Visual geometry, geometric drawings ,
Squares and pentagons |
Extended lines of regular polygons.
Looking at the first intersection of the lines extending from regular polygons which has the shape of a geometric star.
The polygons are arranged in a circle and the pattern repeated all around.
Vesica piscis and regular polygons.
All the vesica piscis in those regular polygons are 4 regular polygons short of a full circle (Odd sided polygons) . In a triange for example the vesica piscis has two triangles , when a full circle would have 6. In a pentagon the vesica has 6 pentagons when the full circle has 10.
Triangle |
Hexagon dissections and the 6n+3 series
I found the inspiration for this ones looking at https://people.missouristate.edu/LesReid/HS175.html , which can also be found at :
https://en.m.wikipedia.org/wiki/File:Equilateral_pentagonal_dissection_of_regular_octadecagon.svg
which is a pentagonal dissection of a 18-gon.
I made this one with a hexagon at the centre and adding nonagons from the radius lines. I then use parallel lines to find the rest of the points needed to complete the drawing.
The following hexagonal dissections/tessellations are made using the nonagon (9-gon) , 15-gon and 21-gon ( 6n+3 series).
Regular polygons with odd numbers.
Looking at the diagonals of regular polygons, in this case odd number regular polygons.
Today I'm looking at the 3 smaller regular polygons that can be constructed radiating from the centre, that can be encased with another bigger regular polygon .Diagonal lines and circles are used to find defining points.
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 |