Euclid's Elements, Book I Table of contents Definitions Definition 1. A point is that which has no part. Definition 2. A line is breadthless length. Definition 3. The ends of a line are points. Definition 4. A straight line is a line which lies evenly with the points on itself. Definition 5. A surface is that which has length and breadth only. Definition 6. The edges of a surface are lines. Definition 7. A plane surface is a surface which lies evenly with the straight lines on itself. Definition 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. Definition 9. And when the lines containing the angle are straight, the angle is called rectilinear. Definition 10. When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. Definition 11. Definition 12. Definition 13. Definition 14. Postulates
The Mathematics of Rainbows | Teaching Mathematics I’ve always been pretty fascinated by rainbows – anyone surely has to admit that it’s truely amazing how nature can construct such a beautiful image. After many years of wondering, I finally decided to spend the evening learning about them. The problem is that when I started to watch a few videos and read a few websites, they gave information without full explanations (is there anything more infuriating?). Don’t get me wrong, the videos are great for about 90% of the information but I felt like a bit more digging was needed to fully grasp the mathematics behind them. I would recommend that before you read this post, you watch the video that I started with. Then pop over to this website I read after watching the video. Question 1: When do rainbows occur? Easy one: When we have cloud, rain or spraying water and UV light. Question 2 (considerably longer answer): How do rainbows occur? Rainbows occur due to a combination of refraction and reflection. i – r + 2π – 2r + i – r = 2π + 2i – 4r
Inscribed and Circumscribed Triangles A circle that circumscribes a triangle is a circle containing the triangle such that the vertices of the triangle are on the circle. A circle that inscribes a triangle is a circle contained in the triangle that just touches the sides of the triangle. Circumscribing a triangle. Here is a method for constructing the circle that circumscribes a triangle. Draw the triangle. Draw the perpendicular bisector to each side of the triangle. Inscribing a triangle. Draw the triangle. Assignment: Draw two triangles of different shapes and then construct the circle that circumscribes them.
Triangle Centers Overview - Math Open Reference Triangle Centers - Overview Thousands of years ago, when the Greek philosophers were laying the first foundations of geometry, someone was experimenting with triangles. They bisected two of the angles and noticed that the angle bisectors crossed. They drew the third bisector and surprised to find that it too went through the same point. They must have thought this was just a coincidence. But when they drew any triangle they discovered that the angle bisectors always intersect at a single point! After some experimenting they found other surprising things. The points where these various lines cross are called the triangle's points of concurrency. Some triangle centers There are many types of triangle centers. In the case of an equilateral triangle, the incenter, circumcenter and centroid all occur at the same point. How many centers does a triangle have? Lots. Other triangle topics General Perimeter / Area Triangle types Triangle centers Congruence and Similarity Congruent triangles
A geometric proof of the impossibility of angle trisection by straightedge and compass One of the most well known problems from ancient Greek mathematics was that of trisecting an angle by straightedge and compass, which was eventually proven impossible in 1837 by Pierre Wantzel, using methods from Galois theory. Formally, one can set up the problem as follows. Define a configuration to be a finite collection of points, lines, and circles in the Euclidean plane. (Straightedge) Given two distinct points in , form the line that connects and , and add it to . We say that a point, line, or circle is constructible by straightedge and compass from a configuration if it can be obtained from after applying a finite number of construction steps. Problem 1 (Angle trisection) Let be distinct points in the plane. Thanks to Wantzel’s result, the answer to this problem is known to be “no” in general; a generic angle cannot be trisected by straightedge and compass. Start with three points .Form the circle with centre and radius , and intersect it with the line . Proof: If of over and around
Fun With Folding After attending a brilliant MoMath talk on Mathematical Origami given by Erik Demaine, I have been folding, cutting, and taping more than I ever thought I would. Here are a few of the ways I have been inspired. In addition, I have also posted some folding photos on my facebook page. Intoduction: Some Basic Mathematical Folding Basic Folds Simple demonstrations of the basic folds: a line through two points; midpoint of a segment; perpendicular bisector of a segment; angle bisector of an angle. Incenter of a Triangle Use basic folds to find the incenter of a triangle! Circumcenter of a Triangle Use basic folds to find the circumcenter of a triangle! Centroid of a Triangle Use basic folds to find the centroid of a triangle! Introduction: The One-Cut Challenge One-Cut Challenge: Triangles Start investigating the one-cut problem by playing around with triangles. One-Cut Challenge: Quadrilaterals Investigate the one-cut problem with squares, rectangles, and other quadrilaterals. Miscellaneous Folding