The Möbius Strip Topology: History, Science, and Applications in Nanotechnology, Materials, and the Arts

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Acknowledgments
Preface
Introduction
Chapter 1: Nineteenth-Century Science Breakthroughs in Europe: Historical Background
	1.1: Timeline of Key Historical Events in Central Europe of the 18th/19th Century
	1.2: The 19th-Century German Ideal of Scientific Education
		1.2.1: Educational Concepts in 19th-Century Central Europe
		1.2.2: Between “Bildung” and “Wissenschaft” in Higher Education
Chapter 2: A. F. Möbius: The Time of His Early Life and Academic Education (1790–1815)
	2.1: August Ferdinand Möbius and the World He Lived in as a Young Boy and School Student
		2.1.1: Childhood of August Ferdinand Möbius (1790–1803)
		2.1.2: August Ferdinand Möbius’s Time as School Student in Schulpforta (1803–09)
	2.2: August Ferdinand Möbius and the World He Lived in as University Student
		2.2.1: A. F. Möbius’s Time in Leipzig (1809–13)
		2.2.2: A. F. Möbius’s Time in Göttingen and Halle (1813–15)
			2.2.2.1: In Göttingen
			2.2.2.2: In Halle
		2.2.3: A. F. Möbius Back at Leipzig University (1815)
Chapter 3: A. F. Möbius: The Time of His Academic Career (1815–68)
	3.1: A. F. Möbius’s Living Situation in Leipzig
	3.2: A. F. Möbius’s Professorship Offers from Outside Saxony
	3.3: A. F. Möbius’s Teaching Activities at Leipzig University
	3.4: The Leipzig Observatory
	3.5: A. F. Möbius’s Scientific Oeuvre and Publications
		3.5.1: Astronomy
		3.5.2: Mathematics
	3.6: A. F. Möbius’s Friendship with Hermann Günther Grassmann
	3.7: A. F. Möbius’s Friendship with Gustav Theodor Fechner
	3.8: The Möbius Strip
	3.9: Development of Möbius’s Conditions of Living
	3.10: Academic Distinctions and Honors of August Ferdinand Möbius—and His Obituary
	3.11: Short Extraction from the Family Saga of August Ferdinand Möbius
	3.12: A. F. Möbius’s Relationship to Politics
Chapter 4: During the Napoleonic Wars in Europe: Eyewitnesses and Victims
	4.1: Testimonies from Contemporary Witnesses of the Napoleonic Wars in Europe (1806–15)
		4.1.1: Madrid Guerilla Fighting the Napoleonic Invasion
		4.1.2: Napoleonic Siege of Vienna and Beethoven’s Dread of Cannon Thunder
		4.1.3: From Naumburg to Schönbrunn: A Failed Attempt of Napoleon’s Assassination
		4.1.4: The Battle of Dresden (August 26–27, 1813)
		4.1.5: The Battle of the Nations in Leipzig (October 16–19, 1813)
		4.1.6: Eyewitnesses of the Battle of the Nations as Documented in the Stadtarchiv Leipzig [3].
	4.2: The Mount Tambora Volcano Eruption in April 1815
Chapter 5: Möbius Strip Topology and More: Elaborations on Mathematics
	5.1: Mathematical Representations of the Möbius Strip
	5.2: Calculation of the “Developable” Shape of the Möbius Strip
	5.3: More about Topology in a Nutshell
		5.3.1: Möbius Strip: An Example of Algebraic Surgery
		5.3.2: From the Möbius Strip to the Klein Bottle: The Rectangle Peg Problem Solved in 2020
	5.4: Euler’s Polyhedron Formula: The Euler Characteristic of Convex Polyhedra without a Hole
	5.5: The Power of Euler’s Formula
	5.6: The Platonic Bodies
	5.7: The Platonic Bodies in Philosophy
	5.8: Generalizations of the Euler Formula for Polyhedra with Holes
		5.8.1: Generalization by L’Huilier
		5.8.2: The Euler–Poincaré Formula
	5.9: More on Topology of Twisted Closed Band Objects: Their Linking, Twist, and Writhe Numbers
	5.10: The Möbius Transformations
	5.11: The Möbius Tetrahedra
	5.12: The Möbius Function, μ(n)
	5.13: The Fibonacci Spiral and the Möbius Strip
	5.14: The Golden Section and the Fibonacci Spiral Geometry
	5.15: Constructing the Golden Rectangle according to the Golden Ratio
	5.16: Spinor Property of the Möbius Strip
	5.17: Utilizing the Spinor Property of Protons to Spectacularly Enhance the ENDOR Effect
