1. Lucas Numbers & Golden Triangles in the Regular Pentagon
Steve Edwards
Southern Polytechnic State Universtiy

2. Area of Regular Plane Figures
Standard Formula
Heron’s Formula

3. Area of Pentagon with side s
Coxeter’s Geometry
Weisstein’s Mathworld
Koshy’s Fibonacci & Lucas Numbers

4. Euclid’s definition of Golden Ratio
When a line is cut in “extreme and mean” ratio, the ratio of the whole to the larger is the same as the ratio of the larger to the smaller.

5. Golden triangles

6. Theorem : A line drawn from one of the larger angles in a golden triangle cuts the opposite side in the golden ratio if and only if the line divides the triangle into obtuse and acute golden triangles.
Create a sequence of dissections by always dividing all the larger triangles in the pentagon

7. 1 3 4
2 1 3
4 7 11
3 4 7
The numbers of triangles of every type are always Lucas Numbers:
2, 1, 3, 4, 7, 11, 18, 29, 47, ...

8. The formula is also true for n non-positive.
Obtuse triangle area
Acute triangle area
7 obtuse 11 acute 18 total
The formula is also true for n non-positive.
Ratio of areas is golden

9. Fibonacci 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Lucas , 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...
2 + 5 = = 11

10. The Fibonacci numbers satisfy the well-known identity
Our area formula is
For n = 1,
Equate the two:
A Lucas identity:

13. Formulas for the area of the regular pentagon

15. Penrose Tiles tile the plane in an aperiodic fashion
Kite Dart