13. Dodecagon-Hexagon-Square

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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. Prepared by circleofsteve.com 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. Prepared by circleofsteve.com 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. Prepared by circleofsteve.com Prepared by circleofsteve.com Prepared by circleofsteve.com .

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by B S Beevers The results so far may be put into a table as follows: Number of Sides 3 4 5 6 7 8 9 10 11 12 Number of rt. angles 1 4 3 5 5 6 7 7 8 9 etting it right Now polygons with 3, 4 and 5 sides are special cases, after which there appears to be a clear pattern for the maximum A familiar investigation is to find the maximum numbers of number of right angles. interior angles of a polygon which can be right angles. This may be investigated by an able pupil as follows: Analytical Approach A triangle can have one right angle. A quadrilateral can have four right angles. For a polygon with n sides, sum of interior angles = 180n - 360 degrees. Pentagon Suppose it has p right angles. Number of degrees remaining for other angles = 180n - 360 Sum of Interior Angles = 540'. - 90p. Four right angles would leave 180', which is impossible. Average size of remaining angles must be less than 360'. So a pentagon has a maximum of three right angles, as shown. Hence: 180n - 360 - 90p < 360 n-p Hence 180n - 360 - 90p < 360n - 360p - 270p < 180n + 360 1 3 Hexagon -p< - (2n + 4) n = 3, 4 and 5 are special cases. Sum of Angles = 720'. For n > 5, the maximum number of right-angles is the 5 right angles = 450', leaving 270'. largest integer less than (2n + 4) So a hexagon can have 5 right-angles, as shown. Note that p cannot equal (2n + 4) or the average remaining angle would equal 360.
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