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

Area of Regular Plane Figures Standard Formula Heron’s Formula

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

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.

Golden triangles

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

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, ...

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

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

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

Formulas for the area of the regular pentagon

Penrose Tiles tile the plane in an aperiodic fashion Kite Dart