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Register a new userGeometry and Physics: An Overview
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David R. Morrison
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We present some episodes from the history of interactions between geometry and physics over the past century.
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In this survey paper we give an historical and at the same time thematical overview of the development of ring geometry from its origin to the current state of the art. A comprehensive up-to-date list of literature is added with articles that treat ring geometry within the scope of incidence geometry.
The book is a unique phenomenon in Russian geometry education. It was first published in 1892; there have been more than 40 revised editions, and dozens of millions of copies (by these parameters it is trailing only Euclids Elements). Our edition is based on 41st edition (the stable edition of Nil Aleksandrovich Glagolev; its been in public domain since 2015). At a few places we reverted changes to the earlier editions; we also made more accurate historical remarks.
David Mumford made groundbreaking contributions in many fields, including the pure mathematics of algebraic geometry and the applied mathematics of machine learning and artificial intelligence. His work in both fields influenced my career at several key moments.
We indicate that Herons formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms, and other elementary 4-dimensional solids.
The main focus of this paper is on models of quartic surfaces, especially so-called complex surfaces. These are special fourth-degree surfaces that Julius Plucker introduced in the 1860s for visualizing the local structure of a quadratic line complex. Pluckers complex surfaces turned out to be closely related to Kummer surfaces and both of these types of quartics are examples of caustic surfaces, which arise in geometrical optics. Indeed, Kummer surfaces represent a natural generalization of the wave surface, first introduced by Augustin Fresnel to explain double refraction in biaxial crystals.
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