Golden Ratio: Algebraic Properties, Geometry, and Applications

All flashcards

How is the golden ratio defined using two positive lengths?

For $a>b>0$, $(a+b)/a=a/b=\varphi$.

Why is the golden ratio irrational?

Because $\varphi=(1+\sqrt{5})/2$ and $\sqrt{5}$ is irrational.

How is the reciprocal of the golden ratio expressed using $\varphi$?

$1/\varphi=\varphi-1\approx0.618$.

What happens to a golden rectangle when a square is removed from its longer side?

The remaining rectangle is similar to the original.

How does the exact golden spiral differ from a Fibonacci-square spiral?

The golden spiral is logarithmic; quarter-circle arcs in Fibonacci squares only approximate it.

Quiz questions

1. A rectangle has side lengths in the ratio $1:\varphi$. What is true after removing a square from its longer side?Answer: The remaining rectangle is similar to the original.
2. Approximately how large is the golden angle, $360^\circ/\varphi^2$?Answer: $137.5^\circ$
3. A logarithmic spiral is a golden spiral when its radius grows by what factor per quarter-turn?Answer: $\varphi$
4. How many vertices does a regular icosahedron have?Answer: 12
5. What evidence best supports a claim that an artwork deliberately uses golden proportions?Answer: Reliable documentation or measurements supporting the claimed proportions.