Abridged Digital Book
|
|
- Anthony Newman
- 5 years ago
- Views:
Transcription
1 SAMPLE
2 SAMPLE Abridged Digital Book First Edition - June 2016 Copyright D3dPuzzles - A Division of LightBe Corp All rights reserved by Bernard F. Dreyer & Pamela Cook Dreyer i
3 Table of Contents Welcome to the 4d3dPuzzles Digital Book Digital Book Navigation Dedication Preface Vision - Mission - Objectives - Commitment Introduction Chapter 1: Geometry - Spatial Dimensions Chapter 2: Moving Between 1D, 2D, 3D & 4D Spaces Chapter 3: Unfolding Geometry Objects Chapter 4: Idea & Concept of the 3d-Puzzle Chapter 5: 4D3dPuzzles EcoSystem Chapter 6: 3d-Puzzles Chapter 7: 3d-Puzzle - Limited Edition Appendices: 1: Euclidian Geometry & Non-Euclidian Geometry 2: Geometry Elements & Objects 3: Spacetime 4: Tesseract - Hypercube - 8 Cell 5: Geometry of the 3d-Puzzle 6: Higher Spacial Dimensions Chapter 8: 4D3dPuzzles Game Apps Chapter 9: 3d-Puzzle - Solid Chapter 10: tesserart: Sculptures & Jewelry Chapter 11: 4D3dPuzzles Digital Media Chapter 12: Target Audiences & Benefits Chapter 13: The 4D3dPuzzles Team Chapter 14: Conclusion Links and Hyperlinks are in Dark Blue Bold in this Sample Abridged Digital Book ii
4 Preface Confusion with 4th Dimension and Space, Time and Spacetime lead us to study to better understand the 3 and 4 Dimensional Spaces. After more studies of basic Geometry & Mathematics, we realized that a 4-Dimension Cube called HyperCube can be unfolded or flattened in our day-to-day 3-Dimensional Space as a 3D Cross. Since the HyperCube is bounded by eight 3- Dimensional Cubes, we invented and designed the concept of a Puzzle made of a group of eight cubes articulated by hinges and arranged in a 2 x 2 x 2 fashion that can be individually rotated in our 3-Dimension Space into a 3-Dimension Cross: We then developed proofs of concept and a number of prototypes of the 3d-Puzzle and related products of the iii
5 Introduction Introduction Introduction Scope & Theme of 4D3dPuzzles The Challenges It is NOT Complicated
6 Introduction Section 1 Scope & Theme of 4D3dPuzzles As eluded in the Preface of this Digital Book we were very curious about Spacial Dimensions greater than 3 and how geometric objects could be unfolded from a Dimensional Space to a lower Dimensional Space - for example from our familiar 3-Dimensional Space to the 2-Dimensional Flat Space; or from a 4-Dimensional Hyper Space to our day-to-day 3-Dimensional Space. This Digital Book reveals in detail what we have discovered regarding spacial geometry and how we applied its concepts to the 3d-Puzzle we invented, designed, developed and produced. The 3dPuzzles are a physical and/or virtual representation of unfolding between Spacial Dimensions. This Digital Book also explores the potential benefits of playing with the 3d-Puzzles, either in their solid format or virtual format on Mobile 4D3d Puzzles Apps. This 4D3dPuzzle Digital Book is bold, wide ranging, provocative and very engaging. It is for you whether you are young, older, have little science education or a lot. You will not want to put it down! With the assistance of Multimedia Content you will get a clear picture and understanding of what is essential to really understand Spatial Dimensions and the 3d-Puzzles. 8-Minutes YouTube Video: 4D3dPuzzles OVERVIEW If you find this topic intriguing, keep reading... 2
7 The Word Cloud below illustrates the main Products, Apps, Services, Benefits and Target Audiences of 4D3dPuzzles. 3
8 Scope & Theme of 4D3dPuzzles Are you curious about the 3-Dimension Space and the 4-Dimension Space? If this Digital Book answers some of your questions, you will be able to add and enter an important new DIMENSION in your life. This Digital Book will definitively challenge and inspire You! Are you ready to learn more? It is now up to you! 4
9 Introduction Section 2 It Is NOT Complicated This Digital Book is an easy to understand introduction to the concepts of Spatial Dimensions, the 3-Dimension Space and the 4-Dimension Space, and how these dimensional spaces relate. These concepts illustrate the idea of the 3d-Puzzles and provide the foundation of the design of such puzzles. The principles are explained in such a way that people with very little knowledge of Mathematics or Geometry will be able to understand them. For those of you who are more adventurous, the Appendices illustrate and explain in much greater details the following: Appendices: 1: Euclidian Geometry & Non-Euclidian Geometry 2: Geometry Elements & Objects 3: Spacetime 4: Tesseract - Hypercube - 8 Cell 5: Geometry of the 3d-Puzzle 6: Higher Spacial Dimensions 5
10 Basics of Geometry & Dimensions Basics of Geometry Basics of Spatial Dimension Relationships Basics of 3-Dimensional Space Basics of 4-Dimensional Space
