Table Designs

I call these "Table Designs" even though they could be used to create stone figures of any size; it just happens that the most likely use of them would be in making tables. For example, I used 90° triangle design for the stone floor replacing Emily's interior garden described here. 

For each of these designs, we start with an isosceles triangle with some apex angle α (alpha) and inscribe a golden triangle into it with dimensions 'x' and 'y,' where x = y*φ. Each of the two sides 's' has a width of y*tan(.5 α). Golden rectangles can be stacked, each transcribed into the triangle bounded by the new base x.

To make a regular polygon, one places copies of the starting triangle at α degrees from each other, where α is a divisor of 360° (e.g., 120°, 90°, 72°, 60° ...)  corresponding to polygons with 360°/α sides (e.g., 3, 4, 5, 6 ...). Below are examples of the triangles and polygons generated in this way.




α = 120°

triangle





α = 90°

square







α  = 72°

pentagon







α = 60°

hexagon











α = 51.43°

heptagon









α = 45°

octagon










α = 40°

nonagon











α = 36°

decagon

Again, to generate a regular polygon, 360°/α has to be an integer. In this case, α = 54° and 360°/54 = 6.667. The ratio isn't an integer, the resulting figure is not a regular polygon.







I call it "wood lily," (Lilium philadelphicum),

 a six-petaled flower.