A Line Puzzle
Imagine an equilateral triangle with sides of one unit length:
The length of the path across the base of the triangle is one unit length. The length of the path over the triangle is two unit lengths. The area of the triangle is √3 / 4.
Imagine two equilateral triangles with sides of one half unit length:
The length of the path across the bases of the triangles is one unit length. The length of the path over the triangles is two unit lengths. The total area of the triangles is √3 / 8.
Imagine four equilateral triangles with sides of one quarter unit length:
The length of the path across the bases of the triangles is one unit length. The length of the path over the triangles is two unit lengths. The total area of the triangles is √3 / 16.
Continue this process creating more and more equilateral triangles. The length of the path across the bases of the triangles will always be one unit length. T he length of the path over the triangles will always be two unit lengths. The to tal area of 2n triangles will always be √3 / 2n + 2.
As the number of equilateral triangles increases, the total area of the triangles gets smaller and smaller. And, the path across the bases of the triangles resembles more and more the path over the triangles.
How can this be when one of that paths is always twice as long as the other?
The length of the path across the base of the triangle is one unit length. The length of the path over the triangle is two unit lengths. The area of the triangle is √3 / 4.
Imagine two equilateral triangles with sides of one half unit length:
The length of the path across the bases of the triangles is one unit length. The length of the path over the triangles is two unit lengths. The total area of the triangles is √3 / 8.
Imagine four equilateral triangles with sides of one quarter unit length:
The length of the path across the bases of the triangles is one unit length. The length of the path over the triangles is two unit lengths. The total area of the triangles is √3 / 16.
Continue this process creating more and more equilateral triangles. The length of the path across the bases of the triangles will always be one unit length. T he length of the path over the triangles will always be two unit lengths. The to tal area of 2n triangles will always be √3 / 2n + 2.
As the number of equilateral triangles increases, the total area of the triangles gets smaller and smaller. And, the path across the bases of the triangles resembles more and more the path over the triangles.
How can this be when one of that paths is always twice as long as the other?
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