11 Basics of Geometry & Dimensions Section 1 Basics of Spatial Dimension Relationships The Dimensions of basic Dimensional Spaces are: 0-Dimension: a Point. 1-Dimension Space: a Line (a Line is made of Points). 2-Dimension Space: a Surface (a Surface is made of Lines). 3-Dimensions Space: a Volume (a Volume is made of Surfaces). Now let s add the 4-Dimension Space: a Tesseract or HyperCube or Polychoron (a Polychoron is made of Volumes whereas a Polyhedron is made of Surfaces) 7
12 Moving Between Dimensional Spaces The Space Landers Lineland & Linelanders Flatland and Flatlanders Earthland and Earthlanders Short Video: Moving Between Dimensional Spaces
13 Moving Between Dimensional Spaces Section 1 The Space Landers In this Chapter you have to really use your imagination. The concepts are however simple and easy to understand. 9
14 10
15 Unfolding Geometry Objects Unfolding 3D Space to 2D Space Unfolding 4D Space to 3D Space HyperCube in Painting
16 Unfolding Geometry Objects Section 1 Unfolding 4D Space to 3D Space The 4-Dimensional HyperCube can be unfolded in our 3-Dimensional Space or world or universe into Eight Cubes making a 3D-Cross, just as a 3D-Cube can be unfolded into a six square Cross in a 2-Dimensional Space. 12
17 Idea & Concept of the 3d-Puzzles Idea behind the 3d-Puzzles Concept of the 3d-Puzzles
18 Idea & Concept of the 3d-Puzzles Section 1 Concept of the 3d-Puzzles Since the IDEA is to represent in the 3D Space the unfolding of the 4-Dimensional HyperCube made of 8-Cells, the 3d-Puzzle consists of eight cubical cells. The 3d-Puzzle initial configuration is represented by a group of Eight Cubes of same size arranged in a 2 x 2 x 2 fashion as a Cubic Honeycomb. 14
19 3d-Puzzles 8-Cube 3d-Puzzle 16-Cube 3d-Puzzle 12-Cube 3d-Puzzle
20 3d-Puzzles Section 1 8-Cube 3d-Puzzle 16
21 3d-Puzzle - Limited Edition
22 3d-Puzzle - Limited Edition The 3d-Puzzle - Limited Edition is made by 3D Printing in one piece in the Factory of the Future. It is a Collection Item that can be purchased on the Online Store It is available in multiple colors. Since the 3d-Puzzle is made in one piece including the articulations between Cubes, the 3D Printing process used is Selective Laser Sintering (SLS). 3d-Puzzle Solution 18
23 4D3d Puzzles Apps
24 4D3d Puzzles Apps Mobile Applications, also called Mobile Apps or simply Apps, are software applications, designed to run on Smartphones and Tablets. Interactions with the App are performed on the touch screen by gestures such a Touch, Swipe, Tilt. Web Applications, also called Web Apps are software applications designed to run on Computer Browsers or on Mobile Browsers. Interactions with the App are performed on Computers by moves and clicks of the Mouse or by moves and clicks on a trackpad or by touch of the screen; and by touch and gestures on Mobiles. Solve the 4D3d-Puzzles on the Screen of your Mobile Solving a 4D3d Puzzle of the Game App Rotate on your Mobile or Computer Screen 3-Dimensional Elements or a Set of Elements of any 4D3d Puzzle to solve it. or your Computer. Categories of the 4D3d Puzzles Game App: The theme is Sci-Fi The 4D3d Puzzles App is available for multiple platforms on the website. 20
25 4D3d Puzzles Apps The 4D3d Puzzles are modeled in 3 Dimensions by software and are rendered on the two dimensional screens/displays of Mobile Devices or Computers. Touch an Image below to start a short YouTube Video Trailer. Each Trailer provides a short overview of a Puzzle Game. NOTE: If you play a Video, to return to the Digital Book where you have left off, tap (or click) Back to ibooks on the top left corner of the Video Screen on a Mobile or close the window on a Mac. The last page of this section provide some general information about the Benefits of the Puzzle Games. 21
26 tesserart: Sculptures & Jewelry
27 Target Audiences & Benefits Target Audience: 3d-Puzzles & 4D3d Apps Benefits of the 3d-Puzzles & 4D3d Puzzles Apps
28 Target Audiences & Benefits Section 1 Benefits of the 3d-Puzzles & 4D3d Puzzles Apps The 3d-Puzzle can be classified as a Sequential Movement Puzzle. Puzzles in this category require a repeated manipulation of the puzzle elements to get the puzzle to a certain target condition or solution. Potential Benefits are specific to the 3d-Puzzles and the 4D3d Puzzles Apps. They apply to a number of the conditions as follows for the targets identified in the previous Section: Conditions: Sensory integration and processing Attention Developmental delays Autism Aspergers Neurological impairment Gross Motor and Fine Motor Visual Perception Visual Motor Integration Coordination/balance/strength 24
29 Appendices Appendix 1: Euclidian & Non-Euclidian Geometry Appendix 2: Geometry Elements & Objects Appendix 3: Spacetime Appendix 4: Tesseract - Hypercube - 8 Cell Appendix 5: Geometry of the 3d-Puzzle Appendix 6: Higher Spacial Dimensions
