This is the Introductory lecture to a beginner’s course in Algebraic Topology given by N J Wildberger of the School of Mathematics and Statistics at UNSW in 2010.
This first lecture introduces some of the topics of the course and three problems.
If you are curious about how to make the interesting flap of paper (Problem 1), the solution can be found here. 🙂
Divine Proportions: Rational Trigonometry to Universal Geometry
Author: N J Wildberger
This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell’s law. Examples over finite fields are also included.
Check out this video on Graham’s Number — The World’s Biggest Number actually used in a mathematical proof! (Featuring Ron Graham himself!)
Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks
Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks–and the profound mathematical ideas behind them–that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today’s mathematical knowledge. For example, the Gilbreath Principle–a fantastic effect where the cards remain in control despite being shuffled–is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat’s last theorem.
Diaconis and Graham are mathematicians as well as skilled performers with decades of professional experience between them. In this book they share a wealth of conjuring lore, including some closely guarded secrets of legendary magicians. Magical Mathematics covers the mathematics of juggling and shows how the I Ching connects to the history of probability and magic tricks both old and new. It tells the stories–and reveals the best tricks–of the eccentric and brilliant inventors of mathematical magic.Magical Mathematics exposes old gambling secrets through the mathematics of shuffling cards, explains the classic street-gambling scam of three-card monte, traces the history of mathematical magic back to the thirteenth century and the oldest mathematical trick–and much more.
Students taking A Maths (Additional Maths) in 2014 should watch this: It may come out in the exam!
This is the newest addition to the new syllabus.
Practical Algebra: A Self-Teaching Guide, Second Edition
This is for Singaporean readers of my blog. Thanks for your reading and support. 🙂
Here is a good recommendation for those who take the train regularly, with chances to win cash prizes.
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Check out this video on the very interesting Mandelbrot Set:
Famously beautiful, the Mandelbrot Set is all about complex numbers. Featuring Dr Holly Krieger from MIT.
The Fractal Geometry of Nature
Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line. The complexity of nature’s shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry, the geometry of fractal shapes.
Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.
More than 80 or 90 per cent of students on four-year direct honours programmes at publicly-funded universities here graduate with honours or the equivalent. But only 60 per cent of those in the three-year arts and social sciences, business and science degree courses at the National University of Singapore (NUS) qualify for the fourth year of study, which allows them to graduate with honours.
To close the gap, NUS is lowering the grade to qualify for the honours year in these three schools, which are among the larger faculties in the university and take in some 3,600 students a year. This means another 10 to 15 per cent – 400 to 500 students- from these three faculties can move on to the fourth year to study for their honours.
Previously, students in the three faculties require a Cumulative Average Point (CAP) of 3.5 and above to qualify for honours study. With the change, they need only 3.2. NUS, though, will stick to its policy of keeping the the three plus one structure. Students who fail to notch up a score of at least 3.2 will have to exit the course.
NUS Provost Tan Eng Chye said the university decided to lower the requirement as the quality of students has gone up over the years. Students need As and Bs to enter most of the courses now. Last year, for example, students needed a ABB to enter the arts and social sciences course and those entering business needed triple As.
Source: NUS makes it easier for students in three faculties to qualify for honours
Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail