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102 results for “Symmetry” · Page 3 of 6
Beautiful Varieties · Lesson
… (square) j = 1728 — the square lattice, with an extra 4-fold symmetry.
String Theory · Lesson
Quantizing the string in the wrong dimension breaks Lorentz symmetry — a quantum anomaly — and the cancellation works out only in 26 dimensions for the bosonic string and 10 for the superstring .
Particle Physics · Lesson
Another consequence of the non-abelian SU(3) gauge symmetry of QCD.
Electromagnetism · Lesson
When combined with symmetry, Gauss's law determines the field from highly symmetric charge distributions in a single step.
Differential Geometry · Lesson
… Paraboloid K > 0 (elliptic) Monkey Saddle K < 0 with 3-fold symmetry Enneper Minimal surface: K ≤ 0 Cylinder K = 0 everywhere space-y-4 Surface Resolution: Gaussian Curvature K rgb(51, 102, 204) K < 0 (Saddle) K = 0 (Flat/Cylindrical) rgb(255, 102, 51) …
Loop Quantum Gravity · Lesson
… on the fixed metric too, so only its isometry subgroup is a symmetry, and that subgroup is what defines particles, energy and the vacuum: Minkowski: isometries = Poincaré ⟹ states labelled by (mass, spin) Generic g: isometries = trivial ⟹ no preferred vacuum, no …
Representation Theory · Lesson
Here we count necklaces up to rotational symmetry (cyclic group Z/n).
The Cube -- A Secret Page · Lesson
The Superflip (all edges flipped, order 2), checkerboard patterns (high symmetry), and many others demonstrate the artistic side of group theory.
Simple Groups · Lesson
Then twist: watch the diagram turn under its symmetry, and see the new family that the turn creates.
Sphere Packing · Lesson
… are beautifully ordered lattices like E₈, or amorphous disordered arrangements with no symmetry at all.
String Theory · Lesson
Note the symmetry: an interval and its complement share the same geodesic, so S(A) = S(complement of A), exactly as required for a pure global state.
Galois Theory · Lesson
Meanwhile, the absolute Galois group Gal(Q-bar/Q) -- the symmetry group of all algebraic numbers -- is one of the most mysterious and important objects in modern mathematics.
Information Theory · Lesson
Not a metric: KL fails the symmetry axiom and the triangle inequality, so it does not metrize the simplex.
Lie Groups · Lesson
Symmetry Breaking & Orbit Types Click any point to see its orbit under SO(2).
Differential Geometry · Lesson
… like a helical ramp Enneper Surface A self-intersecting minimal surface with 2-fold symmetry Scherk's Surface A doubly periodic minimal surface Catalan Surface A minimal surface named after Eugène Catalan, containing a cycloid as a geodesic Henneberg Surface A non-orientable minimal surface Richmond Surface A …
Ramsey Theory · Lesson
… sixteen-vertex colouring above is one of only two that work (up to symmetry), the honest test is to keep it and try to attach a seventeenth vertex without damage.
Mathematical Logic · Lesson
In type theory, this symmetry corresponds to constructors and deconstructors -- the Curry-Howard correspondence between proofs and programs.
Particle Physics · Module overview
… particles in 3D, build Feynman diagrams, watch the Mexican hat potential break symmetry, trace color charge flux tubes, and hunt for the Higgs boson in simulated collider events.
Minimal Surfaces · Lesson
… Rippled Disk A three-lobed wavy loop produces a rippled disk with threefold symmetry Two Circles → Catenoid Two parallel circles are spanned by a catenoid space-y-4 Area: Euler discovers the catenoid Leonhard Euler finds the first non-trivial minimal surface by solving the variational problem …
Beautiful Varieties · Lesson
… and it is a special point, the one place where the torus symmetry stands still.