Search MathAtlas
Explore modules, lessons, demo explanations, and narration. Follow connections between mathematical topics.
102 results for “Symmetry” · Page 2 of 6
Symmetry · Lesson
… bathroom tiles to crystal structures, has one of exactly 17 kinds of symmetry.
Aperiodic Tilings · Lesson
… at the midpoint sits Tile(1, 1), the spectre's skeleton, where the extra symmetry of equal lengths lets reflected copies combine into periodic tilings.
Topology · Lesson
It then turns the pages with the symmetry (z, w) ↦ (e iqφ z, e ipφ w), which multiplies f by e ipqφ .
Algebraic Geometry · Lesson
Find the Barth sextic's 65 nodes and its icosahedral symmetry, and turn on the clipping plane to look inside.
Aperiodic Tilings · Module overview
See also: Symmetry for the wallpaper groups a periodic tiling must belong to, Fractals for other things built by substitution, and Sphere Packing for another question about filling space that stayed open for centuries.
Beautiful Varieties · Module overview
… walk through the show-stoppers of algebraic geometry: world-record singular surfaces with icosahedral symmetry, amoebas and tropical curves, the 3264 conics tangent to five conics, the undrawable Hesse configuration, the ADE singularities, elliptic curves as doughnuts, polygons that are secretly spaces, and the wild ovals …
Random Matrices · Lesson
… and you get a new law with real practical use; drop the symmetry and the eigenvalues leave the line.
Differential Equations · Lesson
Pitchfork Bifurcation Pitchfork bifurcations arise in systems with symmetry.
String Theory · Lesson
Mirror Symmetry Press Swap mirror pair and compare the two shapes and their Hodge numbers.
Group Theory · Lesson
Named after Arthur Cayley, these diagrams reveal symmetry, subgroup structure, and the "shape" of a group in a way that multiplication tables cannot.
Representation Theory · Lesson
… Theory -- the art of turning abstract groups into matrices, where every symmetry becomes a linear transformation you can actually compute with.
Combinatorics · Lesson
Notice the symmetry: C(n, k) = C(n, n-k), because choosing which k to include is equivalent to choosing which n-k to exclude.
Complex Analysis · Lesson
… at z = nπ for integer n Periodic with period 2π Odd symmetry: sin(-z) = -sin(z) Zeros at z = (2n+1)π/2 for integer n Periodic with period 2π cos(z) = sin(z + π/2) Poles at z = (2n+1)π/2 (where cos(z) = 0) Zeros at …
Complex Analysis · Lesson
… rounding errors may occur) All roots of unity: ω = 🔷 Beautiful Symmetry: The nth roots of unity are evenly spaced around the unit circle, forming a perfect regular polygon.
Simple Groups · Lesson
… built from the previous one, and the sporadic groups emerge naturally as symmetry groups at each stage.
Particle Physics · Lesson
Each one comes from a symmetry of nature, a result known as Noether's theorem.
Machine Learning · Lesson
… shows up in physics, signal processing, and ML wherever translations are a symmetry.
String Theory · Lesson
Stacks of Branes and U(N) Gauge Symmetry Raise N to grow the gauge group from U(1) to U(4).
String Theory · Lesson
The Self-Dual Point: Enhanced Symmetry Slide R toward √α′ and watch the rungs merge.
Sphere Packing · Lesson
… function out of modular forms — objects from number theory with extreme symmetry — engineered so that the function and its Fourier transform both vanish to second order at every distance occurring in E₈.