Triangular nested grids

Triangular nested grids:

3,3,3,3... or 5,5,5,...

Visual representation of Fermat numbers

 Images that show a visual representation of Fermat numbers and how those images can be used to create tessellations and other types of mathematical art.

The images are a combination of  visual geometry and visual number theory with an art component.

The images are based on square grids. 

The starting point for the Fermat numbers (3,5,17,257...) is a 3x3 square grid.

Geo Oct25, visual geometry

 

7,9,13,21...

27-gon neusis construction

 The construction of a 27-gon is based on the general construction methods for 6n+3 polygons.

Starting with the nonagon, 18-gon (double the nonagon) , and we can find the defining lines to construct the 27-gon.

27-gon neusis construction

Visual Geometry Sept 25

5,13,125,15133

Generalized Fermat Numbers, visualized

3,5,17 Fermat numbers, visualized

 

Geo June25

Visual geometry, april/may 2025

 

Visual geometry Feb/ March 25

Squares fitted in regular polygons

The process I initially follow , is to start with the regular polygon that I want to insert the squares in, draw a loose point in one of the sides and repeat on the other sides (same length). I draw the squares and move the original point until I find the meeting point of the squares. I use GeoGebra to make the drawings.

6n+3 series, visual geometry

 Nonagon

6n+3 series, visual geometry

Visual geometry Jan25

Visual geometry Nov-Dec24

 

8,24,56,120,248,504,1016,2040,4088,8184,16376

,32760...

Visual geometry Aug-Sep 24

Visual geometry July24

Visual geometry June24

Squares and pentagons

3d geometry: heptagon

 A geometric template to create a 3d geometric figure based on the heptagon:

A 3d shape created with heptagons

Visual geometry May 24

Geometric sphere

Square root 3 and 2

 Nonagons and hexagons circular expansion:

(6n+7)*3

 (6n+7)*3

14,21

(6n+3)*3 series

(6n+3)*3 series

(6n+5)*3 series

 (6n+5)*3 series

CUBES

 

CUBES

Pentagons and squares filling of the plane

 Pentagon and square filling of the plane,

Pentagon and 1/2 square root 3

 

Numbers:

Phi: (√(5) + 1) ÷ 2

1.53884176859

0.19320903898

0.09318924785

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?

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, 

Filling of the plane with pentagons and hexagons

Filling of the plane with pentagons and hexagons

11-gon and triangle

 

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 ,

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

Heptagons

 

Heptagons

15-gon art design , 4n+3

 15-gon art , 4n+3