The Mobius Strip

Dr. August Mobius's Marvelous Band in
Mathematics, Games, Literature, Art, Technology, and Cosmology

Clifford A. Pickover
Thunder's Mouth Press, 2006

Acclaimed popular science writer Clifford Pickover explores the weird world of the shape made famous by M. C. Escher

"Pickover inspires a new generation of da Vincis to build unknown flying machines and create new Mona Lisas." -- Christian Science Monitor

"Bucky Fuller thought big, Arthur C. Clarke thinks big, but Cliff Pickover outdoes them both." -- WIRED

"A perpetual idea machine, Clifford Pickover is one of the most creative, original thinkers in the world today." -- Journal of Recreational Mathematics

Now at Amazon.Com

The Mobius Strip:
A Gateway to Higher Dimensions

The Mobius strip--the common sense-defying continuous loop with only one side and one edge, made famous by the illustrations of M. C. Escher-leads us to some of the strangest spots imaginable. It takes us to a place where the purely intellectual enters our daily world: where our outraged senses, overloaded as they are with grocery bills, the price of gas, and what to eat for lunch, are expected to absorb truly bizarre ideas. And no better guide to our weird universe exists than the brilliant thinker Clifford A. Pickover, the twenty-first century's answer to Buckminster Fuller.

Come along as Pickover traces the origins of the Mobius strip from the mid-1800s, when the visionary scientist Dr. August Mobius described the properties of one-sided surfaces, to the present, where it is an integral part of mathematics, magic, science, art, engineering, literature, and music. It has become a metaphor for change, strangeness, looping, and rejuvenation.

From molecules and metal sculptures to postage stamps, architectural structures, and models of our entire universe, The Mobius Strip gives readers a glimpse both of new ways of thinking and of new worlds entirely, as Pickover travels across cultures and peers beyond our ordinary reality. Lavishly illustrated, The Mobius Strip is an infinite fountain of strange and wondrous forms that can be used to help demonstrate how mathematics has permeated every field of scientific endeavor. Indeed, mathematics can be used to help explain the colors of a sunset or the architecture of our brains, to help us build supersonic aircraft and roller coasters, simulate the flow of Earth's natural resources, explore subatomic quantum realities, and depict faraway galaxies. Following the strange path of the Mobius strip, we learn how mathematics has changed the way we look at the cosmos.

Mobius Transformation
Courtesy of Jos Leys

Mobius Strip with Penrose Tiling
Courtesy of Teja Krasek


Mobius Fly Maze
Courtesy of David Phillips
"Find the path that the four flies take if they all travel the same route without meeting, and without retracing their path until they reach their original position. Keep track of which side of the path you are on."


Bonan-Jeener's Klein Surface 2
Courtesy of Jos Leys


Mystery Woman
wears Mobius earring
Courtesy of Sinisa Rancov


Alexander Horned Sphere
Courtesy of Cameron Brown

Table of Contents


In which we encounter a "hole through a hole in a hole," topology, Mobius strips, the desiccated skull of Mobius, Franz Gall, the recycling symbol, Mobius beer, the Mobius Flip, Max Bill's "Endless Ribbon," Gustavo Mosquera, Eternal Sunshine of a Spotless Mind, Mobius strip metaphors, Mobius strips in religion, Eugene Ionesco's play The Bald Soprano, the cerebral Mobius strip, and the Acme Klein Bottle.

Chapter 1 Mobius Magicians

In which we encounter Mobius illusions, gospel magic, the Mobius treadmill puzzle, Mobius's place in history, and my boyhood introduction to the marvelous band.

Chapter 2 Knots, Civilization, Autism, and the Collapse of Sidedness

In which we encounter ants inside spheres, Mobius dissections, sandwich Mobius strips, Ljubljana ribbons, Lord Kelvin's vortex knots, trefoil knots, New York lawyer and part-time topologist Kenneth Perko, the mystery unknot, Asperger's syndrome, Wolfgang Haken's unimplementable knot algorithm, the triquetra, the occult TV show "Charmed," Led Zeppelin, The Book of Kells, knots in proteins, Borromean rings, knots as catalysts of civilization, the alien knot puzzle, and Mobius aliens.

