Purpose
Increase students. familiarity with the Golden Ratio and its relationship to the Fibonacci sequence
Objectives
Students will be able to construct a Golden Rectangle, compute using the Golden ratio and see Fibonacci
connections.
Activities
- Knowing that Ø= √5/2 + 1/2, show that 1/Ø = √5/2 - 1/2
- Construct a Golden Rectangle. Start with a 2-unit square, mark a segment of length √5, strike an arc ending
to the level of the base and extend the base
- Sketch a graph of quotients of Fibonacci numbers, fn+1/fn. Use the values from the table. Using graph paper,
mark the horizontal scale from 1 to 14 in units of 1 and the vertical axis from 1 to 2 in units of 1/25 (0.04). Put a
dot at each pair (n, fn+1/fn) and connect the dots. Lable Ø on the graph.
- Make a table of values for powers of Ø divided by √5 and find a pattern for the values of the nearest
integer.
| n | Øn/√5 | Nearest Integer
|
|---|
| 1 | Ø/√5 = 0.72360... | 1
|
| 2 | Ø2/√5 = 1.1782... | 1
|
| 3 | Ø3/√5 = 1.89443... | 2
|
| 4 | Ø4/√5 = 3.06525... | ..
|
| 5 | Ø5/√5 = 4.95967... | ..
|
| 6 | Ø6/√5 = | ..
|
| 7 | Ø7/√5 = | ..
|
| 8 | Ø8/√5 = | ..
|
- Fill in the table to find another pattern:
| n | (Øn + 1/Øn)/√5 | Nearest Integer
|
|---|
| 1 | (Ø + 1/Ø)/√5 | 1
|
| 2 | (Ø2 + 1/Ø2)/√5 | 1
|
| 3 | (Ø3 + 1/Ø3)/√5 | 2
|
| 4 | (Ø4 + 1/Ø4)/√5 | ..
|
| 5 | (Ø5 + 1/Ø5)/√5 | ..
|
| 6 | (Ø6 + 1/Ø6)/√5 | ..
|
| 7 | (Ø7 + 1/Ø7)/√5 | ..
|
| 8 | (Ø8 + 1/Ø8)/√5 | ..
|
- Finally, consider the optical illusion called Luis Carroll's paradox.
Does the area really increase from 82 = 64 to 5*13 = 65? Cut up an 8" square and reassemble to find out
.