30 Appendix 2 Geometry Elements & Objects Geometric Object Geometry as a branch of Mathematics considers Objects such as Points, Lines, Triangles, Circles, Hexagons, Spheres, Polyhedra, Topological Spaces and Manifolds to name a few. Geometric Shapes A Geometric Shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a Geometric Object. Moving a Geometric Shape around, enlarging it, rotating it, or reflecting it in a mirror is the same Shape as the original, and not a distinct new Shape. Dimensions In Physics and Mathematics, the Dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a Dimension of 1 because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a Dimension of 2 because two coordinates are needed to specify a point on it. The inside of a cube, a cylinder or a sphere is 3-Dimensional because three coordinates are needed to locate a point within these spaces. 26
31 Geometry Elements & Objects Time A temporal Dimension is a Dimension of Time. Time is often referred to as the "4 th dimension" for this reason, but that is NOT to imply that it is a Spatial Dimension. A temporal Dimension is one way to measure physical change. It is perceived differently from the 3-Dimensional Space in that there is only one of it, and that we cannot move freely in Time but subjectively move in one direction. The best-known example of Time as a Dimension is Einstein's Special Relativity (and extended to General Relativity), which treats perceived Space and Time as components of a 4-Dimensional manifold, known as Spacetime: Science fiction texts often mention the concept of "Dimension" when referring to parallel or alternate universes or other imagined planes of existence. This is derived from the idea that to travel to parallel/alternate universes/ 27
32 Appendix 4 Tesseract - Hypercube - 8 Cell In the field of Mathematics, in Euclidean Geometry, Hypercubes in a 4-Dimensional Space are called Tesseracts. They are also called 3D-Hypercubes or 8-Cell or Octachoron or Polychoron. Projection in 3D Space of a rotating Tesseract The Spaces of the Euclidian Geometry are characterized by: In the 0-Dimension Space, a Point, is contained in, and as a result controlled, by the 1-Dimension Space, a Line. In the 1-Dimension Space, a Segment or Line, is contained in, and as a result controlled, by the 2-Dimension Space, a Polygon (a Surface). In the 2-Dimension Space, a Polygon, is contained in, and as a result controlled, by the 3-Dimension Space, a Polyhedron (a Volume). In the 3-Dimension Space, a Polyhedron, is contained in, and as a result controlled, by the 4-Dimension Space, a Polychoron (a Tesseract). In the 4-Dimension Space, a Polychoron (also called Tesseract, Hypercube, 8-cell, Regular Octachoron, Cubic Prism, and Tetracube) is contained in, and as a result controlled, by the 5-Dimension Space, a Hexadecachoron. In the higher Dimensional Spaces (5, 6, , 11, 12, etc.), it become extremely complicated. 28
33 Tesseract - Hypercube - 8 Cell Of special interest for this Digital Book is the 4D Hypercube or Tesseract According to the Oxford English Dictionary, the word Tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ακτίνες (téssereis aktines or "four rays"), referring to the four lines from each Vertex to other Vertices. The HyperCube or Tesseract is to the Cube as the Cube is to the Square; or, more formally, the Tesseract can be described as a regular convex 4-Polytope. The Tesseract is the Hypercube in, also called the 8-Cell or Octachoron. It is a regular Polytope with mutually perpendicular sides, and is therefore an Orthotope. The figure below shows how the Tesseract is made of 8 Cells (Cubes). 29
34 Appendix 5 Geometry of the 3d-Puzzle To understand the geometry of the 3d-Puzzle and the 4D3d Puzzles Apps the Geometry Nets have to be examined and reviewed. What are a Geometry Nets? In geometry the Net of a Polyhedron (a polyhedron is a solid in 3-Dimensions with flat polygonal faces, straight edges and sharp corners or vertices) is an arrangement of Edge-joined Polygons in the plane which can be folded (along Edges) to become the Faces of the Polyhedron. Cubes and Pyramids are examples of Polyhedron. In other words a Net is a Pattern that you can be cut and folded to make a model of a solid shape. A Net is a 2-Dimensional representation of a 3-Dimensional object (such as a Cube). A Net is also a 3-Dimensional representation of a 4-Dimensional object (such as a Tesseract or Hypercube). Polyhedral Nets are a useful aid to the study of Polyhedron and solid geometry in general, as they allow for physical models of polyhedron to be constructed from material such as thin cardboard or other material. Many different Nets can exist for a given polyhedron, depending on the choices of which Edges are joined and which are separated. There are distinct 11 distinct Nets of a Cube and 261 distinct Nets for a Tesseract. They will be studied in the following pages. 30
35 Geometry of the 3d-Puzzle Net of a Cube The Cube has 11 different Nets. The Net making a 2-Dimensional Cross (in red on the illustration below) is of special interest. 31
Chapter 1 Introduction
Chapter 1 Introduction 1.1 Historical Overview In the ancient Greek, Euclid said that "a point has no dimension at all. A line has only one dimension: length. A plane has two dimensions: length and breadth.