Chapter 3 Mobius the Man

In which we encounter Mobius's family tree, simultaneity in science, Schulpforta, Paul Julius Mobius, Mobius syndrome, Johann Benedict Listing, the "king with five sons" problem, Mobius's mathematical output, Karl August Mobius, the false dawn animal, the Mobius maze puzzle, and Mobius licentiousness.

Chapter 4 Technology, Toys, Molecules, and Patents

In which we encounter the Rhennius machine, Roger Zelazny's Doorways in the Sand, Mobius patents and toys, Mobius molecules, mathematics patents, lemniscates, astroids, Reuleaux triangle drill bits, conveyor belts with twists, surgical retractors, Mobius electrical components and train tracks, knot patents, the metaphysics of shoelaces, chirality, Lipitor, Paxil, Zoloft, Nexium, Thalidomide, Advil, enantiomers, Methanobacterium thermoautotrophicum, Mobius plant proteins that induce labor in African women, Mobius crystals, the Noah's ark puzzle, and Mobius strips in fashion and hair style.

Chapter 5 Strange Adventures in Topology and Beyond

In which we encounter Benoit Mandelbrot, fractals, parameterizations, a conical helix, butterfly curves, paradromic rings, Leonhard Euler, Antoine-Jean Lhuilier, chromatic numbers, projective planes, the four-color theorem, "The Island of Five Colors," Mobius's triangulated band, Johann Listing, homeomorphisms, ghosts, the fourth dimension, Immanuel Kant, Johann Carl Friedrich Zollner, Henry Slade, Alfred Schofield's "Another World," turning spheres and doughnuts inside out, optiverses, the Boy surface, cross-caps, Roman surfaces, the fantastic Mobius function, the Mertens conjecture, the Riemann zeta function, Mobius palindromes, the amazing o, coprimality, graph theory, hexaflexagons, Mobius shorts, Mobius tetrahedra, Mobius triangles, solenoids, Alexander's horned sphere, prismatic doughnuts, perfect square dissections, the squiggle map coloring puzzle, the cannibal torus, the pyramid puzzle, and Mobius in pop culture.

Chapter 6 Cosmos, Reality, Transcendence

In which we encounter nonorientable spaces, more enantiomorphs, dextrocardia with situs inversus, hyperspheres, Klein bottle coffee mugs, the world's largest Klein bottle, the Bonan-Jeener's Klein surface, Immanuel Kant redux, bilaterally symmetric Vernanimalcula guizhouena, the Babylonian god Marduk, Gottfried Wilhelm Leibniz, 3-tori, Max Tegmark, parallel universes, artificial life, simulated universes, John Horton Conway, the Wilkinson Microwave Anisotropy Probe, Gerhardus Mercator, the ekpyrotic model of universe formation and destruction, the Book of Genesis, self-reproducing universes, playing God, the pretzel transformation puzzle, and Mobius cosmoses.

Chapter 7 Games, Mazes, Art, Music, and Architecture

In which we encounter Mobius chess, mazes, Knights' tours, Bishop domination on a Klein bottle chessboard, Mobius stairs and snow sculptures, Mobius buildings, Mobius postage stamps, Max Bill, LEGO sculptures beyond imagination, Mobius gear assemblies, complex knots, Teja Krasek, Mobius strips with Penrose tilings, Mobius music, Johann Sebastian Bach, Arnold Schoenberg, Nicolas Slonimsky, devil configurations, Mobius strips in psychology and human relations, and mazes played on Mobius strips.

Chapter 8 Literature and Movies

In which we encounter the literature of nonorientable surfaces, the "No-Sided Professor," "A. Botts and the Mobius Strip," "Paul Bunyon Versus the Conveyor Belt," "The Wall of Darkness," "A Subway Named Mobius," Gustavo Mosquera, The Secret Life of Amanda K. Woods, The Journey of Mobius and Sidh, The Lobotomy Club, Flatterland, Bana Witt's Mobius Stripper, Dhalgren, Marcel Proust's In Search of Lost Time, Six Characters in Search of An Author, "Time and the Conways," Donnie Darko, Femme Fatale, Mobius the Stripper, Vladimir Nabokov's The Gift, Coleman Dowell's Island People, Daniel Hayes's Tearjerker, Eugene Ionesco's The Bald Soprano, Solvej Balle's According to the Law, John Barth's Lost in the Funhouse, Paul Nahin's "Twisters," Martin Gardner's Visitors from Oz, ants trapped in Jordan curves, and Mobius in the suburbs.

