# Triangles, squares, and pentagons

Two overlapping snub square tilings:

Each vertex of one tiling lies within a triangle of the other tiling. If we partition each square into 45-45-90 isosceles-triangles and glue two of them to each triangle, forming irregular pentagons, we get two overlapping Cairo pentagonal tilings:

The two Cairo tilings are close to being planar duals of each other: they share some vertices, but the vertices that belong only to one tiling correspond one-for-one with the tiles of the other tiling. More standardly, the Cairo tilings are dual to the snub square tilings.

how did you create these images?

Adobe Illustrator. I knew before starting the drawings what I wanted the shapes to look like, though not the colors. I started by drawing an equilateral triangle, grouped it with a rotated-by-180 copy, made another copy of the two triangles rotated by 90, centered them all together, and played with colors and transparency for a while. Then I did four drag-and-copies to end up with 16 copies of the four colored triangles. I used the first figure as a guide to draw a pentagon and did the same sequence of rotates, groups, and copies to make the second figure.

Each vertex of one tiling lies within a triangle of the other tiling. If we partition each square into 45-45-90 isosceles-triangles and glue two of them to each triangle, forming irregular pentagons, we get two overlapping Cairo pentagonal tilings:

The two Cairo tilings are close to being planar duals of each other: they share some vertices, but the vertices that belong only to one tiling correspond one-for-one with the tiles of the other tiling. More standardly, the Cairo tilings are dual to the snub square tilings.

### Comments:

**ohmiee**:

**2007-05-15T06:02:08Z**

how did you create these images?

**11011110**:

**2007-05-15T07:45:48Z**

Adobe Illustrator. I knew before starting the drawings what I wanted the shapes to look like, though not the colors. I started by drawing an equilateral triangle, grouped it with a rotated-by-180 copy, made another copy of the two triangles rotated by 90, centered them all together, and played with colors and transparency for a while. Then I did four drag-and-copies to end up with 16 copies of the four colored triangles. I used the first figure as a guide to draw a pentagon and did the same sequence of rotates, groups, and copies to make the second figure.