https://mathimages.swarthmore.edu/api.php?action=feedcontributions&user=JWallison&feedformat=atomMath Images - User contributions [en]2022-09-27T08:34:47ZUser contributionsMediaWiki 1.31.1https://mathimages.swarthmore.edu/index.php?title=User:JWallison&diff=36254User:JWallison2013-06-11T16:43:32Z<p>JWallison: </p>
<hr />
<div>John Wallison<br />
<br />
I am working on a page with Enri Kina, and we are working on a page about the Pythagorean tree, which is a fractal based on squares, which iterate to form a tree.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
mon ma mes<br />
ton ta tes<br />
son sa ses<br />
notre nos<br />
votre vos<br />
leur leurs</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=User:JWallison&diff=36253User:JWallison2013-06-11T16:40:35Z<p>JWallison: </p>
<hr />
<div>John Wallison<br />
<br />
I am working on a page with [[Enri Kina]], and we are working on a page about the [[Pythagorean tree]], which is a fractal based on squares, which iterate to form a tree.</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36252Pythagorean Tree2013-06-11T16:39:56Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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.<br />
|ImageDesc=[[Image:le pyhto.png|Example pythagorean tree|thumb|300px|left]]<br />
<br />
<br />
<br />
In this image, ''CFD'' is a right triangle. The original square has an area of ''s<sup>2</sup>'', meaning its side length is ''s'', because the area of a square is its side length squared (''s<sup>2</sup>''). This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^2</math>. This means the sum of the areas of the two branched-off squares will always be equal to the original square. This is because the Pythagorean theorem states that the sum of the side lengths squared (in other words, the sum of the areas) equal the hypotenuse, ''s'', squared. These areas are dependent on the side lengths of the right triangle in the middle, which in turn are dependent the angles. <br />
<br />
With the hypotenuse ''s'' and angle ''θ<sub>1</sub>'', the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
|other=Basic Algebra and Geometry, Trigonometry<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36249Pythagorean Tree2013-06-11T16:28:23Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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.<br />
|ImageDesc=[[Image:le pyhto.png|Example pythagorean tree|thumb|300px|left]]<br />
<br />
<br />
<br />
In this image, ''CFD'' is a right triangle. With the hypotenuse ''s'' and angle ''θ<sub>1</sub>'', the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
The original square has an area of ''s<sup>2</sup>'', meaning its side length is ''s''. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^2</math>. This means the sum of the areas of the two branched-off squares will always be equal to the original square. This is because the Pythagorean theorem states that the sum of the side lengths squared (in other words, the sum of the areas) equal the hypotenuse, ''s'', squared. These areas are dependent on the side lengths of the right triangle in the middle, which in turn are dependent the angles. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
|other=Basic Algebra and Geometry, Trigonometry<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36238Pythagorean Tree2013-06-11T16:19:12Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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.<br />
|ImageDesc=[[Image:le pyhto.png|Example pythagorean tree|thumb|300px|left]]<br />
<br />
<br />
<br />
In this image, CFD is a right triangle. With the hypotenuse s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
The original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = s^2, or (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
|other=Basic Algebra and Geometry, Trigonometry<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36233Pythagorean Tree2013-06-11T16:16:27Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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.<br />
|ImageDesc=[[Image:le pyhto.png|Example pythagorean tree|thumb|350px|left]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = s^2, or (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
|other=Basic Algebra and Geometry, Trigonometry<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Masterman_Students%27_Work_2013&diff=36229Masterman Students' Work 20132013-06-11T16:11:10Z<p>JWallison: </p>
<hr />
<div>Welcome to the Masterman Math Images page for 2013! Masterman students and the teachers, professors, and college students working with them can use this space to keep each other up-to-date on projects and feedback.<br />
<br />
==How Does this Page Work?==<br />