A Few Final Words

In which we encounter Stanislaw Ulam, Franz Reuleaux, Georg Bernhard Riemann, Zen koans, Eternal Sunshine of a Spotless Mind, Harlan Brothers, Marjorie Rice, Roger Penrose, Arthur C. Clarke, the Mandelbrot set, an ambiguous ring, and Mobius strips in business and government.

Now at Amazon.Com

Topology is about spatial relationships and glistening shapes that span dimensions. It's the Silly Putty of mathematics. Sometimes, topology is called "rubber-sheet geometry" because topologists study the properties of shapes that don't change when an object is stretched or distorted. The best way for people of all ages to fall in love with topology is through the contemplation of the Mobius strip -- a simple loop with a half twist….

Our nature is to dream, to search, and to wonder about our place in a seemingly lonely cosmos. Perhaps this is a reason that philosophers and writers have speculated about Mobius-shaped universes and higher dimensions, and what their inhabitants might be like. For many young, prospective scientists, the Mobius strip is a launch-pad to more sophisticated geometries and topological exploration….

The Mobius strip is the ultimate metaphor for something simple, yet profound -- something anyone could have discussed centuries prior to its discovery, but didn't. The Mobius strip is a metaphor for magic and mystery, and a perpetual icon that stimulates us to dream new dreams and look for depths even in seemingly shallow waters.

I discuss all kinds of patents involving the Mobius strip
and knots like this one,
which is patented for surgical use!


50 First Dates, 182
Afghan bands, 3
Alexander's sphere, 100
Aliens and Mobius strips, 23-24
Ambiguous ring, 197-198, 208
Applications, 89
April Fool's joke, 71-72
Architecture, 155
Art, 154-170
Artificial life, 134
Asimov, Isaac, 37
Autism and knots, 15

Balle, Solvej, 186
Barth, John, 186
Barycentric calculus, 106-107, 111
Bill, Max, xx, 157
Borromean rings, 20-21
Boy surface, 82-83
Brown, Ronnie, 13
Browne, Cameron, 100-102
Business, 198
Butterfly curves, 64

Calculus, barycentric, 106-107, 111
Cantor set, 98
Celtic knots, 18
Chemistry, 51-58
Chess, 150-153
Chirality, 52-54
Christmas tree, 59, 167
Chromatic number, 69
Civilization and knots, 21
Clarke, Arthur, C., 174-175, 197
Collins, Stanley, 4
Coloring maps, 69-72, 108
Cosmos models, 111-119, 136-141, 193
Crosscap, 82-84
Crumey, Andrew, 188

Darwin, Charles, 36
Dehn, Max, 12
Devil's configuration, 170-171
Dhalgren, 179
Dietrich-Buchecker, Christiane, 54-55
DNA and knots, 17
Dodecahedral space, 131
Donnie Darko, 181
Dowell, Coleman, 184
Drugs, 52-56

Egan, Greg, 136
Einstein, Albert, 119
Ekpyrotic universe, 141
Elliot, Bruce, 188
Enatiomers, 54
Eozoon, 36
Escher, M. C., xx, 158
Eternal Sunshine, 194
Euclid, 117
Euler, Leonhard, 66-67, 74, 90, 103
Eversions, 79-81
Extrinsic geometry, 119

False dawn animal, 35-36
Fashion, Mobius, 60
Fauvel, John, xi, 26
Femme Fatale, 182
Fibonacci numbers, 85
Fiction, 40
Flat space, 138-139
Flood, Raymond, xi
Four-color theorem, 70-72
Fourth dimension, 76, 116-117, 125-127
Fractals, 196, 213

