By Chris McMullen

This colourful, visible advent to the fourth measurement presents a transparent rationalization of the thoughts and diverse illustrations. it's written with a marginally of character that makes this an attractive learn rather than a dry math textual content. The content material is particularly obtainable, but while exact adequate to meet the pursuits of complicated readers. This publication is dedicated to geometry; there aren't any religious or non secular parts to this e-book. may perhaps you get pleasure from your trip into the attention-grabbing global of the fourth dimension!


  • Introduction
  • Chapter zero: what's a Dimension?
  • Chapter 1: Dimensions 0 and One
  • Chapter 2: the second one Dimension
  • Chapter three: three-d Space
  • Chapter four: A Fourth size of Space
  • Chapter five: Tesseracts and Hypercubes
  • Chapter 6: Hypercube Patterns
  • Chapter 7: Planes and Hyperplanes
  • Chapter eight: Tesseracts in Perspective
  • Chapter nine: Rotations in 4D Space
  • Chapter 10: Unfolding a Tesseract
  • Chapter eleven: go Sections of a Tesseract
  • Chapter 12: residing in a 4D House
  • Further Reading
  • Glossary
  • About the Author

Put in your spacesuit, strap in your defense harness, swallow your anti-nausea drugs, and luxuriate in this trip right into a fourth size of area! 10D, 9D, 8D, 7D, 6D, 5D, 4D, 3D, 2nd, 1D, 0D. Blast off!

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Extra resources for A Visual Introduction to the Fourth Dimension (Rectangular 4D Geometry)

Example text

They are labeled A thru P. Here is another way to deduce that a tesseract (full of monkeys) has 16 corners. Consider the binary unit hypercube. By this, I mean that all of the coordinates of the corners are either 0 or 1 (like the binary number system, where all of the digits are 0 or 1). The 1D line has corners at x = 0 and x = 1. The 2D square has corners at (0,0), (1,0), (0,1), and (1,1). The 3D cube has corners at (0,0,0), (1,0,0), (1,1,0), (0,1,0), (0,0,1), (1,0,1), (1,1,1), and (0,1,1). We can deduce that the N-dimensional hypercube (full of monkeys) will have 2N corners.

Breadth (how wide she is, shoulder to shoulder), and depth (front to back, or nose to tail); she is three-dimensional (3D). However, there are different kinds of dimensions, like space and time – or if you want to get exotic, we can talk about the dimensionality of your thoughts or even your body odor. We need to explore the concept of a dimension a little further, so that you'll know exactly which type of dimensions we are discussing in this book. Let's begin with a simple geometric example.

The Second Dimension A plane is 2D, but a 2D world doesn't need to be flat; it could be curved like a sphere or a cylinder. A monkey in a plane could move in two independent directions – north/south or east/west. Similarly, a monkey confined to the surface of a sphere (so just like the monkey in the plane, she can't go up or down) could only travel north/south or east/west. Walking around in an open field is a largely 2D human activity. The following figures show a black hole, intersecting worlds, parallel universes, a bridge, a wormhole, and a curled dimension in 2D.

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