This is a place for the Masterman Math Images students to say what they're working on and thinking about. Students can post what they're proud of, questions they have, areas of confusion, or material on which they want feedback. Mr. Taranta, Diana Patton, and others from both Swarthmore College and the Math Forum will respond to them as quickly as possible. There shouldn't be any long conversations posted on this page; simple explanations or requests and brief responses or statements like, "I posted on your comments page," are all we need here.<br />
<br />
Students, this page has a section for each person or group working on a project this semester. Edit the space under your heading with brief descriptions of your project's current status, your plan going forward, and what kind of support you would like from the people giving feedback. Questions and comments such as, "Please read this over," "Will this work?" and other thoughts are all important to share. Please keep your section of this page neat and up to date. If no one can tell what you're working on, no one will know to send any feedback or tips your way.<br />
<br />
==Helpful Links for Students==<br />
At any time, you can find your current assignment at [[Current Masterman Assignment]].<br />
<br />
While writing, you'll find these links helpful to keep on hand, maybe even open in another tab, while working on your projects. Note that you can get to all these and more through [[Page Building Help]].<br />
===Read these Before you Write '''Anything'''!===<br />
*[[Special:Thumb|Other math images pages!]] - Reading other people's pages is the best way to figure out how to make your own.<br />
*[[Starting a New Page]] - A step-by-step for creating a new image page.<br />
*[[What Makes a Good Math Images Page?]] - The grand overview. This is what really tells you both the why and the how of what you do when you sit down to edit a page or write a new one.<br />
*[[Writing Guide Hit List]] - Short and sweet. This lays out some basics to bear in mind about writing for an audience.<br />
<br />
===Answers to Questions and Self-Editing Tips===<br />
*[[From a Bunch of Old Timers]] - Incredibly helpful. This has the most salient tips from people who have done it all before.<br />
*[[Checklist for writing pages]] - The definitive list. Check this over while you're writing and before you proofread your work.<br />
<br />
===All the Technical Stuff===<br />
*[[Top 5 things you need to know how to do on the wiki]] - Straight to the point. This covers coding for images, "balloons," hide/show, citations, and our feedback system.<br />
*[[Wiki Tricks]] - Goes into more detail. This is a compilation specific to the Math Images wiki, so it covers everything you're most likely to want to do in code, including things for which we've developed our own templates.<br />
*[http://www.mediawiki.org/wiki/Help:Editing Editing Help Page] - The outside source. This is a guide to codes and templates that apply to all wiki sites. Go here when the first two pages don't give you the information you need.<br />
<br />
==Geometry 3 Students==<br />
<br />
===Joel Chacko===<br />
[[User:JChacko|Joel Chacko]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
===Eiman Eltigani===<br />
[[User:Eimaneltigani|Eiman Eltigani]]<br />
<br />
[[Crop Circles]]<br />
<br />
===Constance Lee===<br />
[[User:CLee|Constance Lee]]<br />
<br />
[[Epitrochoids|Link to Epitrochoid page]]<br />
<br />
===Kevin Liu===<br />
[[User:Kevinwoot|Kevin Liu]]<br />
<br />
[[epitrochoids|link to my page]]<br />
<br />
===Rohit Thaiparambil===<br />
[[User:Rohithaip|Rohit Thaiparambil]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
===Charles Wattley===<br />
[[User:CWattley|Charles Wattley]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
===Kevin Yang===<br />
[[User:Kyang16|Kevin Yang]]<br />
<br />
[[Epitrochoids|Link to my Page]]<br />
<br />
===Howard Yuan===<br />
[[User:HYuan|Howard Yuan]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
==Geometry 1 Students==<br />
<br />
===Enri Kina===<br />
[[User:EKina|Enri Kina]]<br />
<br />
Working with John Wallison: <br />
[[Pythagorean Tree]]<br />
<br />
===Ali Landers===<br />
[[User:Alanders|Ali Landers]]<br />
<br />
[[Eternal Knot]]<br />
<br />
===Natalia Nottingham===<br />
[[User:Nnottingham|Natalia Nottingham]]<br />
<br />
[[Spiral Explorations]]<br />
<br />
===Hana Pearlman===<br />
[[User:HPearlman|Hana Pearlman]]<br />
<br />
[[Spiral Explorations]]<br />
<br />
===Eliza Sulea===<br />
[[User:ESulea|Eliza Sulea]]<br />
<br />
[[Romanesco Broccoli]]<br />
<br />
===John Wallison===<br />
[[User:JWallison|John Wallison]]<br />
<br />
Working with Enri Kina: <br />
[[Pythagorean Tree]]<br />
<br />
===Michael Willis===<br />
[[User:MWillis|Michael Willis]]<br />
<br />
[[Fractals With Stars]]<br />
<br />
===Jane Xu===<br />
[[User:JXu99|Jane Xu]]<br />
<br />
[[Spiral Explorations]]<br />
<br />
===Vicky Zheng===<br />
[[User:V.Zheng|Vicky Zheng]]<br />
<br />
[[Circular Spiral Envelope Intersection]]</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Masterman_Students%27_Work_2013&diff=36016Masterman Students' Work 20132013-06-05T18:28:31Z<p>JWallison: </p>