More informationFebruary 07, Dimensional Geometry Notebook.notebook. Glossary & Standards. Prisms and Cylinders. Return to Table of Contents
Prisms and Cylinders Glossary & Standards Return to Table of Contents 1 Polyhedrons 3-Dimensional Solids A 3-D figure whose faces are all polygons Sort the figures into the appropriate side. 2. Sides are
More informationMathematics Assessment Anchor Glossary Grades 3 & 4
Mathematics Assessment Anchor Glossary Grades 3 & 4 The definitions for this glossary were taken from one or more of the following sources: Webster s Dictionary, various mathematics dictionaries, the PA
More informationRoswell Independent School District Grade Level Targets Summer 2010
1 NM Standards Children s Progress Core Standards Target: Possesses a working knowledge of the base ten number system, including ones and tens. Q1 Counts, sketches and represents some numbers. Q2 Counts,
More informationGeometry Vocabulary. acute angle-an angle measuring less than 90 degrees
Geometry Vocabulary acute angle-an angle measuring less than 90 degrees angle-the turn or bend between two intersecting lines, line segments, rays, or planes angle bisector-an angle bisector is a ray that
More informationMATHEMATICS. Y4 Understanding shape Visualise, describe and classify 3-D and 2-D shapes. Equipment
MATHEMATICS Y4 Understanding shape 4501 Visualise, describe and classify 3-D and 2-D shapes Paper, pencil, ruler Equipment Maths Go Go Go 4501 Visualise, describe and classify 3-D and 2-D shapes. Page
More informationAnswer Key: Three-Dimensional Cross Sections
Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection
More informationPolyhedron. A polyhedron is simply a three-dimensional solid which consists of a collection of polygons, joined at their edges.
Polyhedron A polyhedron is simply a three-dimensional solid which consists of a collection of polygons, joined at their edges. A polyhedron is said to be regular if its faces and vertex figures are regular
More informationacute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6
acute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6 angle An angle is formed by two rays with a common end point. Houghton Mifflin Co. 3 Grade 5 Unit
More informationA Physical Proof for Five and Only Five Regular Solids
A Physical Proof for Five and Only Five Regular Solids Robert McDermott Center for High Performance Computing University of Utah Salt Lake City, Utah, 84112, USA E-mail: mcdermott@chpc.utah.edu Abstract
More informationLesson Polygons
Lesson 4.1 - Polygons Obj.: classify polygons by their sides. classify quadrilaterals by their attributes. find the sum of the angle measures in a polygon. Decagon - A polygon with ten sides. Dodecagon
More informationD A S O D A. Identifying and Classifying 3-D Objects. Examples
Identifying Classifying 3-D Objects Examples Have you noticed that many of the products we purchase come in packages or boxes? Take a look at the products below. A) Did you notice that all the sides or
More informationMeasurement 1 PYTHAGOREAN THEOREM. The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of
Measurement 1 PYTHAGOREAN THEOREM Remember the Pythagorean Theorem: The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the other two sides.
More informationVisualising Solid Shapes
VISUALISING SOLID SHAPES 2 7 7 Visualising Solid Shapes Chapter 15 15.1 INTRODUCTION: PLANE FIGURES AND SOLID SHAPES In this chapter, you will classify figures you have seen in terms of what is known as
More informationNets and Drawings for Visualizing Geometry. Unit 1 Lesson 1
Nets and Drawings for Visualizing Geometry Unit 1 Lesson 1 Students will be able to: Represent three-dimensional figures using nets. Make isometric and orthographic drawings. Key Vocabulary: Net Isometric
More informationThe radius for a regular polygon is the same as the radius of the circumscribed circle.
Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
More informationSHAPE AND STRUCTURE. Shape and Structure. An explanation of Mathematical terminology
Shape and Structure An explanation of Mathematical terminology 2005 1 POINT A dot Dots join to make lines LINE A line is 1 dimensional (length) A line is a series of points touching each other and extending
More informationMath 311. Polyhedra Name: A Candel CSUN Math
1. A polygon may be described as a finite region of the plane enclosed by a finite number of segments, arranged in such a way that (a) exactly two segments meets at every vertex, and (b) it is possible
More informationExample: The following is an example of a polyhedron. Fill the blanks with the appropriate answer. Vertices:
11.1: Space Figures and Cross Sections Polyhedron: solid that is bounded by polygons Faces: polygons that enclose a polyhedron Edge: line segment that faces meet and form Vertex: point or corner where
More informationGeometry 10 and 11 Notes
Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into
More informationMath 366 Lecture Notes Section 11.4 Geometry in Three Dimensions
Math 366 Lecture Notes Section 11.4 Geometry in Three Dimensions Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. A sphere is the set of all points at a
More informationExplore Solids
1212.1 Explore Solids Surface Area and Volume of Solids 12.2 Surface Area of Prisms and Cylinders 12.3 Surface Area of Pyramids and Cones 12.4 Volume of Prisms and Cylinders 12.5 Volume of Pyramids and
More informationMathematics Curriculum
6 G R A D E Mathematics Curriculum GRADE 6 5 Table of Contents 1... 1 Topic A: Area of Triangles, Quadrilaterals, and Polygons (6.G.A.1)... 11 Lesson 1: The Area of Parallelograms Through Rectangle Facts...
