Can somebody PLEASE help me to understand Baravelle Spirals? Even the dummy/novice info on the National Curve Bank site isn't helping me to understand any better how exactly this works...it's really cool, but I'm having a hard time understanding the math behind it.
Question about geometric series: Baravelle Spirals
- Thread starter riverjib
- Start date
Baravelle spirals are about infiinite sums. See all those triangles?. They spiral infintiely toward the center. Let's say the first one on the outside has side length 1. Therefore, it's area is 1/2. The next triangle will have area 1/4, the next 1/8, etc.
\(\displaystyle \L\\S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+........\)
Factor out 1/2:
\(\displaystyle \L\\S=\frac{1}{2}(1+\underbrace{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...}_{\text{this is S}})\)
So, we have \(\displaystyle \L\\S=\frac{1}{2}(1+S)\)
S=1
It's a geometric series with sum 1. The total area of the infinite number of triangles sums to 1.
\(\displaystyle \L\\\sum_{n=1}^{\infty}\frac{1}{2^{n}}=\frac{1}{1-\frac{1}{2}}=1\)
\(\displaystyle \L\\S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+........\)
Factor out 1/2:
\(\displaystyle \L\\S=\frac{1}{2}(1+\underbrace{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...}_{\text{this is S}})\)
So, we have \(\displaystyle \L\\S=\frac{1}{2}(1+S)\)
S=1
It's a geometric series with sum 1. The total area of the infinite number of triangles sums to 1.
\(\displaystyle \L\\\sum_{n=1}^{\infty}\frac{1}{2^{n}}=\frac{1}{1-\frac{1}{2}}=1\)