<hr />
<div>Welcome to the Masterman Math Images page for 2013! Masterman students and the teachers, professors, and college students working with them can use this space to keep each other up-to-date on projects and feedback.<br />
<br />
==How Does this Page Work?==<br />
This is a place for the Masterman Math Images students to say what they're working on and thinking about. Students can post what they're proud of, questions they have, areas of confusion, or material on which they want feedback. Mr. Taranta, Diana Patton, and others from both Swarthmore College and the Math Forum will respond to them as quickly as possible. There shouldn't be any long conversations posted on this page; simple explanations or requests and brief responses or statements like, "I posted on your comments page," are all we need here.<br />
<br />
Students, this page has a section for each person or group working on a project this semester. Edit the space under your heading with brief descriptions of your project's current status, your plan going forward, and what kind of support you would like from the people giving feedback. Questions and comments such as, "Please read this over," "Will this work?" and other thoughts are all important to share. Please keep your section of this page neat and up to date. If no one can tell what you're working on, no one will know to send any feedback or tips your way.<br />
<br />
==Helpful Links for Students==<br />
At any time, you can find your current assignment at [[Current Masterman Assignment]].<br />
<br />
While writing, you'll find these links helpful to keep on hand, maybe even open in another tab, while working on your projects. Note that you can get to all these and more through [[Page Building Help]].<br />
===Read these Before you Write '''Anything'''!===<br />
*[[Special:Thumb|Other math images pages!]] - Reading other people's pages is the best way to figure out how to make your own.<br />
*[[Starting a New Page]] - A step-by-step for creating a new image page.<br />
*[[What Makes a Good Math Images Page?]] - The grand overview. This is what really tells you both the why and the how of what you do when you sit down to edit a page or write a new one.<br />
*[[Writing Guide Hit List]] - Short and sweet. This lays out some basics to bear in mind about writing for an audience.<br />
<br />
===Answers to Questions and Self-Editing Tips===<br />
*[[From a Bunch of Old Timers]] - Incredibly helpful. This has the most salient tips from people who have done it all before.<br />
*[[Checklist for writing pages]] - The definitive list. Check this over while you're writing and before you proofread your work.<br />
<br />
===All the Technical Stuff===<br />
*[[Top 5 things you need to know how to do on the wiki]] - Straight to the point. This covers coding for images, "balloons," hide/show, citations, and our feedback system.<br />
*[[Wiki Tricks]] - Goes into more detail. This is a compilation specific to the Math Images wiki, so it covers everything you're most likely to want to do in code, including things for which we've developed our own templates.<br />
*[http://www.mediawiki.org/wiki/Help:Editing Editing Help Page] - The outside source. This is a guide to codes and templates that apply to all wiki sites. Go here when the first two pages don't give you the information you need.<br />
<br />
==Geometry 3 Students==<br />
<br />
===Joel Chacko===<br />
[[User:JChacko|Joel Chacko]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
===Eiman Eltigani===<br />
[[User:Eimaneltigani|Eiman Eltigani]]<br />
<br />
[[Crop Circles]]<br />
<br />
===Constance Lee===<br />
[[User:CLee|Constance Lee]]<br />
<br />
[[Epitrochoids|Link to Epitrochoid page]]<br />
<br />
===Kevin Liu===<br />
[[User:Kevinwoot|Kevin Liu]]<br />
<br />
[[epitrochoids|link to my page]]<br />
<br />
===Rohit Thaiparambil===<br />
[[User:Rohithaip|Rohit Thaiparambil]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
===Charles Wattley===<br />
[[User:CWattley|Charles Wattley]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
===Kevin Yang===<br />
[[User:Kyang16|Kevin Yang]]<br />
<br />
[[Epitrochoids|Link to my Page]]<br />
<br />
===Howard Yuan===<br />
[[User:HYuan|Howard Yuan]]<br />
<br />
[[Three Dimensional Pythagorean Tree]]<br />
<br />
==Geometry 1 Students==<br />
<br />
===Enri Kina===<br />
[[User:EKina|Enri Kina]]<br />
<br />
Working with John Wallison<br />
[[Pythagorean Tree]]<br />
<br />
===Ali Landers===<br />
[[User:Alanders|Ali Landers]]<br />
<br />
[[Eternal Knot]]<br />
<br />
===Natalia Nottingham===<br />
[[User:Nnottingham|Natalia Nottingham]]<br />
<br />
[[Spiral Explorations]]<br />
<br />
===Hana Pearlman===<br />
[[User:HPearlman|Hana Pearlman]]<br />
<br />
[[Spiral Explorations]]<br />
<br />
===Eliza Sulea===<br />
[[User:ESulea|Eliza Sulea]]<br />
<br />
