Hyperbolic geometry | Spheres | Polyhedra
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space. The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra β a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a circumscribed sphere. Using linear programming, it is possible to test whether a given polyhedron has an ideal version, in polynomial time. Every two ideal polyhedra with the same number of vertices have the same surface area, and it is possible to calculate the volume of an ideal polyhedron using the Lobachevsky function. The surface of an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique ideal polyhedron. (Wikipedia).
What are four types of polygons
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
What is the definition of a regular polygon and how do you find the interior angles
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
What is the difference between a regular and irregular polygon
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Geometric Techniques in Knot Theory - Jessica S. Purcell
Jessica S. Purcell Brigham Young University; von Neumann Fellow, School of Mathematics October 20, 2015 https://www.math.ias.edu/seminars/abstract?event=83224 We will discuss methods of decomposing knot and link complements into polyhedra. Using hyperbolic geometry, angled structures, a
From playlist Geometric Structures on 3-manifolds
Boris Springborn: Discrete Uniformization and Ideal Hyperbolic Polyhedra
CATS 2021 Online Seminar Boris Springborn, Technical University of Berlin Abstract: This talk will be about two seemingly unrelated problems: 00:46:00 A discrete version of the uniformization problem for piecewise flat surfaces, and 00:35:48 Constructing ideal hyperbolic polyhedra with p
From playlist Computational & Algorithmic Topology (CATS 2021)
Jessica Purcell: Structure of hyperbolic manifolds - Lecture 3
Abstract: In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions
From playlist Topology
Tropical Geometry - Lecture 4 - GrΓΆbner Bases and Tropical Bases | Bernd Sturmfels
Twelve lectures on Tropical Geometry by Bernd Sturmfels (Max Planck Institute for Mathematics in the Sciences | Leipzig, Germany) We recommend supplementing these lectures by reading the book "Introduction to Tropical Geometry" (Maclagan, Sturmfels - 2015 - American Mathematical Society)
From playlist Twelve Lectures on Tropical Geometry by Bernd Sturmfels
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
What are the names of different types of polygons based on the number of sides
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Lecture 15: General & Edge Unfolding
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 lecture begins with describing polyhedron unfolding for convex and nonconvex polygons. Algorithms for shortest path solutions
From playlist MIT 6.849 Geometric Folding Algorithms, Fall 2012
Gerard Cornuejols: Dyadic linear programming
A finite vector is dyadic if each of its entries is a dyadic rational number, i.e. if it has an exact floating point representation. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. This is joint work with Ahmad Abdi, Bertrand Guenin and Levent
From playlist Workshop: Continuous approaches to discrete optimization
Lecture 17: Alexandrov's Theorem
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 lecture addresses the mathematical approaches for solving the decision problem for folding polyhedra. A proof of Alexandrov's
From playlist MIT 6.849 Geometric Folding Algorithms, Fall 2012
What is the difference between convex and concave
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Jessica Purcell - Lecture 2 - Fully augmented links and circle packings
Jessica Purcell, Monash University Title: Fully augmented links and circle packings Fully augmented links form a family of hyperbolic links that are a playground for hands-on hyperbolic geometry. In the first part of the talk, Iβll define the links and show how to determine their hyperboli
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Nexus Trimester - John Walsh (Drexel University)
Rate Regions for Network Coding: Computation, Symmetry, and Hierarchy John Walsh (Drexel University) February 17, 2016 Abstract: This talk identifies a number of methods and algorithms we have created for determining fundamental rate regions and efficient codes for network coding proble
From playlist Nexus Trimester - 2016 - Fundamental Inequalities and Lower Bounds Theme
What is the difference between a regular and irregular polygons
π Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Thin Groups and Applications - Alex Kontorovich
Analysis and Beyond - Celebrating Jean Bourgain's Work and Impact May 21, 2016 More videos on http://video.ias.edu
From playlist Analysis and Beyond