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.
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.
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| 27-gon neusis construction |
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.
A geometric template to create a 3d geometric figure based on the heptagon:
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| A 3d shape created with heptagons |
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,
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.
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| Lines of one in an octagon |
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| The nonagon and the square root 3 |
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| The nonagon and lines of 1 , 2 and square root 3. The circle in red has a radius of square root 3. |
