In the 19th century, pure mathematics research reached a climax in Germany, and Carl Friedrich Gauss (1777–1855) was an epochal example. August Ferdinand Möbius (1790–1868) was his doctoral student whose work was profoundly influenced by him. In the 18th century, it had been mostly the French school of applied mathematics that enabled the rapid developments of science and technology in Europe. How could this shift happen? It can be argued that the major reasons were the devastating consequences of the Napoleonic Wars in Central Europe, leading to the total defeat of Prussia in 1806. Immediately following, far-reaching reforms of the entire state system were carried out in Prussia and other German states, also affecting the educational system. It now guaranteed freedom of university teaching and research. This attracted many creative people with new ideas enabling the “golden age” of pure mathematics and fundamental theory in physical sciences.
Möbius’ legacy reaches far into today’s sciences, arts, and architecture. The famous one-sided Möbius strip is a paradigmatic example of the ongoing fascination with mathematical topology. This is the first book to present numerous detailed case studies on Möbius topology in science and the humanities. It is written for those who believe in the power of ideas in our culture, experts and laymen alike.
Author(s): Klaus Möbius, Martin Plato, Anton Savitsky
Publisher: Jenny Stanford Publishing
Year: 2022
Language: English
Pages: 924
City: Singapore
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
