The Visual Guide to Extra Dimensions Volume 1: Visualizing the Fourth Dimension, Higher-Dimensional Polytopes, and Curved Hypersurfaces

203 Pages·2008·17.37 MB·English

by  Chris McMullen

The Visual Guide to Extra Dimensions
Volume 1: Visualizing the Fourth Dimension, Higher-Dimensional Polytopes,
and Curved Hypersurfaces
Copyright (c) 2008 Chris McMullen
All rights reserved. This includes the right to reproduce any portion of this book
in any form.
www.faculty.lsmsa.edu/CMcMullen
Custom Books
Nonfiction science physics
Nonfiction science mathematics / geometry
ISBN: 1438298927
EAN-13: 9781438298924
Volume 1
Visualizing the Fourth Dimension, Higher-Dimensional
Polytopes, and Curved Hypersurfaces
Volume 2
The Physics of the Fourth and Higher Dimensions,
Compactification, and Current and Upcoming Experiments
to Detect Extra Dimensions
The Visual Guide to Extra Dimensions
Volume 1
Visualizing the Fourth Dimension, Higher-Dimensional Polytopes,
and Curved Hypersurfaces
Chris McMullen, Ph.D.
Contents Introduction
Dedications
Chapter 0: The Known Dimensions
0.0 What Is a Dimension?
0.1 The First Dimension
0.2 The Second Dimension
0.3 The Third Dimension
Chapter 1: Common Objections
1.0 Is Time the Fourth Dimension?
1.1 Where Are the Extra Dimensions?
1.2 Why Study Extra Dimensions?
1.3 How Can Extra Dimensions Be Detected?
Chapter 2: Visualizing the Fourth Dimension
2.0 The Fourth Dimension
2.1 Projective Geometry
2.2 Drawing the Fourth Dimension
2.3 Constructing the Fourth Dimension
2.4 Higher-Dimensional Perspective
2.5 Lower-Dimensional Analogies
2.6 Snapshots of the Fourth Dimension
Chapter 3: Higher-Dimensional Polytopes
3.0 Polygons, Polyhedra, and Polychora
3.1 Anatomy of a Tesseract
3.2 Hypercube Patterns
3.3 Anatomy of a Pentachoron
3.4 Simplex Patterns
3.5 Other Polytopes
Chapter 4: Curved Hypersurfaces
4.0 Curves
4.1 Surfaces
4.2 Hypersurfaces
4.3 Anatomy of a Glome
4.4 Hyperspheres
4.5 Quadric Hypersurfaces
4.6 Hypertor i
Chapter 5: A Hypothetical Hyperuniverse
5.0 Hyperuniverse NCF4D-U
5.1 The Physics of NCF4D-U
5.2 Humanoids in the Fourth Dimension
5.3 Higher-Dimensional Chirality
5.4 Construction in the Fourth Dimension
5.5 Life in NCF4D-U
Volume 2 Contents
References and Further Reading
Puzzle Solutions
Author’s Qualifications
Introduction
My fascination with extra dimensions began when I was in high school and
encountered one of Rudy Rucker’s books on the fourth dimension during one of
my trips to the math and science shelves of a bookstore. I remember staying up
into the wee hours contemplating the ana and kata directions, drawing
hypercubes and hyperspheres, and solving the puzzles from Rudy Rucker’s
book.
I was fortunate to combine my interest in extra dimensions with my
expertise in physics during my doctoral studies at Oklahoma State University:
Some papers in the 1990’s had stimulated interest in large string-inspired extra
dimensions in the particle physics community. I earned my Ph.D. in physics
when an explosion of professional research was published in this field, and have
continued to publish papers on the collider phenomenology of large extra
dimensions ever since.
Recalling my appreciation for Rudy Rucker’s works, I wanted to make my
own humble contribution to readers with an interest or background in math or
physics who are curious about extra dimensions. I hope that my work will
engage the interest of some readers who share my curiosity for the fourth and
higher dimensions.
This first volume is dedicated toward a geometric extra dimension very
much like the three known dimensions. It begins by surveying interesting
features of 1D and 2D worlds, since we can understand many higher-
dimensional concepts through analogies in the lower dimensions. Some common
objections to studies of a fourth dimension are addressed, especially to convey
the message that times have changed: There is good motivation for the
possibility of superstring-inspired extra dimensions that are much larger than
originally thought – large enough that current and upcoming experimental
searches are underway with prospects for detecting them.
One of the main goals of the first volume is to develop techniques for
visualizing extra dimensions; numerous illustrations, many of which are novel,
aim to aid in this process. Another goal is to thoroughly analyze some
fundamental 4D objects, especially the tesseract and the glome. The final chapter
of this volume looks at a hypothetical hyperuniverse, in which these
visualization techniques and basic geometric objects are applied. This chapter
serves as a useful bridge between the higher-dimensional geometry considered in