date: 2025-02-24
title: "2.24 Math Diary"
status: UNFINISHED
author:
  - AllenYGY
tags:
  - NOTE
publish: false

2.24 Math Diary

Post-Lecture Diary: Appreciation of Applied Mathematics – The Golden Ratio

Today’s lecture in the Appreciation of Applied Mathematics course was truly fascinating. The Golden Ratio, a concept that has been admired and studied for centuries, was the focus of discussion, and it was exciting to see its beauty unfold both mathematically and conceptually. Here's a summary and my reflections from the session:

The Golden Ratio: What is it?

The lecture began by introducing the Golden Ratio, also known by the symbol (phi). It arises from dividing a line segment into two parts such that the ratio of the whole length to the longer part is the same as the ratio of the longer part to the shorter part.

In simpler terms, if you have two segments and (where ), the Golden Ratio satisfies this proportion:

This unique value, , approximates to 1.6180339887…, an irrational number that continues infinitely without repeating.

Derivation of the Golden Ratio

To solidify our understanding, the professor walked us through the mathematical derivation of starting from the above relationship. Here’s how it went:

Step 1: Define the proportion

We define the relationship:

Let (normalize the larger segment to simplify the math) and (let represent the shorter segment as a proportion of the longer). Substituting, the proportion becomes:

Simplify:

Step 2: Rearrange into a quadratic equation

Multiply through by (assuming ) to eliminate the fraction:

Simplify:

Step 3: Solve the quadratic equation

Using the quadratic formula:

Here, , , and . Substituting these values:

Since , we take the positive root:

This value is the reciprocal of the Golden Ratio , so:

Applications of the Golden Ratio

After deriving , the lecture transitioned into its applications. It was amazing to learn how the Golden Ratio appears almost everywhere! For instance:

Art and Architecture: From the Parthenon in Greece to Da Vinci’s "Vitruvian Man," the Golden Ratio has been used to create visually pleasing proportions.
Nature: Spirals in sunflowers, pinecones, and seashells often follow the Fibonacci sequence, which is closely related to .
Mathematics: The Fibonacci sequence results in ratios that converge to as the sequence progresses:

It was breathtaking to see how something as abstract as a mathematical ratio finds expressions in our physical and aesthetic world.

Reflections and Thoughts

I really enjoyed the derivation process today. It reminded me how mathematics often reveals unexpected connections between seemingly unrelated fields. It’s incredible how —just a number—captures such universal harmony.

What struck me the most was the universality of the Golden Ratio, tying together nature, art, and science. My main takeaway is that mathematics is not just about numbers and equations; it’s a lens to understand the beauty and order of the universe.

I’m inspired to dig deeper into how the Fibonacci sequence and the Golden Ratio show up in the natural world. Perhaps I’ll look at spirals more closely the next time I see shells or flowers.