The use of mathematics to read the book of nature. About Kepler and snowflakes

“Philosophy is written in that great book which ever lies before our eyes – I mean the universe – but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometric figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth,” wrote Galileo (Il Saggiatore, chapter 6, p. 4). In 1611, the mathematician Johannes Kepler,a contemporary of Galileo and voracious reader of the book of the world, wrote his shortest and most surprising book, The Six-Cornered Snowflake: A New Year’s Gift. “Even as I write these things, it has begun to snow again, and more thickly than before. I have been attentively observing the tiny particles of snow, and although they were all falling with pointed radii, they were of two kinds. Some were exceedingly small, with varying numbers of radii that spread in every direction and were plain, without tufts or striations. These were most delicate, but at the same time joined together at the center in a somewhat larger droplet; and they were the majority. Sprinkled among them were the rarer, six-cornered snowflakes” (Kepler, 1611). This text by Kepler, little known outside the physics and mathematics community, marked a milestone in the use of mathematics to understand a part of the physical world that surrounds us. With this text as a map, this article covers part of the terrain explored by geometry, from the 3rd century AD until today

The Introduction to Geometry by Qustā ibn Lūqā: Translation and Commentary

The paper contains an English translation with commentary of the Introduction to Geometry by the Christian mathematician, astronomer and physician Qustā ibn Lūqā. This elementary work was written in Baghdad in the ninth century A.D. It consisted of circa 191 questions and answers, of which 186 are extant today. The Arabic text has been published in a previous volume of Suhayl by Youcef Guergour, on the basis of the two extant Arabic manuscripts. The Introduction to Geometry consists mainly of material which QusÐā collected from Greek sources, some of which are now lost. Most of chapter 2 of the Jumal al-Falsafa by Abu Abdallah al-Hindi (12th century) was directly copied from QusÐā’s Introduction

Quantum Theory as Symmetry Broken by Vitality

I summarize a research program that aims to reconstruct quantum theory from a fundamental physical principle that, while a quantum system has no intrinsic hidden variables, it can be understood using a reference measurement. This program reduces the physical question of why the quantum formalism is empirically successful to the mathematical question of why complete sets of equiangular lines appear to exist in complex vector spaces when they do not exist in real ones. My primary goal is to clarify motivations, rather than to present a closed book of numbered theorems, and consequently the discussion is more in the manner of a colloquium than a PRL.Comment: 31 pages, 1 figure; the tone and topic of a Chris Fuchs samizdat, but 1/100th the size; v4: one passage reordered and another expanded in the pursuit of clarit

The transition from high school mathematics to engineering mathematics

Abstract: Mathematics is an essential course in the study of engineering. It can be argued that mathematics is the backbone of engineering. It is important for educators to have an understanding of the varying backgrounds of students and the way in which this affects their learning. This information will have an impact on teaching methods in the classroom which will ensure that they are inclusive and not exclusive. This is especially true in South Africa where we have a range of schools with different standards even though the final examination is the same. There seems to be a gap that exists between high school and first year engineering mathematics programs. First year engineering mathematics programs seem to present school learners joining the program with significant problems. This article attempted to identify whether there is a mathematics knowledge gap in South Africa. What the impact of this gap on engineering students was and who was responsible to address this problem. This study evaluated the differences between matric mathematics and university engineering mathematics 1. BTech and first year students interviewed and their opinion solicited with regard to the existence of a knowledge gap between matric mathematics and engineering mathematics 1. The pass rate for engineering mathematics 1 was compared over the past four years (2014 to 2017) to determine influence of CAPS on the engineering mathematics 1 results

monomials and polynomials the long march towards a definition

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