# Mathematics

### The geometry of planetary orbits and ellipses

We all "know" that Newton showed that planetary orbits are elliptical because they are subject to the inverse square law of gravity. This document explains Newton's proof and contains a comprehensive presentation of the Euclidean geometry of ellipses that is used in the proof.

### A gentle introduction to Lean

Lean is a proof assistant that can be used to formalize mathematical proofs.

Bagrut exams (in Hebrew)

Solutions to all the questions on the Israeli Bagrut examinations in mathematics (806 / 581) from years ayin-dalet to ayin-het. A separate document contains a rewritten and expanded version of the solutions to the problems in probability.

Mosteller's challenging problems in probability

This is a "reworking" of Fifty Challenging Problems in Probability with Solutions by Fredrick Mosteller: simplifying the problems, dividing them into subproblems, providing hints and including additional detail, calculations and diagrams.

### Markov chain simulations

Simulations written in Python 3 (with graphs) of problems in Markov chains: gambler's ruin, random walk (1D, 2D, 3D), Ehrenfest model, two-state process.

### A functional approach to teaching trigonometry

This document is a guide for teachers that examines teaching trigonometry initially as functions defining by "winding" a thread around the unit circle. The Hebrew document was written by Avital Elbaum-Cohen, at the Department of Science Teaching, Weizmann Institute of Science. The English document is my translation and adaptation of her work.

### Geometry without words

An A2 sized poster with 15 theorems of secondary-school geometry displayed visually in color, but without words.

### The many guises of induction

A tutorial on mathematical induction.

### Minesweeper is NP-Complete

A detailed presentation of Richard Kaye's  result that the minesweeper game is NP-complete.

### Three-dimensional rotations

Tutorial on Euler angles and quaternions for describing rotations in three dimensions.

### TikZ examples

Txamples of TikZ diagrams for Euclidean geometry: (1) the intersection of the perpendicular bisectors of a triangle is the center of the circumscribed circle; (2) the intersection of the medians of a triangle divide the medians in the ration 2:1; (3) Ptolemy's theorem relating the lengths of the diagonals and the lengths of the sides of a quadrilateral that is circumscribed by a circle; (4) Ramanujan's construction 355/113, an excellent approximation to pi.