19-gons |
There are many ways of constructing/drawing geometric pictures.
This one is based on a repetition of a image . This image is then pasted in some particular points , in this case on the defining points of a regular polygon.
A lot of this images have been constructed before and are sometimes called polygon dissections. It is quite common in geometric drawing to reach to the same image using different ways of thinking.
Hexagons
What do the numbers 3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99... when it comes to regular polygons have in common?
A visual construction of a regular nonagon with compass and marked ruler (neusis construction of a nonagon).
Start with a hexagon and triangle as a base,
√2+1 lines, one being the side of the regular octagon. |
√ 2 |
√ 2 |
a2 + b2 = c2. (√ 2+1)2 +(1)2 = c2 |
Lines of one |
a2 + b2 = c2. (√2/2)2 + (√2/2+1)2 =c2 |
Lines of 2 |
Mapping regular polygons(from my point of view) is mainly about measuring lines and segments of diagonal lines of regular polygons. It is also about looking for patterns occurring in regular polygons. In an octagon we can easily find ratios of square root 2, finding ratios and patterns in other polygons is much more difficult.
Lines of one in an octagon |