Geometric dissection | Polyhedra
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other. Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal.A polyhedron can be cut up and reassembled to tile space if and only if its Dehn invariant is zero, so having Dehn invariant zero is a necessary condition for being a space-filling polyhedron. The Dehn invariant of a self-intersection-free flexible polyhedron is invariant as it flexes. The Dehn invariant is zero for the cube but nonzero for the other Platonic solids, implying that the other solids cannot tile space and that they cannot be dissected into a cube. All of the Archimedean solids have Dehn invariants that are rational combinations of the invariants for the Platonic solids. In particular, the truncated octahedron also tiles space and has Dehn invariant zero like the cube. The Dehn invariants of polyhedra are not numbers. Instead, they are elements of an infinite-dimensional tensor space. This space, viewed as an abelian group, is part of an exact sequence involving group homology. Similar invariants can also be defined for some other dissection puzzles, including the problem of dissecting rectilinear polygons into each other by axis-parallel cuts and translations. (Wikipedia).
Lorentz Covariance VS Lorentz Invariance: What's the Difference? | Special Relativity
In special relativity, Lorentz covariance and Lorentz invariance are two very important concepts. But what exactly are these concepts? In this video, we will find out! Contents: 00:00 Definitions 00:51 Examples If you want to help us get rid of ads on YouTube, you can support us on Patr
From playlist Special Relativity, General Relativity
Symmetry in Physics | Noether's theorem
▶ Topics ◀ Global / Local Symmetries, Continuous / Discrete Symmetries ▶ Social Media ◀ [Instagram] @prettymuchvideo ▶ Music ◀ TheFatRat - Fly Away feat. Anjulie https://open.spotify.com/track/1DfFHyrenAJbqsLcpRiOD9 If you want to help us get rid of ads on YouTube, you can support us on
From playlist Symmetry
Daniel Pomerleano: Degenerations from Floer cohomology
The lecture was held within the framework of the Hausdorff Trimester Program: Symplectic Geometry and Representation Theory. Abstract: I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log Calabi-Yau varieties to the Sta
From playlist HIM Lectures: Trimester Program "Symplectic Geometry and Representation Theory"
Bertrand Eynard - An overview of the topological recursion
The "topological recursion" defines a double family of "invariants" $W_{g,n}$ associated to a "spectral curve" (which we shall define). The invariants $W_{g,n}$ are meromorphic $n$-forms defined by a universal recursion relation on $|\chi|=2g-2+n$, the initial terms $W_{0,1}$
From playlist Physique mathématique des nombres de Hurwitz pour débutants
An introduction to Invariant Theory - Harm Derksen
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From playlist Mathematics
Matthias Goerner's 3D print: http://shpws.me/SZbN Countdown d24: https://youtu.be/U0soSn7BojQ Matthias' version of the construction of the polyhedron: http://www.unhyperbolic.org/sydler.html Demonstration of the Wallace–Bolyai–Gerwien theorem by Dima Smirnov and Zivvy Epstein: https://dmsm
From playlist 3D printing
http://www.teachastronomy.com/ Deduction is a way of combining observations or statements made in science logically. Deduction provides a very strong way of connecting observations with a conclusion. Typically we start with premises and combine them to draw conclusions. For example, if
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MIT 6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Fall 2012 View the complete course: http://ocw.mit.edu/6-849F12 Instructor: Erik Demaine This class focuses on hinged dissections. Examples of hinged dissections and several built, reconfigurable applications are offere
From playlist MIT 6.849 Geometric Folding Algorithms, Fall 2012
Abstract Analogues of Flux as Symplectic Invariants - Paul Seidel
Paul Seidel Massachusetts Institute of Technology November 16, 2012 For more videos, visit http://video.ias.edu
From playlist Mathematics
Commutative algebra 4 (Invariant theory)
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. This lecture is an informal historical summary of a few results of classical invariant theory, mainly to show just how complic
From playlist Commutative algebra
From playlist Linear Algebra Ch 8 (updated Jan2021)
Allison Moore - Essential Conway spheres and Floer homology via immersed curves
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Allison Moore, Virginia Commonwealth University Title: Essential Conway spheres and Floer homology via immersed curves Abstract: We consider the problem of whether Dehn surgery along a knot in the three-sphere produces an
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
Cheuk Yu Mak: Spherical Lagrangian submanifolds and spherical functors
The lecture was held within the framework of the Hausdorff Trimester Program: Symplectic Geometry and Representation Theory. Abstract: Spherical twist is an auto equivalence of a category whose definition is motivated from the Dehn twist along a Lagrangian submanifold inside a symplectic
From playlist HIM Lectures: Trimester Program "Symplectic Geometry and Representation Theory"
Yi Xie - Surgery, Polygons and Instanton Floer homology
June 20, 2018 - This talk was part of the 2018 RTG mini-conference Low-dimensional topology and its interactions with symplectic geometry Many classical numerical invariants (including Casson invariant, Alexander polynomial and Jones polynomial) for 3-manifolds or links satisfy surgery fo
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Taming the hydra: the Word Problem, Dehn functions, and extreme integer compression - Timothy Riley
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From playlist Mathematics
Classification of 2-manifolds and Euler characteristic | Differential Geometry 26 | NJ Wildberger
We describe the important classification of compact, oriented 2-manifolds, and the relation with the topological invariant called the Euler characteristic. The idea is to work combinatorially, by decomposing a 2-manifold into polygon pieces which are glued, or identified, along common edge
From playlist Differential Geometry
Ignat Soroko - Groups of type FP: their quasi-isometry classes and homological Dehn functions
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Ignat Soroko, Louisiana State University Title: Groups of type FP: their quasi-isometry classes and homological Dehn functions Abstract: There are only countably many isomorphism classes of finitely presented groups, i.e.
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
Hyperbolic 3-Manifold Stories - Professor Robert Meyerhoff (Boston College)
A brief historical outline of selected results on hyperbolic 3-manifolds will be given, leading up to recent research. Topics: low-volume manifolds, hyperbolic Dehn Filling spaces, Chern-Simons invariant.
Simplify the Negation of Statements with Quantifiers and Predicates
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From playlist Symbolic Logic and Proofs (Discrete Math)
Veering Dehn surgery - Saul Schleimer
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From playlist Mathematics