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First Edition - June 2016

Copyright © 2016 - 4D3dPuzzles - A Division of LightBe CorpAll rights reserved

Abridged Digital Book

by

Bernard F. Dreyer & Pamela Cook Dreyer

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Table of ContentsWelcome 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

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

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

Links and Hyperlinks are in Dark Blue Bold in this Sample Abridged Digital Book

Preface

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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

Introduction

IntroductionIntroduction

Scope & Theme of 4D3dPuzzlesThe ChallengesIt is NOT Complicated

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

Section 1Introduction

Scope & Theme of 4D3dPuzzles

The Word Cloud below illustrates the main Products, Apps, Services,Benefits and Target Audiences of 4D3dPuzzles.

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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

Scope & Theme of 4D3dPuzzles

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:

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Section 2Introduction

It Is NOT Complicated

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

Basics of GeometryBasics of Spatial Dimension RelationshipsBasics of 3-Dimensional SpaceBasics of 4-Dimensional Space

Basics of Geometry & Dimensions

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 HyperCubeor Polychoron (a Polychoron is made of Volumes whereas a Polyhedron is made of Surfaces)

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Basics of Spatial Dimension RelationshipsSection 1Basics of Geometry & Dimensions

The Space LandersLineland & LinelandersFlatland and FlatlandersEarthland and EarthlandersShort Video: Moving Between Dimensional Spaces

Moving Between Dimensional Spaces

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The Space LandersSection 1Moving Between Dimensional Spaces

In this Chapter you have to really use your imagination.The concepts are however simple and easy to understand.

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Unfolding 3D Space to 2D SpaceUnfolding 4D Space to 3D SpaceHyperCube in Painting

Unfolding Geometry Objects

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.

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Unfolding 4D Space to 3D SpaceSection 1Unfolding Geometry Objects

Idea behind the 3d-PuzzlesConcept of the 3d-Puzzles

Idea & 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.

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Concept of the 3d-PuzzlesSection 1Idea & Concept of the 3d-Puzzles

8-Cube 3d-Puzzle16-Cube 3d-Puzzle12-Cube 3d-Puzzle

3d-Puzzles

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8-Cube 3d-PuzzleSection 13d-Puzzles

3d-Puzzle - Limited Edition

The 3d-Puzzle - Limited Edition is made by 3D Printing in one piece in theFactory of the Future.

It is a Collection Item that can be purchased on the Online StoreIt 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).

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3d-Puzzle - Limited Edition

3d-Puzzle

Solution

4D3d Puzzles Apps

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Solve the 4D3d-Puzzles on the Screen of your Mobile

or your Computer.

The 4D3d Puzzles App is avai lable for mult iple platforms on the website.

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 applicationsdesigned 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.

Solving a 4D3d Puzzle of the Game AppRotate on your Mobile or Computer Screen 3-Dimensional Elements

or a Set of Elements of any 4D3d Puzzle to solve it.

Categories of the 4D3d Puzzles Game App: The theme is Sci-Fi

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4D3d Puzzles Apps

The 4D3d Puzzles are modeled in 3 Dimensions by software and are renderedon 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.

tesserART: Sculptures & Jewelry

Target Audience: 3d-Puzzles & 4D3d AppsBenefits of the 3d-Puzzles & 4D3d Puzzles Apps

Target Audiences & Benefits

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

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Benefits of the 3d-Puzzles & 4D3d Puzzles AppsSection 1Target Audiences & Benefits

Appendix 1: Euclidian & Non-Euclidian GeometryAppendix 2: Geometry Elements & ObjectsAppendix 3: SpacetimeAppendix 4: Tesseract - Hypercube - 8 CellAppendix 5: Geometry of the 3d-PuzzleAppendix 6: Higher Spacial Dimensions

Appendices

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.

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Geometry Elements & Objects

Appendix 2

Time

A temporal Dimension is a Dimension of Time. Time is often referred to as the "4th 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/

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Geometry Elements & Objects

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, 7 .... 10, 11, 12, etc.), it become extremely complicated.

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Tesseract - Hypercube - 8 Cell

Appendix 4

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 Spaceof a rotating Tesseract

Of special interest for this Digital Book is the 4D Hypercube or TesseractAccording 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).

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Tesseract - Hypercube - 8 Cell

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.

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Geometry of the 3d-Puzzle

Appendix 5

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.

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Geometry of the 3d-Puzzle