In this chapter, we develop patterns on a 12-6-4 grid. To begin, we will first construct the grid of dodecagons, hexagons and squares. Next, we overlay patterns using the PIC method.
Begin with a horizontal line and 3 circles as shown. This defines 6 points on the circumference of the center circle.
Now several lines are drawn for reference. Begin with the diagonals connecting points on the center circle, then proceed to draw the inner segments making a 6-pointed star. Also, the central rectangle is bolded which will contain the final unit that can be tiled.
The size of the hexagon can now be found as follows: First draw 2 circles about the midpoints of the two vertical edges of the unit. Then draw a circle, centered in the middle, that is tangent to these circles.
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Then draw a circle of that radius at the 6 points shown below.
Where these circles intersect the reference lines drawn, we can identify the sides of hexagons to inscribe.
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Now by filing in the squares between the hexagons the dodecagons will appear.
Prepared by circleofsteve.com The final grid appears below with the reference lines removed and the midpoints marked from which the incidence pattern can be drawn.
Prepared by circleofsteve.com Several angles can naturally be identified based on the existing geometry. For example, 63.5 connects the midpoint of a square side with an opposing corner.
Prepared by circleofsteve.com Continuing through all the polygons produces the following. Visible is the underlying grid and overlay design.
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