Problem 395

Problem 395

Pythagorean tree

The Pythagorean tree is a fractal generated by the following procedure:

Start with a unit square. Then, calling one of the sides its base (in the animation, the bottom side is the base):

  1. Attach a right triangle to the side opposite the base, with the hypotenuse coinciding with that side and with the sides in a 3-4-5 ratio. Note that the smaller side of the triangle must be on the ‘right’ side with respect to the base (see animation).
  2. Attach a square to each leg of the right triangle, with one of its sides coinciding with that leg.
  3. Repeat this procedure for both squares, considering as their bases the sides touching the triangle.
    The resulting figure, after an infinite number of iterations, is the Pythagorean tree.

It can be shown that there exists at least one rectangle, whose sides are parallel to the largest square of the Pythagorean tree, which encloses the Pythagorean tree completely.

Find the smallest area possible for such a bounding rectangle, and give your answer rounded to 10 decimal places.




  1. 给基底所对的边粘连上一个三边长之比为3-4-5的直角三角形,其中基底所对的边为斜边。注意直角三角形较短的直角边始终处在相对基底来说的右侧(参见动画演示)。
  2. 给直角三角形的两条直角边分别粘连上一个正方形,该直角边充当正方形的其中一条边。
  3. 对这两个正方形重复以上操作,并将正方形与之粘连的直角边作为基底。