Most Interesting Math Pythagorean Tree A tree drawn by Pythagoras using the Pythagorean Theorem

The Pythagorean tree is an infinitely repeating graph drawn by Pythagoras using the Pythagorean Theorem, and is also known as a tree because the overall graph is shaped like a ? Pythagorean tree? , but due to the overlap limit, the real Pythagorean tree has a finite area of 6 by 4. Follow me on this site to see it!

What is a Pythagorean tree?

Although math is very boring, but scientists can always find unlimited fun, Pythagorean tree is by the ancient Greek mathematician Pythagoras, the use of the collinear theorem drawn an infinite repetition of the figure, when repeated enough times, it will form the shape of a tree, so it is also known as ? the hook and strand tree?

The right triangle and the three squares extending from its three sides have some magical features, such as the area of the right triangle is less than or equal to 1/4 of the area of the larger square and greater than or equal to 1/2 of the area of the smaller square, and the two smaller squares are equal to the area of the larger square, and the sum of the areas of all the smaller squares at the same time is equal to the area of the largest square.

Simple drawing of the Pythagorean tree

As we all know, the collinearity theorem is the sum of the squares of the two right sides of a right triangle is equal to the square of the hypotenuse, Pythagoras utilized this point in the initial large square, made the two congruent small squares, and so on, infinite repetition of a variety of different sizes of the squares, the formation of a dense ? Pythagorean tree?

Since the interiors of the three squares form an isosceles right triangle, it follows by the collinearity theorem that the side lengths of the small squares are ?2/2 of those of the large square, and this is repeated ad infinitum by repeating the above process for the small squares. If one assumes that the side length of one of the large squares is 1, at the nth addition, 2n small squares are added, and the side length of each small square is ?2/2, then the area added at each addition is 2n?(?2)=1.

Is the Pythagorean tree infinite?

Theoretically, the Pythagorean tree can be repeated indefinitely because the area of the Pythagorean tree tends to be infinite after setting n in the formula of the appeal to an infinite number of times. The area of the Pythagorean tree would also be more dense, but in reality this is not the case.

Because all the resulting small squares overlap each other when n is greater than 5, the area of the Pythagorean tree is actually finite. So the Pythagorean tree can really only grow in a 6?4 square, but of course the exact value is not very easy to work out.

Variants of the Pythagorean tree

The original Pythagorean tree had unequal angles between the large and small squares, so one variant of the Pythagorean tree changed the angles so that when the angle between the original large and small squares was changed to 60 degrees, the triangle in the center became an equilateral triangle, and the lengths of the sides of each of the squares were equal.

But this variant is also finite, like the normal Pythagorean tree, and overlap occurs when it reaches the fourth step, ending up with a large hexagon full of squares with equal sides.

There are quite a few other interesting phenomena in mathematics, besides the Pythagorean tree, there is the 123 black hole where the result is always 123, and the world's most magical number, 142857, which is the result of mathematical ingenuity.