[[Romanesco Broccoli]]<br />
<br />
===John Wallison===<br />
[[User:JWallison|John Wallison]]<br />
<br />
Working with Enri Kina<br />
[[Pythagorean Tree]]<br />
<br />
===Michael Willis===<br />
[[User:MWillis|Michael Willis]]<br />
<br />
[[Fractals With Stars]]<br />
<br />
===Jane Xu===<br />
[[User:JXu99|Jane Xu]]<br />
<br />
[[Spiral Explorations]]<br />
<br />
===Vicky Zheng===<br />
[[User:V.Zheng|Vicky Zheng]]<br />
<br />
[[Circular Spiral Envelope Intersection]]</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36015Pythagorean Tree2013-06-05T18:26:34Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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.<br />
|ImageDesc=[[Image:le pyhto.png]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = s^2, or (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
|other=Basic Algebra and Geometry, Trigonometry <br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36013Pythagorean Tree2013-06-05T18:24:34Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:le pyhto.png]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = s^2, or (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36012Pythagorean Tree2013-06-05T18:24:11Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:le pyhto.png]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths:<br />
<br />
<math>a^2 = (s * cos\theta)^2</math><br />
<math>b^2 = (s * sin\theta)^2</math><br />
<br />
Which means:<br />
<br />
<math>a^2 + b^2 = s^2, or (s * cos\theta)^2 + (s * sin\theta)^2 = s^2</math><br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36006Pythagorean Tree2013-06-05T18:19:51Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:le pyhto.png]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, square the side lengths<br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36005Pythagorean Tree2013-06-05T18:18:59Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:le pyhto.png]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>. <br />
<br />
To find the areas of the 2 branched-off squares, multiply the <br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36003Pythagorean Tree2013-06-05T18:14:43Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:le pyhto.png]]<br />
<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. This means that <math>b = s(sin\theta)</math>. The other side length, a, can be found similarly: <math>cos\theta = \frac{a}{s}</math>, or <math>a = s(cos\theta)</math>.<br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36001Pythagorean Tree2013-06-05T18:11:06Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:le pyhto.png]]<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. <br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=36000Pythagorean Tree2013-06-05T18:09:28Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:lepyhto.png]]<br />
In this image, the original square has an area of <math>s^2</math>, meaning its side length is s. This can be put into the Pythagorean theorem: <math>a^2 + b^2 = s^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 s and angle <math>\theta</math>, the length of side b can be found with <math>sin\theta = \frac{b}{s}</math>. <br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=File:Le_pyhto.png&diff=35996File:Le pyhto.png2013-06-05T18:07:11Z<p>JWallison: Pythagorean tree with variables</p>
<hr />
<div>Pythagorean tree with variables</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35994Pythagorean Tree2013-06-05T18:05:04Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
<br />
|ImageDesc=<br />
[[Image:pythagoreantree.png]]<br />
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.<br />
<br />
If <math>\theta</math> = 60º, then sin<math>\theta</math> = <math>\frac{b}{6}</math> = 5.196...<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35993Pythagorean Tree2013-06-05T18:04:28Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|This animation shows how the angles of the triangle affect the shape of the tree.|thumb|350px|left]]<br />
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. <br />
<br />
|ImageDesc=<br />
[[Image:pythagoreantree.png]]<br />
<br />
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.<br />
<br />
If <math>\theta</math> = 60º, then sin<math>\theta</math> = <math>\frac{b}{6}</math> = 5.196...<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35990Pythagorean Tree2013-06-05T18:01:06Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|thumb|350px|left]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|ImageDesc=<math>\frac{1}{x}</math><br />
[[Image:pythagoreantree.png]]<br />
<br />
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.<br />
<br />
If <math>\theta</math> = 60º, then sin<math>\theta</math> = <math>\frac{b}{6}</math> = 5.196...<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35762Pythagorean Tree2013-05-29T18:27:36Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif|thumb|350px|left]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|ImageDesc=<math>\frac{1}{x}</math><br />
