Euclidean solid geometry | Rotational symmetry | Lie groups | Rotation in three dimensions

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable, so it is in fact a Lie group. It is compact and has dimension 3. Rotations are linear transformations of and can therefore be represented by matrices once a basis of has been chosen. Specifically, if we choose an orthonormal basis of , every rotation is described by an orthogonal 3 × 3 matrix (i.e., a 3 × 3 matrix with real entries which, when multiplied by its transpose, results in the identity matrix) with determinant 1. The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3). The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its representations are important in physics, where they give rise to the elementary particles of integer spin. (Wikipedia).

Visual Group Theory, Lecture 2.2: Dihedral groups

Cyclic groups describe the symmetry of objects that exhibit only rotational symmetry, like a pinwheel. Dihedral groups describe the symmetry of objects that exhibit rotational and reflective symmetry, like a regular n-gon. The corresponding dihedral group D_n has 2n elements: half are rota

From playlist Visual Group Theory

7 Rotation of reference frames

Ever wondered how to derive the rotation matrix for rotating reference frames? In this lecture I show you how to calculate new vector coordinates when rotating a reference frame (Cartesian coordinate system). In addition I look at how easy it is to do using the IPython notebook and SymPy

From playlist Life Science Math: Vectors

Link: https://www.geogebra.org/m/D4hmNy9M

From playlist 3D: Dynamic Interactives!

From playlist GeoGebra 3D

The Quaternion Symmetry Group – Vi Hart

From playlist G4G11 Videos

Lecture 06: 3D Rotations and Complex Representations (CMU 15-462/662)

Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz2emSh0UQ5iOdT2xRHFHL7E Course information: http://15462.courses.cs.cmu.edu/

From playlist Computer Graphics (CMU 15-462/662)

Linear Algebra for Computer Scientists. 14. 3D Transformation Matrices

Most real time animated computer games are based on 3 dimensional models composed of thousands of tiny primitive shapes such as triangles, and each vertex in a model is encoded as a vector. In this computer science video you will learn how matrices are used to transform these vectors in th

From playlist Linear Algebra for Computer Scientists

Rotating 3d graph with xy-plane

From playlist 3d graphs

This clip gives describes a rotation matrix in 2D. The clip is from the book "Immersive Linear Algebra" available at http://www.immersivemath.com.

From playlist Chapter 6 - The Matrix

Toronto Deep Learning Series, 11-Feb-2019 https://tdls.a-i.science/events/2019-02-11 TENSOR FIELD NETWORKS: ROTATION- AND TRANSLATION-EQUIVARIANT NEURAL NETWORKS FOR 3D POINT CLOUDS We introduce tensor field neural networks, which are locally equivariant to 3D rotations, translations, a

From playlist Math and Foundations

Week 7 - Symmetry and Equivariance in Neural Networks - Tess Smidt

More about this lecture: https://dl4sci-school.lbl.gov/tess-smidt Deep Learning for Science School: https://dl4sci-school.lbl.gov/agenda

From playlist ML & Deep Learning

Tess Smidt: "Euclidean Neural Networks for Emulating Ab Initio Calculations and Generating Atomi..."

Machine Learning for Physics and the Physics of Learning 2019 Workshop I: From Passive to Active: Generative and Reinforcement Learning with Physics "Euclidean Neural Networks* for Emulating Ab Initio Calculations and Generating Atomic Geometries *also called Tensor Field Networks and 3D

From playlist Machine Learning for Physics and the Physics of Learning 2019

Christian Ott: Modeling the Death of Massive Stars

PROGRAM: NUMERICAL RELATIVITY DATES: Monday 10 Jun, 2013 - Friday 05 Jul, 2013 VENUE: ICTS-TIFR, IISc Campus, Bangalore DETAL Numerical relativity deals with solving Einstein's field equations using supercomputers. Numerical relativity is an essential tool for the accurate modeling of a wi

From playlist Numerical Relativity

Ara Sedrakyan - Three dimensional Ising model as a non-critical string theory

I will discuss the sign factor problem in the 3D gauge Ising model, present the corresponding fermionic model on random surfaces, which leads to the formulation of non-critical fermionic string theory on the basis of induced Dirac action. I will demonstrate how the sign factor model is lin

From playlist 100…(102!) Years of the Ising Model

Categorical aspects of vortices (Lecture 3) by Niklas Garner

PROGRAM: VORTEX MODULI ORGANIZERS: Nuno Romão (University of Augsburg, Germany) and Sushmita Venugopalan (IMSc, India) DATE & TIME: 06 February 2023 to 17 February 2023 VENUE: Ramanujan Lecture Hall, ICTS Bengaluru For a long time, the vortex equations and their associated self-dual fie

From playlist Vortex Moduli - 2023

Excel Hash - Interactive Bug Menu

This is my entry in the Excel Hash competition using the FREQUENCY and MAX Functions, plus 3D Models and Form Controls. VOTE for your favourite entry here: https://tinyurl.com/y9hmvelp WATCH the other awesome entries from my fellow Excel MVPs in the playlist here: https://www.youtube.com

From playlist Excel Hash 2018

Live CEOing Ep 117: Chemistry Functions in Wolfram Language

Watch Stephen Wolfram and teams of developers in a live, working, language design meeting. This episode is about Chemistry Functions in the Wolfram Language.

From playlist Behind the Scenes in Real-Life Software Design

Orbit retrieval, with applications to cryo-electron microscopy

From playlist Fall 2018 Symbolic-Numeric Computing