More informationStandard 2.0 Knowledge of Geometry: Students will apply the properties of one-,
VSC - Mathematics Print pages on legal paper, landscape mode. Grade PK Grade K Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Geometry: Students will apply the properties of one-, two-,
More informationCurriculum Correlation Geometry Cluster 3: Geometric Relationships
ON Master 19a 20.3 compose pictures, designs, shapes, and patterns, using two-dimensional shapes; predict and explore reflective symmetry in two-dimensional shapes (e.g., visualize and predict what will
More information11.4 Three-Dimensional Figures
11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called
More informationCourse Number: Course Title: Geometry
Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line
More informationMathematics Scope & Sequence Geometry
Mathematics Scope & Sequence Geometry Readiness Standard(s) First Six Weeks (29 ) Coordinate Geometry G.7.B use slopes and equations of lines to investigate geometric relationships, including parallel
More information6 Mathematics Curriculum
New York State Common Core 6 Mathematics Curriculum GRADE GRADE 6 MODULE 5 Table of Contents 1 Area, Surface Area, and Volume Problems... 3 Topic A: Area of Triangles, Quadrilaterals, and Polygons (6.G.A.1)...
More informationMultiply using the grid method.
Multiply using the grid method. Learning Objective Read and plot coordinates in all quadrants DEFINITION Grid A pattern of horizontal and vertical lines, usually forming squares. DEFINITION Coordinate
More informationHow to print a Hypercube
How to print a Hypercube Henry Segerman One of the things that mathematics is about, perhaps the thing that mathematics is about, is trying to make things easier to understand. John von Neumann once said
More informationZipper Unfoldings of Polyhedral Complexes
Zipper Unfoldings of Polyhedral Complexes Erik D. Demaine Martin L. Demaine Anna Lubiw Arlo Shallit Jonah L. Shallit Abstract We explore which polyhedra and polyhedral complexes can be formed by folding
More informationHANDS ON ENQUIRY IN TEACHING OF TETRAHEDRON
HANDS ON ENQUIRY IN TEACHING OF TETRAHEDRON Sibawu Witness Siyepu Cape Peninsula University of Technology siyepus@cput.ac.za INTRODUCTION The National Curriculum Statement advocates that learners should
More informationKey Concept Euler s Formula
11-1 Space Figures and Cross Sections Objectives To recognize polyhedrons and their parts To visualize cross sections of space figures Common Core State Standards G-GMD.B.4 Identify the shapes of two-dimensional
More informationLesson 9. Three-Dimensional Geometry
Lesson 9 Three-Dimensional Geometry 1 Planes A plane is a flat surface (think tabletop) that extends forever in all directions. It is a two-dimensional figure. Three non-collinear points determine a plane.
More informationCTI, November 19, 2015
Consider a large cube made from unit cubes 1 Suppose our cube is n n n Look at the cube from a corner so that you can see three faces How many unit cubes are in your line of vision? Build a table that
More informationThe Game of Criss-Cross
Chapter 5 The Game of Criss-Cross Euler Characteristic ( ) Overview. The regions on a map and the faces of a cube both illustrate a very natural sort of situation: they are each examples of regions that
More informationGrade 6 Math Circles. Spatial and Visual Thinking
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles October 31/November 1, 2017 Spatial and Visual Thinking Centre for Education in Mathematics and Computing One very important
More informationToday we will be exploring three-dimensional objects, those that possess length, width, and depth.
Lesson 22 Lesson 22, page 1 of 13 Glencoe Geometry Chapter 11.1 3-D figures & Polyhedra Today we will be exploring three-dimensional objects, those that possess length, width, and depth. In Euclidean,
More informationMath 6: Geometry 3-Dimensional Figures
Math 6: Geometry 3-Dimensional Figures Three-Dimensional Figures A solid is a three-dimensional figure that occupies a part of space. The polygons that form the sides of a solid are called a faces. Where
More informationLesson 22: Surface Area
Student Outcomes Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals, specifically focusing on pyramids. They use polyhedron nets
More informationGeometry Workbook WALCH PUBLISHING
Geometry Workbook WALCH PUBLISHING Table of Contents To the Student..............................vii Unit 1: Lines and Triangles Activity 1 Dimensions............................. 1 Activity 2 Parallel
More informationUnit 4 Reasoning about shape. Year 4. Five daily lessons. Autumn term. Unit Objectives. Link Objectives
Unit 4 Reasoning about shape Five daily lessons Year 4 Autumn term (Key objectives in bold) Unit Objectives Year 4 Describe and visualise 3-D and 2-D shapes, Page 102 including the tetrahedron and heptagon.