[[Image:pythagoreantree.png]]<br />
<br />
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. The length of a a can be found use the <math>\theta</math><br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35757Pythagorean Tree2013-05-29T18:23:30Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|ImageDesc=<math>\frac{1}{x}</math><br />
[[Image:pythagoreantree.png]]<br />
<br />
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. The length of a<math>\theta</math><br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35745Pythagorean Tree2013-05-29T18:10:29Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|ImageDesc=<math>\frac{1}{x}</math><br />
[[Image:pythagoreantree.png]]<br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=File:Pythagoreantree.png&diff=35744File:Pythagoreantree.png2013-05-29T18:08:46Z<p>JWallison: </p>
<hr />
<div></div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35743Pythagorean Tree2013-05-29T18:08:15Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|ImageDesc=<math>\frac{1}{x}</math><br />
<br />
<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=User:JWallison&diff=35735User:JWallison2013-05-29T17:48:20Z<p>JWallison: </p>
<hr />
<div>John Wallison<br />
<br />
I am working on a page with [Enri Kina], and we are working on a page about the [Pythagorean tree], which is a fractal based on squares, which iterate to form a tree.</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35578Pythagorean Tree2013-05-22T18:28:04Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions<br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35576Pythagorean Tree2013-05-22T18:17:16Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions <br />
|Image=Le pytho.jpg<br />
|ImageIntro=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.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35574Pythagorean Tree2013-05-22T18:08:59Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions <br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree is a fractal that is created out of squares. The space between the squares in each iteration creates a right triangle.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35571Pythagorean Tree2013-05-22T18:02:07Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree, in 2 Dimensions <br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35570Pythagorean Tree2013-05-22T18:01:33Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree<br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle.<br />
|ImageDescElem=[[Image:Output_ANIihE.gif]]<br />
<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35569Pythagorean Tree2013-05-22T17:56:51Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree<br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle.<br />
|ImageDesc=[[Image:Output_ANIihE.gif]]<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35568Pythagorean Tree2013-05-22T17:56:28Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree<br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle. [[Image:Output_ANIihE.gif]]<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|ImageDesc=[[Image:Output_ANIihE.gif]]<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35567Pythagorean Tree2013-05-22T17:55:29Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree<br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle. [[Image:Output_ANIihE.gif]]<br />
This animation shows how the angles of the triangle affect the shape of the tree.<br />
|other=Basic Algebra<br />
|AuthorName=Enri Kina and John Wallison<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35388Pythagorean Tree2013-05-02T12:42:37Z<p>JWallison: </p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree<br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle.<br />
|other=Basic Algebra<br />
|Field=Algebra<br />
|Field2=Fractals<br />
|InProgress=Yes<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=Pythagorean_Tree&diff=35362Pythagorean Tree2013-04-24T18:01:48Z<p>JWallison: New page: {{Image Description |ImageName=Pythagorean Tree |Image=Le pytho.jpg |ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle. |Field=Algebra |InProgress=No }}</p>
<hr />
<div>{{Image Description<br />
|ImageName=Pythagorean Tree<br />
|Image=Le pytho.jpg<br />
|ImageIntro=A Pythagorean Tree, based off of a 45º-45º-90º triangle.<br />
|Field=Algebra<br />
|InProgress=No<br />
}}</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=File:Le_pytho.jpg&diff=35358File:Le pytho.jpg2013-04-24T17:58:07Z<p>JWallison: The Pythagorean Tree, based off of a 45º-45º-90º right triangle.</p>
<hr />
<div>The Pythagorean Tree, based off of a 45º-45º-90º right triangle.</div>JWallisonhttps://mathimages.swarthmore.edu/index.php?title=User:JWallison&diff=35281User:JWallison2013-04-10T18:32:51Z<p>JWallison: New page: John Wallison</p>
<hr />
<div>John Wallison</div>JWallison