Games, 146-153
Gardner, Martin, xxii, 3, 10, 71, 94, 174, 187, 196
Gauss, Carl, 28, 31, 73, 118
God, 142
Government, 198
Graph theory puzzle, 59
Graph theory, 93

Haken, Wolfang, 16
Handedness, 52-54
Hayes, Daniel, 184
Helix, 63
Hexaflexagons, 94
History and knots, 21
Hoidn, Phoebe, 17-18
Hole through hole, xv
Homeomorphisms, 74-75, 199
Horned sphere, 100-102
Hyperbolic universe, 139
Hyperspace,  116-117, 125-127
Hypersphere, 119, 137

Infinite universe, 141
Inflation, 140
Intrinsic geometry, 119
Inventions, 39-60, 148
Ionesco, Eugene, 185

Jordan curves, 189
Josipovici, Gabriel , 183

Kalata, 56
Kells, Book of, 19
Kelvin, William, 16
Klein bottles, 69, 82, 105, 120-124, 129, 152-153, 160, 187-188
Kleinian group, 213
Knight's tours, 151, 155
Knots, 12-18, 77-78, 165
	Borromean rings, 20-21
	Celtic, 18
	Chinese, 21
	Chromatic number, 69
	Civilization and, 21
	Cutting, 14, 65
	DNA, 17
	Figure-eight, 16, 160
	Fourth dimension, 77, 116
	History and, 21
	Homeomorphisms, 75, 100
	Kells, Book of, 18
	Mathematics, and 22
	Molecules, 54-55
	Patents, 50-51, 148
	Perko, 14
	Proteins, 18, 56
	Puzzle, 22
	Sailors, 21
	Subatomic particles, 17-18
	Trefoils, 12-18, 75, 162-163
	Triquetra, 18
	Unknot, 15
	Vortex atom theory, 16-17

Krasek, Teja, 166-167

Longtin, Tom, 161-164
Language, 210
Lego, 158-160
Leibniz, Gottfried, 128
Leys, Jos, 14, 20, 213
Licentiousness, 38
Life, 134-135
Lipson, Andrew, 158-160
Lissajous curves, 63
Listing, Johann, 28, 73
Literature, 173-189
Luther, Martin, 26, 30
Ljubljana ribbon, 12

Magic, 2-4
Mandelbrot, Benoit, 196-197
Manifolds, 128, 137
Map coloring, 33, 69-72, 108
Matrix, 133-134
Maxwell, James, 17
Mazes, 37, 146-149, 209, 216
Mertens function, 87
Mirror worlds, 115-116
Mirror molecules, 52-54
Mirrored organs, 116
Mirrors, 125-127
Mobidromes, 91
	Applications, 89
	Bundles, 212, 215
	Dualities, 212, 215
	Fiber bundles, 212, 215
	Function, 84-89
	Groups, 212
	Inversion formulas, 33, 212, 214
	Nets, 33, 212
	Statics, 33, 212
	Tetrahedra, 96
	Transform, 214
	Transformations, 33, 212-213
	Triangles, 97
Mobius, August, 25-35
	Skull, xvii		
	Place in history, 6, 197
Mobius, Paul, xvii, xviii, 27
Mobius, Karl August, 35-36
Mobius strip
	Aliens, 23-24
	Architecture, 155
	Art, xx, 154-170
	Beer, xix
	Business, 198
	Chemistry, 51-58
	Chess, 150-153
	Christmas tree, 59, 167
	Coloring a side, 8
	Cosmos, 143-144
	Cutting, 9-10, 65
	Fashion, 60
	Fiction, 40
	Government, 198
	Language, 210
	Lego, 158-160
	Licentiousness, 38
	Listing, Johann, 28
	Literature, 173-189
	Magical use, 2-4
	Map coloring, 33, 69-72
	Mazes, 37, 146-149, 209, 216
	Molecules, 51-58
	Movies, 173-189, 194
	Music, 168-170
	Parametric equations, 64
	Patents, 39-60, 148
	Penrose tiles, 166-167
	Pop culture, 109
	Proteins, 56
	Psychology, 172
	Religious use, 4
	Sandwich, 11
	Science metaphor, 36-37
	Shorts, 95-96
	Skiing, xix
	Snow sculpture, 168
	Square dissections, 104-105
	Stairs, 154-155
	Stamps, 156-157
	Suburbia, 190
	Technology, 39-60
	Toys, 147-149
	Train tracks, 58
	Triangulated, 72
	Triple-thick, 10
	Twist, number of, 11-12, 65
Mobius syndrome, 27-28
Molecular shapes, 51-58
	See also Knots and Mobius strip
Mosquera, Gustavo, 176
Movies, 173-189, 194
Multiple universes, 131-136, 141
Multiverse, 131-133
Music, 168-170
Nabokov, Vladimir, 184
Nahin, Paul, 187
Nelson, James, 3