More informationVocabulary. Term Page Definition Clarifying Example. cone. cube. cylinder. edge of a threedimensional. figure. face of a polyhedron.
CHAPTER 10 Vocabulary The table contains important vocabulary terms from Chapter 10. As you work through the chapter, fill in the page number, definition, and a clarifying example. cone Term Page Definition
More information25. How would you make the octahedral die shown below?
304450_ch_08_enqxd 12/6/06 1:39 PM Page 577 Chapter Summary 577 draw others you will not necessarily need all of them. Describe your method, other than random trial and error. How confident are you that
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 24 Solid Modelling Welcome to the lectures on computer graphics. We have
More informationGeometry. Professor Harms Minnesota State University Moorhead Feb. 24 th, 2014
Geometry Professor Harms Minnesota State University Moorhead Feb. 24 th, 2014 Engaging those Calvins in your class The van Hiele model of thinking in Geometry The student recognizes, names, compares and
More informationClass 4 Geometry. Answer the questions. For more such worksheets visit (1) The given figure has line segments.
ID : in-4-geometry [1] Class 4 Geometry For more such worksheets visit www.edugain.com Answer the questions (1) The given figure has line segments. (2) How many curved lines can be found in the given figure?
More informationa 3-dimensional solid with a circular base and a curved surface that meets at a point
q. Super Solids Whole Class or Small Group Geometric Vocabulary reproducible (2 per student) (pg. 20) Super Solids reproducible (pg. 24) Make photocopies of the Geometric Vocabulary (2 per student) and
More informationQuestion. Why is the third shape not convex?
1. CONVEX POLYGONS Definition. A shape D in the plane is convex if every line drawn between two points in D is entirely inside D. Convex 6 gon Another convex 6 gon Not convex Question. Why is the third
More informationInstructional Alignment Chart
CLUSTER HEADING: STANDARD: N/A CLUSTER HEADING: Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). STANDARD: K.G.3 Identify shapes as
More informationUnit 7: 3D Figures 10.1 & D formulas & Area of Regular Polygon
Unit 7: 3D Figures 10.1 & 10.2 2D formulas & Area of Regular Polygon NAME Name the polygon with the given number of sides: 3-sided: 4-sided: 5-sided: 6-sided: 7-sided: 8-sided: 9-sided: 10-sided: Find
More informationCaught in a Net. SETTING THE STAGE Examine and define faces of solids. LESSON OVERVIEW. Examine and define edges of solids.
Caught in a Net LESSON FOCUS Using informal geometric vocabulary to describe physical objects and geometric figures. Constructing mental and physical images of common geometric figures. Classifying geometric
More informationSOLIDS.
SOLIDS Prisms Among the numerous objects we see around us, some have a regular shape while many others do not have a regular shape. Take, for example, a brick and a stone. A brick has a regular shape while
More informationUnderstand the concept of volume M.TE Build solids with unit cubes and state their volumes.
Strand II: Geometry and Measurement Standard 1: Shape and Shape Relationships - Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes,
More informationNumber/Computation. addend Any number being added. digit Any one of the ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9
14 Number/Computation addend Any number being added algorithm A step-by-step method for computing array A picture that shows a number of items arranged in rows and columns to form a rectangle associative
More information5.1 Any Way You Slice It
SECONDARY MATH III // MODULE 5 MODELING WITH GEOMETRY 5.1 Students in Mrs. Denton s class were given cubes made of clay and asked to slice off a corner of the cube with a piece of dental floss. Jumal sliced
More informationPatterns on Triply Periodic Uniform Polyhedra
Patterns on Triply Periodic Uniform Polyhedra Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/
More informationMATH DICTIONARY. Number Sense. Number Families. Operations. Counting (Natural) Numbers The numbers we say when we count. Example: {0, 1, 2, 3, 4 }
Number Sense Number Families MATH DICTIONARY Counting (Natural) Numbers The numbers we say when we count Example: {1, 2, 3, 4 } Whole Numbers The counting numbers plus zero Example: {0, 1, 2, 3, 4 } Positive
More informationWhat is dimension? An investigation by Laura Escobar. Math Explorer s Club
What is dimension? An investigation by Laura Escobar Math Explorer s Club The goal of this activity is to introduce you to the notion of dimension. The movie Flatland is also a great way to learn about
More information1 Appendix to notes 2, on Hyperbolic geometry:
1230, notes 3 1 Appendix to notes 2, on Hyperbolic geometry: The axioms of hyperbolic geometry are axioms 1-4 of Euclid, plus an alternative to axiom 5: Axiom 5-h: Given a line l and a point p not on l,
More informationIdentifying and Classifying Angles and Shapes
Grade 5 Mathematics, Quarter 2, Unit 2.1 Identifying and Classifying Angles and Shapes Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Describe, compare, and classify
More information3D shapes introduction
3D shapes introduction 2D shapes have 2 dimensions width and height. They re flat. height 3D shapes have 3 dimensions height, width and depth. Sometimes we call them solids. When we draw them, we often
More informationEscher-type Tessellations and Pull-up Polyhedra: Creative Learning for the Classroom
Bridges 2010: Mathematics, Music, Art, Architecture, Culture Escher-type Tessellations and Pull-up Polyhedra: Creative Learning for the Classroom E.B. Meenan* and B.G. Thomas School of Education* and School
More informationMATHEMATICS. Y4 Understanding shape Visualise 3-D objects and make nets of common solids. Equipment
MATHEMATICS Y4 Understanding shape 4502 Visualise 3-D objects and make nets of common solids Equipment Paper, pencil, boxes, range of 3-D shapes, straws and pipe cleaners or 3-D model construction kits.
