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1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General RiemannianPowerPoint Presentation

1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian

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1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian

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1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian

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MATERIALS SCIENCE

&

ENGINEERING

Part of

A Learner’s Guide

AN INTRODUCTORY E-BOOK

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh

http://home.iitk.ac.in/~anandh/E-book.htm

SPACE

1D, 2D, 3D…. nD

Euclidian, Spherical, Hyperbolic

General Riemannian

SPACE

- A simplistic point of view is that space is a perfect inert vacuum and matter resides in that space.
- The real picture is more complicated, with actual space deviating from the ‘smooth picture’ at the very small lengthscales. Space at these lengthscales is teeming with virtual particles which fleetingly come into existence.
- There are unsolved questions regarding the number of dimensions and curvature of space we live in → though it is assumed to be 3D Euclidean locally* → At the scale of the universe it is understood to be non-Euclidean
- Many of the theories of physics which describe nature (e.g. the string theories) require higher dimensions (10 or more in some of them!). Dimensions higher than 3 are supposed to be ‘compactified’
- Apart from the possibility of higher spatial dimensions, in Einstein’s description of gravity, the 3D space is intertwined with time into a 4D ‘space-time unit’.
- In some theories, the structure of glasses and quasicrystals are described by hyper-dimensional constructs.

* refer next slide

SPACE continued…

- Gaussian curvature is the product of two orthogonal curvatures
- Mean curvature is the average of two orthogonal curvatures
- Space can be:Euclidean (flat) → Zero (0) Gaussian Curvature Spherical→ Positive (+) Gaussian Curvature Hyperbolic→ Negative () Gaussian Curvature
- If for a Hyperbolic surface the mean curvature is zero then the surface is called a Minimal Surface

Emphasis though it is assumed that space is: 3D → in general it can be nD Euclidean → it can be Non-Euclidean with local variations in the curvature of space