# Exploring The 4th Dimension Using Animations

The following paragraphs and Animations will explore certain aspects of the 4th Dimension using Animation techniques. The open source software packages, OpenGL and the OpenGL Utility Toolkit (GLUT), were extensively used to create the Animations. The OpenGL and GLUT software are configured to display a 3D Perspective View on the monitor screen. The 3D Perspective View shows closer objects as larger than more distant objects which makes it easier to visualize what is going on in the Animations. The Animations are driven by a Translational and Rotational Math Engine (TRME) that is constantly updating positions and orientations of the displayed objects. First an array of Points representing the objects that are being studied is generated and displayed in the 3D Perspective View. This array of Points is then processed using standard Matrix Rotation and Translation methods for an incremental angle and an incremental distance to calculate new locations for all the Points and these are displayed in the 3D Perspective View. The processing and display operation is repeated for a series of incremental angles and incremental distances to animate the objects under study. Note that the TRME can calculate full 4D Translations and Rotations which are displayed in the Animations using a method called Axis Sharing as will be fully described.

We will use the classic analogy approach where an Animation for a 2D Observer studying 3D Space will be shown and then an analogous Animation for a 3D Observer studying 4D Space will be shown. The Axis Sharing method combined with the special way objects are constructed effectively produces slices or cross sections into the Hyper Spaces under study. Using analogies and slices is not new but the Animations present these methods in a new and dynamic way. Also, the concept of Dimensional Points allows Observers to graphically see the Hyper Space connections of objects, that must be understood when thinking about higher dimensions. And as a final note, this study is not about the Time dimension from Relativity, where they like to say that Time is the fourth dimension.

The Animations are Animated GIF files. The GIF format is a good cross-platform choice but may load slow on some systems. If you have a good modern high speed Internet connection then the load time should only be a few seconds.

Thank You

Steve Klinko

4/24/15

Last Update 4/25/15

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CONTENTS:

Using Axis Sharing To Represent Points in Hyper Space

Hyper Factoring And Zero Factoring An Axis

Constructing Fundamental Objects In Point World

2D And 3D Hyper Space Rotation Analogies

Sphere In 2D And Hyper Sphere In 3D Rotation Analogies

Mathematical Points And Dimensional Points

Rotations On A Line And On A Plane Of Dimensional Points

Flatness of nD Space relative to (n+1)D Space