More informationCCM6+ Unit 12 Surface Area and Volume page 1 CCM6+ UNIT 12 Surface Area and Volume Name Teacher Kim Li
CCM6+ Unit 12 Surface Area and Volume page 1 CCM6+ UNIT 12 Surface Area and Volume Name Teacher Kim Li MAIN CONCEPTS Page(s) Unit 12 Vocabulary 2 3D Figures 3-8 Volume of Prisms 9-19 Surface Area 20-26
More informationUNIT 12. Volume and Surface Area CCM6+ Name: Math Teacher: Projected Test Date: Vocabulary 2. Basics of 3-D Figures 3 8
UNIT 12 Volume and Surface Area 2016 2017 CCM6+ Name: Math Teacher: Projected Test Date: Main Concept(s) Page(s) Vocabulary 2 Basics of 3-D Figures 3 8 Volume of Rectangular Prisms and Finding Missing
More informationGeometry Practice. 1. Angles located next to one another sharing a common side are called angles.
Geometry Practice Name 1. Angles located next to one another sharing a common side are called angles. 2. Planes that meet to form right angles are called planes. 3. Lines that cross are called lines. 4.
More informationSOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier
Mathematics Revision Guides Solid Shapes Page 1 of 15 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier SOLID SHAPES Version: 1. Date: 10-11-2015 Mathematics Revision Guides Solid
More informationGeometry Unit 10 Note Sheets Date Name of Lesson. 1.6 Two-Dimensional Figures Areas of Circles and Sectors
Date Name of Lesson 1.6 Two-Dimensional Figures 11.3 Areas of Circles and Sectors Quiz 11.1 Areas of Parallelograms and Triangles 11.2 Areas of Trapezoids, Rhombi and Kites 11.4 Areas of Regular Polygons
More informationEngage NY Lesson 15: Representing Three-Dimensional Figures Using Nets
Name: Surface Area & Volume Packet Engage NY Lesson 15: Representing Three-Dimensional Figures Using Nets Classwork Cereal Box Similarities: Cereal Box Differences: Exercise 1 1. Some of the drawings below
More informationInvestigations in Number, Data, and Space for the Common Core 2012
A Correlation of Investigations in Number, Data, and Space for the Common Core 2012 to the Common Core State s with California Additions s Map Kindergarten Mathematics Common Core State s with California
More informationGeometry Vocabulary Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and
More informationTHE PLATONIC SOLIDS BOOK DAN RADIN
THE PLATONIC SOLIDS BOOK DAN RADIN Copyright 2008 by Daniel R. Radin All rights reserved. Published by CreateSpace Publishing 3-D renderings were created on a thirteen-year-old Macintosh computer using
More informationAcademic Vocabulary CONTENT BUILDER FOR THE PLC MATH GRADE 1
Academic Vocabulary CONTENT BUILDER FOR THE PLC MATH GRADE 1 : academic vocabulary directly taken from the standard STANDARD 1.2(C) use objects, pictures, and expanded and standard forms to represent numbers
More informationseen something like it many times when playing video games.
Cakes and Pancakes Translating and Stacking Two-Dimensional Figures.2 Learning Goals In this lesson, you will: Apply translations to two-dimensional plane figures to create three-dimensional solids. Describe
More informationPOSITION, DIRECTION AND MOVEMENT Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Use mathematical
POSITION, DIRECTION AND MOVEMENT Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Use mathematical Use mathematical Describe positions on a Identify, describe and vocabulary to describe vocabulary to describe
More information3 Identify shapes as two-dimensional (lying in a plane, flat ) or three-dimensional ( solid ).
Geometry Kindergarten Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). 1 Describe objects in the environment using names of shapes,
More informationBracken County Schools Curriculum Guide Geometry
Geometry Unit 1: Lines and Angles (Ch. 1-3) Suggested Length: 6 weeks Core Content 1. What properties do lines and angles demonstrate in Geometry? 2. How do you write the equation of a line? 3. What affect
More informationDimension Theory: Road to the Fourth Dimension and Beyond
Dimension Theory: Road to the Fourth Dimension and Beyond 0-dimension Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own
More informationA Study of the Rigidity of Regular Polytopes
A Study of the Rigidity of Regular Polytopes A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Helene
More informationGeometry Surface Area and Volume of Pyramids and Cones.
