Scaling multiplies lengths by the same factor and produces a similar figure. It preserves angles and ratios of lengths of corresponding line segments. Animation shows what happens to perimeters, areas, and volumes under scaling, with various applications from real life.
Although pi is the ratio of circumference to diameter of a circle, it appears in many formulas that have nothing to do with circles. Animated sequences dissect a circular disk and transform it to a rectangle with the same area as the disk. Animation shows how Archimedes estimated pi using perimeters of approximating polygons.
The Theorem of Pythagoras: Several engaging animated proofs of the Pythagorean theorem are presented, with applications to real-life problems and to Pythagorean triples. The theorem is extended to 3-space, but does not hold for spherical triangles.
Sines and cosines occur as rectangular coordinates of a point moving on a unit circle, as graphs related to vibrating motion, and as ratios of sides of right triangles. They are related by reflection or translation of their graphs. Animation demonstrates the Gibbs phenomenon of Fourier series.
This video focuses on trigonometry, with special emphasis on the law of conies and the law of sines, together with applications to The Great Survey of India. The history of surveying instruments is outlined, from Hero’s dioptra to modern orbiting satellites.
Animation relates the sine and cosine of an angle with chord lengths of a circle, as explained in Ptolemy’s Almagest. This leads to elegant derivations of addition formulas, with applications to simple harmonic motion.
Animation shows how the Cartesian equation changes if the graph of a polynomial is translated or subjected to a vertical change of scale. Zeros, local extrema, and points of inflection are discussed. Real-life examples include parabolic trajectories and the use of cubic splines in designing sailboats and computer-generated teapots.
This video describes a remarkable engineering work of ancient times: excavating a one-kilometer tunnel straight through the heart of a mountain, using separate crews that dug from the two ends and met in the middle. How did they determine the direction for excavation? The program gives Hero's explanation (ca. 60 A.D.), using similar triangles, as well as alternative methods proposed in modern times.
Early History of Mathematics: This video traces some of the landmark developments in the early history of mathematics, from Babylonian calendars on clay tablets produced 5000 years ago, to the introduction of calculus in the seventeenth century.
