Nonagons (6n+3)

 

21-gons (6n+3)

 

The 6n+3 series

 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?

hexagon art

Hexagon art

Construction of a nonagon with compass and marked ruler

 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, 

The nonagon

 Connecting the crossing points of nonagons (dot to dot)

54-gon

 54-gon

9-gon, 18-gon, 27-gon, 54-gon

9-gon, 18-gon, 27-gon, 54-gon

13-gon

 13-gon

13-gon

13-gon

Nonagon art

Nonagon art

Filling of the plane with pentagons and hexagons

Filling of the plane with pentagons and hexagons

11-gon and triangle

 

Mapping of regular polygons :decagon

Mapping of regular polygons :decagon

Mapping of regular polygons : dodecagon

Mapping of regular polygons : dodecagon 

Mapping of regular polygons: octagon

 

√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 of regular polygons : hexagon

Mapping of regular polygons : hexagon

Mapping of regular polygons : pentagon

 The lines in a pentagon : one and the golden ratio ,

20°

 

Mapping of regular polygons :square

 

Mapping of regular polygons :square

Mapping of regular polygons : triangle

 

Mapping of regular polygons

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

49- gon

 49-gon and heptagon, squared numbers

14-gons

 

Decagons

Decagons

Heptagons

 

Heptagons

15-gon art design , 4n+3

 15-gon art , 4n+3

The nonagon and the square root 3

The nonagon and the square root 3

 

The nonagon and lines of 1 , 2 and square root 3. The circle in red has a radius of square root 3.

Nonagon and square root 3

 

The nonagon and the square root of 3

17-gon and square root 3

 

17-gon , 34-gon

 

17-gon , 34-gon

23-gon and triangles

 

23-gon and triangles

23-gon

 23-gon

23-gon

23-gon and hexagons

23-gon and squares

heptagon and nonagon

 

10-gon and 15-gon

 

The nonagon

 THE REGULAR NONAGON

The regular nonagon or 9-gon has 9 lines of equal length arranged around a middle point.

Figure 1:The nonagon