Difference between revisions of "Pythagorean Tree"
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|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]] | |ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]] | ||
The Pythagorean Tree begins with a square that has a right triangle branching off of it. The hypotenuse of the triangle must always be the one that is directly connected to the square. When the right triangle is created, the legs of said triangle then become one of the sides of two brand new squares. Important to note is that the length of the legs is not changed during this creation, so the squares are smaller than the big one. The sum of the areas of the two smaller squares is equal to the area of the big square. The interesting thing about the tree is that the right triangle can have any valid value of the non right angles. When the angles of the triangle are changed, one is made bigger, and the other is made smaller. Length of sides corresponds to measure of angles, so the sides change too. Since the leg is bigger, the square created using that leg is also bigger, creating the illusion of a tilt. | The Pythagorean Tree begins with a square that has a right triangle branching off of it. The hypotenuse of the triangle must always be the one that is directly connected to the square. When the right triangle is created, the legs of said triangle then become one of the sides of two brand new squares. Important to note is that the length of the legs is not changed during this creation, so the squares are smaller than the big one. The sum of the areas of the two smaller squares is equal to the area of the big square. The interesting thing about the tree is that the right triangle can have any valid value of the non right angles. When the angles of the triangle are changed, one is made bigger, and the other is made smaller. Length of sides corresponds to measure of angles, so the sides change too. Since the leg is bigger, the square created using that leg is also bigger, creating the illusion of a tilt. | ||
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In this image, the original square has an area of 36, meaning its side length s = 6. This can be put into the Pythagorean theorem, <math>a^2 + b^2 = c^2</math>, to get <math>a^2 + b^2 = 6^2</math>. This means the sum of the areas of the two branched-off squares will always be equal to the original square. These areas are dependent on the side lengths of the right triangle in the middle, which in turn are dependent the angles. With the hypotenuse 6 and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{6}</math>. If <math>\theta</math> = 60º, then b would equal, about, 5.196. | In this image, the original square has an area of 36, meaning its side length s = 6. This can be put into the Pythagorean theorem, <math>a^2 + b^2 = c^2</math>, to get <math>a^2 + b^2 = 6^2</math>. This means the sum of the areas of the two branched-off squares will always be equal to the original square. These areas are dependent on the side lengths of the right triangle in the middle, which in turn are dependent the angles. With the hypotenuse 6 and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{6}</math>. If <math>\theta</math> = 60º, then b would equal, about, 5.196. | ||
Revision as of 13:05, 5 June 2013
Pythagorean Tree, in 2 Dimensions |
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Pythagorean Tree, in 2 Dimensions
- A Pythagorean Tree is a fractal that is created out of squares. The space between the squares in each iteration creates a right triangle. The top line of the square becomes the hypotenuse of the triangle above it.
Basic Description
The Pythagorean Tree begins with a square that has a right triangle branching off of it. The hypotenuse of the triangle must always be the one that is directly connected to the square. When the right triangle is created, the legs of said triangle then become one of the sides of two brand new squares. Important to note is that the length of the legs is not changed during this creation, so the squares are smaller than the big one. The sum of the areas of the two smaller squares is equal to the area of the big square. The interesting thing about the tree is that the right triangle can have any valid value of the non right angles. When the angles of the triangle are changed, one is made bigger, and the other is made smaller. Length of sides corresponds to measure of angles, so the sides change too. Since the leg is bigger, the square created using that leg is also bigger, creating the illusion of a tilt.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Basic Algebra
In this image, the original square has an area of 36, meaning its side length s = 6. This can be put into the Pythagorean theorem, , to get . This means the sum of the areas of the two branched-off squares will always be equal to the original square. These areas are dependent on the side lengths of the right triangle in the middle, which in turn are dependent the angles. With the hypotenuse 6 and angle , the length of side b can be found with . If = 60º, then b would equal, about, 5.196.
If = 60º, then sin = = 5.196...
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