A Pythagorean spiral, also known as the spiral of Theodorus, is a special mathematical spiral. To construct it, start with a right triangle with a base and side of one unit each. Then add a second right triangle that has the hypotenuse of the first triangle as its base and a side one unit long. Continue adding triangles in the same manner until the spiral completes a full circle or reaches the size you want it to be.
Choose a theme you want to use when decorating your Pythagorean spiral. You may want to use the spiral shape to create an image from nature, like a nautilus shell or the arm of a spiral galaxy, or you may simply choose to create a geometric design or spectrum of rainbow colors.
Find a picture that looks like what you want to create if you're having trouble visualizing it. You may be able to find pictures of natural spirals in the pages of a nature magazine, or you could look through a magazine on design, home decor or ancient decorations to find inspiration.
Draw the design you've chosen on your Pythagorean spiral with colored pencils. Emphasize the continuity and contrast in the growing triangles by coloring them according to the same theme, but with slight changes that develop as the spiral extends. For example, you could have a geometric design that gets a little bigger in each triangle or change the color slightly with each new triangle so that the overall effect is a rainbow or smooth gradient.
Consider adding decorations outside the spiral as well. If you choose a geometric design, you can add similar designs to frame the spiral. If you choose an image from nature, you could add external details, like a starry background to a galaxy or the head of the nautilus or snail.