GRADE LEVEL VOCABULARY DRAWN FROM SBAC ITEM SPECIFICATIONS VERSION 1.1 JUNE 18, 2014

VERSION 1.1 JUNE 18, 2014 MATHEMATICS GRADE LEVEL VOCABULARY DRAWN FROM SBAC ITEM SPECIFICATIONS PRESENTED BY: WASHINGTON STATE REGIONAL MATH COORDINATORS Smarter Balanced Vocabulary - From SBAC test/item

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

### Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg

Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg Objectives My unit for the Delaware Teacher s Institute is a 10 th grade mathematics unit that focuses on teaching students

### Pre-Algebra - MA1100. Topic Lesson Objectives. Variables and Expressions

Variables and Expressions Problem Solving: Using a Problem-Solving Plan Use a four-step plan to solve problems. Choose an appropriate method of computation. Numbers and Expressions Use the order of operations

### Students will understand 1. use numerical bases and the laws of exponents

Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan

### Bedford Public Schools

Bedford Public Schools Grade 7 Math In 7 th grade, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding

### Alabama Course of Study Mathematics Geometry

A Correlation of Prentice Hall to the Alabama Course of Study Mathematics Prentice Hall, Correlated to the Alabama Course of Study Mathematics - GEOMETRY CONGRUENCE Experiment with transformations in the

### Glossary. angles, adjacent Two angles with a common side that do not otherwise overlap. In the diagram, angles 1 and 2

Glossary 998 Everyday Learning Corporation absolute value The absolute value of a positive number is the number itself. For example, the absolute value of 3 is 3. The absolute value of a negative number

### 12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. SOLUTION: ANSWER: 1017.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 3. sphere: area of great circle = 36π yd 2 We know that the area of a great circle is r.. Find 1. Now find the surface area.

### 56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

### Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

### Grade 7 Mathematics, Quarter 4, Unit 4.1. Probability. Overview

Grade 7 Mathematics, Quarter 4, Unit 4.1 Probability Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Understand how to use counting techniques to solve problems involving

### ( ) 2 = 9x 2 +12x + 4 or 8x 2 " y 2 +12x + 4 = 0; (b) Solution: (a) x 2 + y 2 = 3x + 2 " \$ x 2 + y 2 = 1 2

Conic Sections (Conics) Conic sections are the curves formed when a plane intersects the surface of a right cylindrical doule cone. An example of a doule cone is the 3-dimensional graph of the equation

Journal for Mathematics and the Arts, Vol. X, No. X, Month 2007, xxx-xxx Three-dimensional finite point groups and the symmetry of beaded beads G. L. FISHER* AND B. MELLOR California Polytechnic State

### Every 3-dimensional shape has three measurements to describe it: height, length and width.

Every 3-dimensional shape has three measurements to describe it: height, length and width. Height Width Length Faces A face is one of the flat sides of a three-dimensional shape. Faces A cuboid has 6 flat

### Objectives/Outcomes. Materials and Resources. Grade Level: Title/Description of Lesson. Making a 3-dimensional sculpture using a 2dimensional patterns

V Viissuuaall & &P Peerrffoorrm miinngg A Arrttss P Prrooggrraam m,, S SJJU US SD D A Arrttss & &M Maatthh C Coonnnneeccttiioonnss Title/Description of Lesson Making a 3-dimensional sculpture using a 2dimensional

### Appendix A. Comparison. Number Concepts and Operations. Math knowledge learned not matched by chess

Appendix A Comparison Number Concepts and Operations s s K to 1 s 2 to 3 Recognize, describe, and use numbers from 0 to 100 in a variety of familiar settings. Demonstrate and use a variety of methods to

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples:

Solid Geometry Solid Geometry is the geometry of three-dimensional space, the kind of space we live in. Three Dimensions It is called three-dimensional or 3D because there are three dimensions: width,

### FCAT Math Vocabulary

FCAT Math Vocabulary The terms defined in this glossary pertain to the Sunshine State Standards in mathematics for grades 3 5 and the content assessed on FCAT in mathematics. acute angle an angle that

### 2.9 3-D Solids and Nets

2.9 3-D Solids and Nets You need to decide whether or not to use the words shape (2-d) and solid (3-d) interchangeably. It can be helpful to avoid saying shape when referring to a 3-d object. Pupils will

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### Topological Treatment of Platonic, Archimedean, and Related Polyhedra

Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 1534-1178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic

### 8 th Grade Pre-Algebra Textbook Alignment Revised 4/28/2010

Revised 4/28/2010 September Lesson Number & Name GLCE Code & Skill #1 N.MR.08.10 Calculate weighted averages such as course grades, consumer price indices, and sports ratings. #2 D.AN.08.01 Determine which

### Symmetry Operations and Elements

Symmetry Operations and Elements The goal for this section of the course is to understand how symmetry arguments can be applied to solve physical problems of chemical interest. To achieve this goal we

COURSE OVERVIEW The geometry course is centered on the beliefs that The ability to construct a valid argument is the basis of logical communication, in both mathematics and the real-world. There is a need

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Michigan Grade Level Content Expectations for Mathematics, Grade 8, Correlated to Glencoe s Michigan Pre-Algebra

Eighth Grade Mathematics Grade Level Content Expectations Michigan Pre-Algebra 2010 Michigan Grade Level Content Expectations for Mathematics, Grade 8, Correlated to Glencoe s Michigan Pre-Algebra Lessons

### Year 1 Maths Expectations

Times Tables I can count in 2 s, 5 s and 10 s from zero. Year 1 Maths Expectations Addition I know my number facts to 20. I can add in tens and ones using a structured number line. Subtraction I know all

### 7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

### TERMINOLOGY Area: the two dimensional space inside the boundary of a flat object. It is measured in square units.

SESSION 14: MEASUREMENT KEY CONCEPTS: Surface Area of right prisms, cylinders, spheres, right pyramids and right cones Volume of right prisms, cylinders, spheres, right pyramids and right cones the effect

### Surface Area and Volume

UNIT 7 Surface Area and Volume Managers of companies that produce food products must decide how to package their goods, which is not as simple as you might think. Many factors play into the decision of

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (2005), Chap. 3, and Atkins and Friedman, Chap. 5.

Chapter 5. Geometrical ymmetry Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (005), Chap., and Atkins and Friedman, Chap. 5. 5.1 ymmetry Operations We have already

### II. Geometry and Measurement

II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics

### Constructing Geometric Solids

Constructing Geometric Solids www.everydaymathonline.com Objectives To provide practice identifying geometric solids given their properties; and to guide the construction of polyhedrons. epresentations