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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/fibonacci
课程评论：没有评论
课程名称：斐波那契数与黄金比例 课程概述：本课程深入探讨斐波那契数、黄金比例及其相互关系。这些主题通常不在传统数学课程中教授，但却蕴含着许多引人入胜的结果，对于高级中学生而言，依然易于理解。课程的最后部分将解释斐波那契数为何在自然界中意外地出现，例如向日葵头部的螺旋数目。 课程大纲： 1. **斐波那契：它就像 1, 1, 2, 3 一样简单** - 描述：学习斐波那契数和黄金比例之间的关系，推导著名的宾奈特公式，该公式通过黄金比例及其倒数的幂提供了斐波那契数的显式公式，使我们能够计算第 n 个斐波那契数，而无需求和前面的数列。 2. **恒等式、求和与矩形** - 描述：学习斐波那契 Q 矩阵和卡西尼恒等式，了解卡西尼恒等式如何成为著名的拼图谬论和斐波那契骗局的基础。推导前 n 个斐波那契数的和以及前 n 个斐波那契数平方的和的公式。同时，展示如何构造黄金矩形，以及这如何形成美丽的螺旋方块图像。 3. **最非理性的数字** - 描述：学习黄金螺旋和斐波那契螺旋。由于斐波那契数与黄金比例之间的关系，斐波那契螺旋最终收敛于黄金螺旋。介绍连分数的概念，揭示黄金比例是最难以用有理数来近似的非理性数，并定义黄金角，使用它来建模向日葵头部的生长，从而解释斐波那契数在向日葵中的意外出现。 下载讲义：[点击这里](https://www.math.ust.hk/~machas/fibonacci.pdf) 观看宣传视频：[点击这里](https://youtu.be/VWXeDFyB1hc)
Name:Fibonacci: It's as easy as 1, 1, 2, 3
Description:We learn about the Fibonacci numbers, the golden ratio, and their relationship. We derive the celebrated Binet's formula, which gives an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprocal. This formula can be used to calculate the nth Fibonacci number without having to sum the preceding terms in the sequence.
Name:Identities, sums and rectangles
Description:We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for the famous dissection fallacy, the Fibonacci bamboozlement. A dissection fallacy is an apparent paradox arising from two arrangements of different area from one set of puzzle pieces. We also derive formulas for the sum of the first n Fibonacci numbers, and the sum of the first n Fibonacci numbers squared. Finally, we show how to construct a golden rectangle, and how this leads to the beautiful image of spiraling squares. This image is a drawing of a sequence of squares, each with side lengths equal to the golden ratio conjugate raised to an integer power, creating a visually appealing and mathematically intriguing pattern.
Name:The most irrational number
Description:We learn about the golden spiral and the Fibonacci spiral. Because of the relationship between the Fibonacci numbers and the golden ratio, the Fibonacci spiral eventually converges to the golden spiral. You will recognize the Fibonacci spiral because it is the icon of our course. We next learn about continued fractions. To construct a continued fraction is to construct a sequence of rational numbers that converges to a target irrational number. The golden ratio is the irrational number whose continued fraction converges the slowest. We say that the golden ratio is the irrational number that is the most difficult to approximate by a rational number, or that the golden ratio is the most irrational of the irrational numbers. We then define the golden angle, which is related to the golden ratio, and use it to model the growth of a sunflower head. The use of the golden angle in the model allows a fine packing of the florets, and results in the unexpected appearance of the Fibonacci numbers in the sunflower.
Learn the mathematics behind the Fibonacci numbers, the golden ratio, and how they are related. These topics are not usually taught in a typical math curriculum, yet contain many fascinating results that are still accessible to an advanced high school student. The course culminates in an explanation of why the Fibonacci numbers appear unexpectedly in nature, such as the number of spirals in the head of a sunflower. Download the lecture notes: https://www.math.ust.hk/~machas/fibonacci.pdf Watch the promotional video: https://youtu.be/VWXeDFyB1hc