Organs, mirrored 116
Orientability, 29, 82-83, 94, 114, 139

Palindromes, 90
Paradromic rings, 65
Parallel universes, 131-136, 141
Parametric equations, 62-65
Patents, 39-60, 148
Penrose triangle, 208
Penrose tiling, 166-167, 196
Perko knots, 14
Phillips, Dave, 37, 209, 216
Phrenology, xviii
Pi, 86, 91-92
Pirandello, Luigi, 180-181
Polyhedron, 66-67
Pop culture, 109
Pretzel puzzle, 143
Priestly, John, 181
Primes, 86, 92
Prismatic doughnuts, 102-103
Projective plane, 69, 82, 105, 120, 129
Proteins, 18, 55
Proust, Marcel, 179-180
Psychology, 172
	Ambiguous ring, 197-198, 208
	Ant planet, 189-190
	Devil's configuration, 170-171
	Graph, 59
	Knot, 22
	Maze, 37, 209, 216
	Pretzel, 143
	Pyramid, 109
	Squiggle coloring, 108
	Treadmill, 5	
Pyramid puzzle, 109

Recycling symbol, xviii-xix
Religion, 4
Reuleaux triangle, 42-43, 192
Riemann, Bernhard, 117, 193
Ring, ambiguous, 197-198, 208
Rivers, 92
Robinson, John, 13
Roman surface, 82-83
Rzepa, Henry, 57

Sailors and knots, 21
Sauvage, Jean-Pierre, 54-55
Scharein, Rob, 164-165
Schoenberg, Arnold, 169
Shoelaces, 51
Simanek, Donald, 208
Simulations, reality, 133-136
Simultaneous discovery, 29
Slapenarski surface, 174
Slonimsky, Nicolas, 169
Snow sculpture, 168
Solenoid, 97-99
Sphere eversions, 79-80
Sphere, horned, 100-102
Square number, 84
Squareful numbers, 85
Squiggle map coloring, 108
Stairs, 154-155
Stamps, 156-157
Star Trek, 177
Stasiak, Andrzej, 17-18
Stephens, Nicky, 154-155
Steward, Ian, 1, 6, 145, 191
String theory, 90
Suburbia, 190
Subway, 176
Surgical knots, 51
Symmetry, 125-126

Technology, 39-60
Tegmark, Max, 132
Thalidomide, 54
Three-torus, 128-130, 140
Time travel, 182-183
Tissandier, Gaston, 3
Topology, xvi, 61-84
Torus eversions, 79-81 
Torus, 69, 79-82, 105, 128-130, 154
Toys, 147-149
Treadmill puzzle, 5
Train tracks, 58
Trefoil knot, 12-17, 75, 162-163
Triangle, Reuleaux,  42-43, 192
Triangles, 117-118
Triquetra, 18
Twists in Mobius strips, 11-12, 65

Universe models, 111-119, 136-141, 193
Unknot, 15
Upson, William, 175

Vortex atoms and knots, 16-17

Walba, David, 52-53
Wilson, Robin, xi
Witt, Bana, 179

Zelazny, Roger, 40
Zollner, Johann, 77-79

High-tech assortment of
Mobius Christmas tree ornaments
Courtesy of Teja Krasek


Mobius strip on
Caltrate package


"We Have Died and Gone to Mobius Heaven"
by Teja Krasek & Cliff Pickover