Geometry 11.6 Surface Area and Volume of Pyramids and Cones mbhaub@mpsaz.org 11.6 Essential Question How do you find the surface area and volume of a pyramid or a cone? Geometry 1.3 Surface Area of Pyramids
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quantile Measure: Skill or Concept: 560Q Use manipulatives, pictorial representations, and appropriate vocabulary (e.g., face, edge, vertex, and base) to identify and compare properties
More informationRochester City School District Kindergarten Mathematics Performance Based Assessment RUBRIC. 12 Tasks for a total of 80 points
Rochester City School District Kindergarten Mathematics Performance Based RUBRIC Tasks for a total of 80 points Task Rubric Know number names and the count sequence. K.CC. Count to 00 by ones and tens.
More information3D shapes types and properties
3D shapes types and properties 1 How do 3D shapes differ from 2D shapes? Imagine you re giving an explana on to a younger child. What would you say and/or draw? Remember the surfaces of a 3D shape are
More informationRight Angle Triangle. Square. Opposite sides are parallel
Triangles 3 sides ngles add up to 18⁰ Right ngle Triangle Equilateral Triangle ll sides are the same length ll angles are 6⁰ Scalene Triangle ll sides are different lengths ll angles are different Isosceles
More informationUNIT 3 CIRCLES AND VOLUME Lesson 5: Explaining and Applying Area and Volume Formulas Instruction
Prerequisite Skills This lesson requires the use of the following skills: understanding and using formulas for the volume of prisms, cylinders, pyramids, and cones understanding and applying the formula
More informationCSG obj. oper3. obj1 obj2 obj3. obj5. obj4
Solid Modeling Solid: Boundary + Interior Volume occupied by geometry Solid representation schemes Constructive Solid Geometry (CSG) Boundary representations (B-reps) Space-partition representations Operations
More information1 The Platonic Solids
1 The We take the celebration of Dodecahedron Day as an opportunity embark on a discussion of perhaps the best-known and most celebrated of all polyhedra the Platonic solids. Before doing so, however,
More informationAnoka Hennepin K-12 Curriculum plan
Anoka Hennepin K-12 Curriculum plan Department: Elementary Math Unit Title: Packages and Polygons (Blue Book, Geo and Measurement) Triangles and Beyond (Blue Book, Geo and Measurement) Everyday Math: Volume
More informationThree-Dimensional Figures
Three-Dimensional Figures The number of coins created by the U.S. Mint changes each year. In the year 2000, there were about 28 billion coins created and about half of them were pennies!.1 Whirlygigs for
More informationPractice A Introduction to Three-Dimensional Figures
Name Date Class Identify the base of each prism or pyramid. Then choose the name of the prism or pyramid from the box. rectangular prism square pyramid triangular prism pentagonal prism square prism triangular
More informationCurvature Berkeley Math Circle January 08, 2013
Curvature Berkeley Math Circle January 08, 2013 Linda Green linda@marinmathcircle.org Parts of this handout are taken from Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill
More informationUnit 1, Lesson 1: Tiling the Plane
Unit 1, Lesson 1: Tiling the Plane Let s look at tiling patterns and think about area. 1.1: Which One Doesn t Belong: Tilings Which pattern doesn t belong? 1 1.2: More Red, Green, or Blue? m.openup.org//6-1-1-2
More informationScott Foresman Investigations in Number, Data, and Space Content Scope & Sequence Correlated to Academic Language Notebooks The Language of Math
Scott Foresman Investigations in Number, Data, and Space Content Scope & Sequence Correlated to Academic Language Notebooks The Language of Math Grade 5 Content Scope & Sequence Unit 1: Number Puzzles
More informationPrinciples and Standards for School Mathematics. Content Standards. Process Standards. Emphasis across the Grades. Principles
1 Navigating through Geometry Grades 3-5 Principles and Standards for School Mathematics Presented by Dr. Karol L. Yeatts Navigations Writer Navigating through Algebra Grades 3-5 Navigating through Number
More informationPRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES
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,
More informationStudents construct nets of three dimensional objects using the measurements of a solid s edges.
Student Outcomes Students construct nets of three dimensional objects using the measurements of a solid s edges. Lesson Notes In the previous lesson, a cereal box was cut down to one of its nets. On the
More informationThe figures below are all prisms. The bases of these prisms are shaded, and the height (altitude) of each prism marked by a dashed line:
Prisms Most of the solids you ll see on the Math IIC test are prisms or variations on prisms. A prism is defined as a geometric solid with two congruent bases that lie in parallel planes. You can create
More informationREGULAR TILINGS. Hints: There are only three regular tilings.
REGULAR TILINGS Description: A regular tiling is a tiling of the plane consisting of multiple copies of a single regular polygon, meeting edge to edge. How many can you construct? Comments: While these
More information