	5.18: Outlook: Quantum Computing with Spin 1/2 Spinor Systems
		5.18.1: Schrödinger’s Cat Paradoxon
		5.18.2: Progress in Quantum Computing
		5.18.3: Future Prospects for Large-Scale Quantum Computing
Chapter 6: A. F. Möbius and His Time: Elaborations on Astronomy
	6.1: The 18th/19th Centuries Scientific Renaissance and Astronomical Revolution in Germany
	6.2: German Astronomical Instrumentation of the 18th and 19th Centuries
	6.3: Fraunhofer’s Refracting Telescopes
	6.4: Astronomical Discoveries by A. F. Möbius’s Contemporaries and Their Instrumentation
		6.4.1: Carl Friedrich Gauss (1777–1855)
			6.4.1.1: C. F. Gauss, A. von Humboldt, W. Weber, A. F. Möbius, an unparalleled quartet in 19th century science
			6.4.1.2: The 2019 anniversaries of A. von Humboldt, C. F. Gauss, and A. F. Möbius, and the latest news on Earth’s magnetic field variations
		6.4.2: Friedrich Wilhelm Argelander (1799–1875)
		6.4.3: Friedrich Wilhelm Bessel (1784–1846)
			6.4.3.1: Friedrich Wilhelm Bessel at the Observatories of Lilienthal (1806–10) and Königsberg (1810–46)
			6.4.3.2: Friedrich Wilhelm Bessel and the prediction of eclipses
			6.4.3.3: Solar eclipse of May 29, 1919, and the first test of Einstein’s relativity theory
			6.4.3.4: The “Einstein Myth” and a few questions asked by critical science historians
			6.4.3.5: Newtonian calculations of light deflections by celestial masses, Johann Georg von Soldner (1776–1833), and the banishing of Einstein’s relativity theory by Nazi Germany
		6.4.4: Johann Elert Bode (1747–1826)
		6.4.5: Johann Franz Encke (1791–1865)
		6.4.6: Johann Gottfried Galle (1812–1910)
		6.4.7: Karl Ludwig Harding (1765–1834)
		6.4.8: Friedrich Wilhelm Herschel (1738–1822) and Caroline Herschel (1750–1848)
		6.4.9: Heinrich Wilhelm Matthias Olbers (1758–1840)
		6.4.10: Johann Hieronymus Schroeter (1745–1816)
		6.4.11: Friedrich Georg Wilhelm von Struve (1793–1864)
		6.4.12: Franz Xaver von Zach (1754–1832)
	6.5: August Ferdinand Möbius’s Activities as Astronomer in Leipzig
	6.6: Honors in Remembrance of August Ferdinand Möbius
	6.7: News (2018) from the NASA Lunar Reconnaissance Orbiter (LRO) Mission
		6.7.1: NASA History and the History of US Civilian–Military Relations in Space
		6.7.2: The NASA Lunar Reconnaissance Orbiter (LRO) Mission
	6.8: Breaking News (December 08, 2018) from The New York Times: China Launched the First-Ever Surface Mission (“Chang’e 4”) to the Moon’s Far Side
		6.8.1: Breaking News (December 2020) from China’s “Chang’e 5 Moon Mission” which Collected and Brought 2 Kg of Rock Samples from the Moon’s Surface Back to Earth
		6.8.2: Fifty Years Anniversary Celebrations (2019) of the Apollo 11 Mission, First Man on the Moon
		6.8.3: Postscriptum to the Apollo 11 Mission, First Man on the Moon
	6.9: News (2016) about Space-Based Infrared Telescope Probe Rosetta and Its Long Journey to the Comet Churyumov-Gerasimenko
	6.10: Breaking News (2019): Saturn’s Rings and Moons Revealed in Unprecedented Detail by Cassini Spacecraft Telescopes and Mass Spectrometers
		6.10.1: Breaking News (2019): Saturn’s Rings as Unveiled by Cassini
		6.10.2: Breaking News (2019): Saturn’s Moons Titan and Enceladus as Unveiled by Cassini
	6.11: Evidence from the Herschel Space Observatory of a Twisted Ring Structure of Dense Galactic Matter Orbiting the Center of the Milky Way
	6.12: Breaking News from November 13, 2019: The James Webb Space Telescope is Finally Ready
Chapter 7: Möbius Strip Topology: Applications from Chemistry
	7.1: Erich Hückel and His Molecular Orbital Concept of Aromaticity in Organic Chemistry
	7.2: Brief Historical Overview of the Development of Quantum Chemistry in Germany
		7.2.1: Development before World War II
