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

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

My fascination with extra dimensions began when I was in high school and encountered one of Rudy Ruckers 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 Ruckers 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 1990s 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 Ruckers 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 the first volume and the higher-dimensional physics developed in the second volume.

This book is primarily conceptual, for the benefit of readers who do not have a strong background in the mathematics of particle physics or superstring theory, yet there is a mathematical component to this book, since it is anticipated that most readers will have some interest or background in mathematics. It will be desirable to have a good handle on the fundamentals of algebra, geometry, and graphing. Where it is deemed useful to discuss more advanced mathematics, higher-level concepts are developed from these starting levels.

Nonetheless, more emphasis is placed on the concepts than on the math, since this is intended to be an enjoyable book on an interesting topic, which should not read like a textbook. This book is very detailed and technical conceptually in an effort to help stimulate and engage the interest of such mathematically-minded readers.

Several illustrations are intended to challenge readers visually. For example, several illustrations combine together to show a hypercube unfold, the various 3D cross sections of a hypercube are drawn, the 3D projection is depicted for a rotating hypercube, higher-dimensional cylinders and tori are graphed, and a higher-dimensional staircase is drawn. Puzzles scattered throughout the book offer additional challenges.

The second volume looks at higher-dimensional mathematics, higher-dimensional force laws related to Gausss law, the important issue of compactification, current and upcoming experimental searches for extra dimensions, and a little background in spacetime, quantum mechanics, and string theory.

May you enjoy the book as much as I enjoyed reading my first book on extra dimensions.

Dedications

I would like to thank the many teachers who inspired my interest in math and science, my passion for teaching, and many of the good qualities that I strive for in my teaching, research, mentorship, and life. I particularly appreciate those teachers who were excellent motivators and mentors and those whose courses were very challenging, rigorous, and thought-provoking. One of my favorites is Dr. Robert Chianese, who taught a fascinating course on Cold War Literature so much so that I have since re-read many of the assigned readings, putting forth much more effort after the credit had already been earned. I had the good fortune of taking a few courses with Dr. Duane Doty, who is a superb mentor, motivator, and teacher. My Masters thesis, under the guidance of Dr. Miroslav Peric, immensely helped to develop my confidence in challenging myself with research. Teachers such as Dr. A.C. Cadavid, Dr. Peter Collas, and Dr. Robert Park helped to instill my interest in rigorous mathematics, and inspired the high level of mathematics that I strive to incorporate into the courses that I teach. Dr. K.S. Babu is the ideal model for a command of knowledge in ones area of expertise. I am very grateful for the opportunities that I have had to collaborate with my mentor, Dr. S. Nandi, on multiple research projects, in addition to being a student in the excellent courses that he taught. I would also like to mention my high school geometry teacher, Mr. Ratkovic, who is not only an exceptional teacher, but who identified gifted students in his classes and found ways to engage their interest and challenge their minds while still teaching to the rest of the class. I appreciate that so much that it is one of my primary reasons that I accepted my current position at the Louisiana School for Math, Science, and the Arts, and it serves as a strong source of my motivation to challenge young minds in math and physics.

I must also thank my family for their encouragement and always believing in my ability, including my mom, dad, grandma, and aunt. I am also very grateful for the invaluable support of my wife, whose Masters thesis was also on the subject of collider phenomenology of large extra dimensions.

I dedicate this book to all those who have had a positive impact on my life.

The Known Dimensions