Science Notes

by Greg Egan

• Symmetries and the 24-cell | Average Shadows | Ball Bearings in a Hypersphere | Black Hole Passing | Born Rigid Motion | Catacaustics | Resonant Modes of a Conical Cavity | The Doppler Shift Triangle Law | Conic Section Orbits | Double-Crossing Orbits | Dihedral Angles | The Ellipse and the Atom | Find The Fake Coin | The Finite Fall | General Relativity in 2+1 dimensions | Howell’s Moving Orbits | Klein's Quartic Curve | Paparazzo vs Bodyguards | Light Mill applet | Littlewood applet | LRL made easy | The Mermin-Peres Magic Square Game | There Are No Corkscrew Orbits | Peeling the Octonions | Ordered Sums | Optimised Origami | Weak-field GR near a planar mass | Polar Orbits Around Binary Stars | Planes, Spheres and Pseudospheres | The Rindler Horizon | Rotating Elastic Rings, Disks and Hoops | Constant Weight Rollercoasters | Steffen’s Polyhedron | Superpermutations | Symmetric Waves | Symplectic applet | The Tell-Tale Board | Exact values of regular 10j symbols
• The Book of All Skies | “Border Guards” | Diaspora | Dichronauts | “Didicosm” | Distress | Essays | Foundations | Horror Stories | Incandescence | Miscellaneous Fiction | Morphotrophic | “Oceanic” | Orthogonal | Permutation City | “The Planck Dive” | Quarantine | Scale | Schild’s Ladder | Science Notes | Teranesia | Zendegi
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This section contains notes on some miscellaneous scientific matters.

• Symmetries and the 24-cell
  • The 24-cell from symmetries in three dimensions
  • Symmetries of the 24-cell
• Average Shadows
  • Cubes
  • Other Polyhedra
  • Average Heights
  • Non-convex solids
• Ball Bearings in a Hypersphere
  • General Considerations
  • An Example Based on the 4-cube
  • The Dimension of the Solution Space
    • The Solution Space in R³
    • The Solution Space in S³
    • The 4-cube Example
  • Exploiting Symmetry
    • Representations and Invariant Subspaces
    • Schur’s Lemma
    • Projections into Isotypic Subspaces
    • Interpreting the Isotypic Degrees of Freedom
      • The 3-dimensional Irrep
      • The 4-dimensional Irrep
    • The Geometry of the Solutions
    • Three Cheers for Frobenius!
    • The 120-cell Example
  • Arbitrary Manifolds
  • Infinite Lattices
    • The Lattice Z³
      • The Geometry of the Gears
    • The Lattice Zⁿ
  • References
• Black Hole Passing
• Born Rigid Motion
  • Hyperbolic rotation
  • Generalising the construction
  • Ordinary rotation
  • Mixed rotation
  • Null rotation
• Catacaustics, Resultants and Kissing Conics
  • Catacaustics of Parameterised Curves
  • Catacaustics of Polynomial Curves
  • Kissing Conics
  • Resultants and Discriminants
  • Finding the Catacaustic with Resultants
  • References
• Resonant Modes of a Conical Cavity
  • Field solutions
  • Energy, pressure and forces
  • Currents, heating and Q factors
  • Proof of zero force for any shape of cavity
  • References
• A Simple Asymmetric Resonant Cavity
• The Doppler Shift Triangle Law
• Conic Section Orbits
  • Notation
  • The Whispering Gallery Proof
  • The Pins, String and Pencil Proof
  • The Focus-Directrix Proof
  • From Shape To Force Law
  • The Squared Harmonic Oscillator Proof
  • Dandelin Spheres
    • The Pins, String and Pencil Property
    • The Focus-Directrix Property
    • The Whispering Gallery Property
    • The Inverse Kinetic Energy Pictured as a Square
  • References
• Double-Crossing Orbits
  • Newtonian orbits
    • Returning to the same point
    • Distinct points
  • References
• Dihedral Angles
  • Basic formulas
  • Application to Regular Polyhedra
  • Generalisation to Regular Polytopes
• The Ellipse and the Atom
  • The Velocity Circle
  • Orbits with Equal Energies
  • Quantum Spin in Three Dimensions
  • Quantum Spin in Four Dimensions
  • Fock’s Momentum Space Projection
  • References
• Find The Fake Coin
  • Analysis of the problem
  • CoinBot
• The Finite Fall
  • Time to the Horizon
  • Time to the Singularity
  • Round Trip Rescue
  • What Times are Infinite?
  • References
• General Relativity in 2+1 dimensions
  • Basic concepts
  • Multi-particle systems
  • Collisions
• Null Rotations in 2+1 dimensions
  • Five roads to null rotations
  • Properties of null rotations
• Howell’s Moving Orbits
• Klein’s Quartic Curve
• Klein’s Quartic Equation
  • Topology
  • Symmetries
  • Visualising the full solution
  • Parameterising the solution
• Paparazzo vs Bodyguards
• Light Mill applet
• Hot Plate applet
• Light Mill results
  • Locked rotor
  • Varying the rotor’s outer radius
• Hot Plate results
  • Displacement vs particle number
  • Force on locked plate vs particle number
• Littlewood applet
  • Palette
  • Screenshots
• Littlewood videos
• LRL made easy
  • The Laplace-Runge-Lenz vector from Noether’s theorem
• The Mermin-Peres Magic Square Game
  • The Magic Square Game
  • The Quantum Mechanics of Spin
  • The Mermin-Peres Square
  • The 24-cell of Eigenvectors
  • References
• There Are No Corkscrew Orbits
  • Numerical Simulations
  • Analytical Calculations
  • Problems with the paper
  • References
• Peeling the Octonions
• Ordered Sums
• Optimised Origami
• Weak-field GR near a planar mass
  • Preliminaries
  • Acceleration towards the plane
  • Tidal effects
  • References
• Polar Orbits Around Binary Stars
  • References
• Planes, Spheres and Pseudospheres
  • Assumptions
  • Gaussian Curvature
  • Intrinsically Flat Surfaces
    • Planes
    • Cylinders
    • Cones
    • Tangent surfaces
    • Ruled surfaces
    • Rectifying developables
    • Möbius strips
      • Möbius strips and Klein’s bottles in four dimensions
  • Intrinsically Curved Surfaces
    • Surfaces of Revolution
    • Surfaces Built From Helices
      • Dini’s surface
      • Other Surfaces Built From Helices
    • The sine-Gordon Equation
      • Exact solutions of the sine-Gordon Equation
    • Choose your own geodesic
  • Appendix: Computing Gaussian Curvature
    • Gaussian curvature of an embedded surface
    • Gaussian curvature from a metric
  • References
• The Rindler Horizon
  • Preliminaries
  • Static situations
  • Free fall
  • Dynamic situations
  • String unreeled at a constant rate
  • Conclusions
• Relativistic Elasticity
• Rotating Elastic Rings, Disks and Hoops
  • Preliminaries
  • Rings and disks
  • Hoops
  • Energy and angular momentum
  • Pulsations, vibrations and stability
    • Pulsations of a Newtonian hoop
    • Pulsations of a relativistic hoop
    • Vibrations of a Newtonian hoop
    • Vibrations of a relativistic hoop
  • Limitations of the model
• Newtonian Hoop applet
• Constant Weight Rollercoasters
  • Rockets, Rollercoasters and Cyclotrons
    • Rollercoaster Trajectories
    • Tipped-over Cyclotron Trajectories
  • Symmetries and Conserved Quantities
    • Symmetries of the Euler-Lagrange Equations
    • The Tipped-over Cyclotron Lagrangian
    • Symmetry and the Rollercoaster
  • References
• Steffen’s Polyhedron
• Superpermutations
  • The Recursive Algorithm
  • Minimal Length
  • Superpermutations as Paths through a Graph
  • The Williams Construction
  • Local Rules
  • 2-cycle graphs
  • Breaking News
  • Breaking News 2
  • References
• Symmetric Waves
  • Solid Harmonics and Spherical Harmonics
  • Dodecahedral Wave Functions
  • Tetrahedral Wave Functions
  • Four-dimensional Harmonics
  • References
• Symplectic applet
• The Tell-Tale Board
• Exact values of regular 10j symbols

