# Iris Folding Computer Design

By far, the easiest way to create your own iris folding pattern is to use a pencil and a ruler and draw concentric shapes each smaller than the previous. But if you are one of those people who like to draw on a computer, you will need a way to calculate two things:

1) the size of the inner square

2) the angle to rotate the inner square.

The equations below will allow you to calculate the size of the inner shape and the angle of rotation. Armed with that info, you can generate the inner shapes and rotate them so they fit into a nice spiral. Two solutions are provided: the first is for a square and the second is suitable for other shapes such as a triangle, pentagon, hexagon, and so forth.

### Calculations for a Square Template

Start with a square of size L and assume you want to draw the dot distance K from the corner.

You want to determine the length of side C and angle of rotation a.

**Side Length of Inner Square**

The side lenght of the inner square can be determined with Pythagoras’ theorem:

a

^{2}= b

^{2}+ c

^{2}

For example, if you want to start with a square of size 3 inch and you want to draw the dot 0.5 inches from the corner, then L = 3, K = 0.5 and the inner square length C is 2.55 inches.You would

*think*that if you start with a square of size 3″ and you decrease its size by 0.5″ then the inner square would be 2.5″. This is not true because you are rotating the inner square so the size needs to be a little large that expected.

**Angle of Rotation**

The angle of rotation can be determined with the trigonometry tanφ = y/x

**Summary**

Start with first square of size 3 inch.

Make a second square size 2.55 inch, rotate it 11.3 degrees.

Repeat calculation with L = 2.25 and K = 0.5

Continue until innermost square is close to or less than 0.5 inches wide

### Calculations for Triangle, Pentagon, Hexagon, and More

Shape | Angle c |

triangle | 60 degrees |

square | 90 degrees |

pentagon | 108 degrees |

hexagon | 120 degrees |

heptagon | 128.57 degrees |

octagon | 135 degrees |

nonagon | 140 degrees |

**Side Length of Inner Triangle**

The side length of the inner triangle can be determined with the law of cosines:

a

^{2}= b

^{2}+ c

^{2}– 2abcos(c)Note that angle c is 90 degrees for a square, so cos(90) = 0; this means the -2abcos(c) part becomes zero and the equation reduces to a

^{2}= b

^{2}+ c

^{2}which is the same as the instructions above for a square. [The law of cosines reduces to the pythagorean theorem for right triangles.]

cos(60) for triangle

cos(90) for square

cos(108) for pentagon

cos(120) for hexagon

cos(128.57) for heptagon

cos(135) for octagon

cos(140) = nonagon

**Angle of Rotation**

__A__=

__B__=

__C__

sin(a) sin(b) sin(c)

Again, remember that sin(c) is different depending on the shape you are working with:

sin(90) for square

sin(108) for pentagon

sin(120) for hexagon

sin(128.57) for heptagon

sin(135) for octagon

sin(140) = nonagon