Dividing a 90 Degree Angle into Five Equal Angles

Chris Author

How to divide a ninety degree angle into five equal ( 18˚ ) angles in 10 easy steps.

When it comes to designing five-fold Islamic geometric patterns, being able to establish 18˚ angles is essential to the pattern's construction. The method outlined below uses dynamic symmetry and by association the golden ratio to divide a 90˚ angle into five equal angles of 18˚.

Steps 1 & 2

Step 1 - Draw a square. Include diagonal lines from the bottom left of the square to the top-right of the square, and from the top-left of the square to the bottom-right of the square. Divide the square into four smaller squares.

Step 2 - Place your compass in the bottom-right corner of the square and scribe an arc from the bottom-left corner to the top-right corner.

Note: The radius must equal the length of the square's side.

Steps 3, 4 & 5

Step 3 – Draw a line from the bottom-left corner of the square to halfway up the right-side of the square.

Step 4 - Scribe an arc centred half-way up the right side of the square. Start the arc from the bottom-right corner and swing it up until it overshoots the diagonal line drawn in Step 3.

Step 5 - Set the radius of the compass from the bottom-left of the square to the intersection point of the diagonal and arc drawn in Steps 3 and 4 respectively. Then place the point of the compass in each corner of the square and scribe an arc. The arcs must span two adjacent sides as illustrated above.

Steps 6, 7 & 8

Step 6 - Establish the height of a line to where the bottom two arcs drawn in Step 5 intersect the sides of the square, then draw the line by connecting the intersection points.

Step 7 - From the bottom left and top right corners, scribe two arcs from the point at which the horizontal line drawn in Step 6 intersects the diagonals drawn in Step 1. Scribe both arcs to the left and top sides of the square respectively.

Step 8 - Draw two lines from the bottom-right corner of the square to where the arcs drawn in Step 7 intersect the left-side and top-side of the square.

Steps 9 & 10

Acknowledgement: Many thanks to Mohamad Aljanabi for sharing the Traditional Method of Islamic geometric pattern construction that was used in this tutorial.

Categories: : Dynamic Symmetry, Geometry, Golden Ratio, Islamic Geometry