Other scientific topics are discussed in the sections of this web site relating to particular works of science fiction, in the notes for some of my applets, and in four essays I wrote on the Foundations of modern physics.

• Symmetries and the 24-cell | Average Shadows | Ball Bearings in a Hypersphere | Black Hole Passing | Born Rigid Motion | Catacaustics | Resonant Modes of a Conical Cavity | The Doppler Shift Triangle Law | Conic Section Orbits | Double-Crossing Orbits | Dihedral Angles | The Ellipse and the Atom | Find The Fake Coin | The Finite Fall | General Relativity in 2+1 dimensions | Howell’s Moving Orbits | Klein's Quartic Curve | Paparazzo vs Bodyguards | Light Mill applet | Littlewood applet | LRL made easy | The Mermin-Peres Magic Square Game | There Are No Corkscrew Orbits | Peeling the Octonions | Ordered Sums | Optimised Origami | Weak-field GR near a planar mass | Polar Orbits Around Binary Stars | Planes, Spheres and Pseudospheres | The Rindler Horizon | Rotating Elastic Rings, Disks and Hoops | Constant Weight Rollercoasters | Steffen’s Polyhedron | Superpermutations | Symmetric Waves | Symplectic applet | The Tell-Tale Board | Exact values of regular 10j symbols
• The Book of All Skies | “Border Guards” | Diaspora | Dichronauts | “Didicosm” | Distress | Essays | Foundations | Horror Stories | Incandescence | Miscellaneous Fiction | Morphotrophic | “Oceanic” | Orthogonal | Permutation City | “The Planck Dive” | Quarantine | Scale | Schild’s Ladder | Science Notes | Teranesia | Zendegi
• Back to home page | Site Map | Side-bar Site Map