		7.2.2: Development after World War II
		7.2.3: The Tragic Story of Hans Hellmann
	7.3: Edgar Heilbronner and his MO Concept of Möbius Aromaticity in Organic Chemistry
		7.3.1: Edgar Heilbronner: The Man of Science
		7.3.2: Edgar Heilbronner: The Man of Culture
	7.4: Topology of Twisted Ribbon Molecules, the DNA Example
	7.5: Quantifying the Topology of Twisted Molecules: Twist, Writhe, and Linking Number
	7.6: Examples of Molecules with Möbius Strip Topology
		7.6.1: Aromatic Möbius Strip Molecules: Annulene Systems
		7.6.2: Aromatic Möbius Strip Molecules: Expanded Porphyrin Systems
	7.7: Basics about NMR, EPR, and ENDOR Spectroscopies at High Magnetic Fields
		7.7.1: NMR versus EPR Spectroscopy at High Magnetic Fields
		7.7.2: ENDOR at High Magnetic Fields
	7.8: Recent EPR Experiments on Möbius-Type Expanded Porphyrin Molecules
		7.8.1: Möbius–Hückel Topology Switching in Expanded Porphyrins: EPR, ENDOR, and DFT Studies of Doublet and Triplet Open-Shell Systems
			7.8.1.1: Introduction
			7.8.1.2: Experimental
			7.8.1.3: Theoretical
			7.8.1.4: Results and Discussion
	7.9: Cyclotides as Example of Cyclic Proteins with Trefoil Knot and Möbius Strip Topology
		7.9.1: More about Knot Topology of Lines in Relation to Möbius Strip Topology of Surfaces
		7.9.2: Cyclotides in Medical Therapy Applications
		7.9.3: Oral Activity of a Nature-Derived Cyclic Peptide for the Treatment of Multiple Sclerosis
Chapter 8: The Möbius Strip Topology: Applications from Physics and Nanomaterials
	8.1: Quantization of the Hall Effect
	8.2: A Primer on Topological Insulators
	8.3: Graphene-Based Topological Insulators
	8.4: Spin States of Möbius Graphene Nanoribbons
	8.5: Designing Topological Insulators from Graphene Nanoribbons
	8.6: Laser Pulses Create Topological State in Graphene
Chapter 9: Möbius Strip Topology: Highlights from the Arts and Architecture
	9.1: Möbius Strips before Möbius: By the Example of Baroque Music
	9.2: Pre-Möbius One-Sidedness: By the Example of Visual Arts and Architecture
	9.3: More Twisted Bands from Roman Mosaics, the Ouroboros and Infinity
	9.4: Mathematics and Architecture, Symmetry and Beauty in the Arts and Sciences
		9.4.1: Influence of Mathematics on Art and Architecture over the Centuries
		9.4.2: Symmetry and Beauty
			9.4.2.1: Interplay of Aesthetics and Science
			9.4.2.2: Symmetry and Architecture
			9.4.2.3: Symmetry and Poetry
	9.5: Möbius Strip Topology in Literature, Theater, and Film
		9.5.1: “La Banda de Moebius” Theater Play by Antonio Toga, Neza (Mexico City), 2020
		9.5.2: “The Parallel World,” Theater Play about People in the Quantum World, a Twin Production by Kay Voges in Berlin and Dortmund, 2018
		9.5.3: “The Bald Soprano,” Theater Play by Eugène Ionesco, 1950
		9.5.4: “Kata-Omoi” (“One-Sided Love”), Mystery Novel by Keigo Higashino, 2001
		9.5.5: “A Subway Named Mobius,” Short Story by Armin Joseph Deutsch, 1950, and “Moebius,” a Film by Gustavo Daniel Mosquera R, 1996
		9.5.6: “Möbius,” Film by Éric Rochant, 2013
	9.6: Möbius Topology in Sculpture
		9.6.1: Max Bill (1908–94), Switzerland
		9.6.2: Keizo Ushio (Born 1951), Japan
		9.6.3: Robert R. Wilson (1914–2000), the United States
	9.7: Möbius Strip Topology: Highlights from Modern Architecture
		9.7.1: A Fundamental Problem of Architecture when Designing Buildings in Three-Dimensional Space with Several Floors Interconnected by a Möbius Strip Surface
		9.7.2: Max Reinhardt Haus in Berlin by Peter Eisenman (1992)
		9.7.3: Möbius House at Het Gooi Near Amsterdam by Ben van Berkel, 1993
		9.7.4: Phoenix International Media Center in Beijing by Shao Weiping, 2004
		9.7.5: Lucky Knot Bridge in Changsha (China) by NEXT Architects with their Chief Architects John van de Water (Dutch) and Jiang Xiaofei (Chinese), 2013
Index