Three Dimensional Pythagorean Tree

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3-D Pythagorean Tree
A 3-D Pythagorean Tree is a geometric figure that uses cubes connected to the sides of an isosceles right triangular prism to create a shape not unlike that of a tree.

Contents

Basic Description

The basic structure of a Pythagorean tree is that of squares, with side lengths equal to the triangle side lengths, connected to the sides of the Pythagorean Triangle. Ours is an isosceles right triangle therefore, the iterations are symmetrical. When applied to three dimensions, the equations and relationships become more complicated. We made this image using Google Sketchup. Sketchup allowed us to make perfectly accurate cubes by manually changing side lengths and copying pieces to other areas of the tree to ensure similar dimensions. Also, we were able to find endpoints, intersections, midpoints, and lines of symmetry while using Sketchup, making the process of constructing the tree much more simple and the results much more accurate.

First, to create the first iteration of the tree, we created a square that would be used as the base. Then, we copied the square and put the same exact square on top of the existing one. Then, using the center of the second square, we connected the center with the vertices of the first square, making a triangle.

Image:Screen_shot_2013-06-17_at_1.59.10_PM.png

Then, we removed the excess parts to just have the first square and triangle. This becomes the first iteration. We then created four smaller squares off the first triangle to create a square on the side. Since sketchup would not allow sideways squares to be created, we needed to connect the points of the four smaller squares.

Image:Screen_shot_2013-06-17_at_2.04.51_PM.png

So there is now a square on the side of the triangle. We then used this method on the other side to create another square. This method was really useful since it already created another triangle on the second square.

Image:Screen_shot_2013-06-17_at_2.11.57_PM.png

We kept using this method to develop the completed pythagorean triangle. To make it 3-D, we just raised the squares to become cubes and deleted the triangles.

Image:Screen_shot_2013-06-17_at_2.24.14_PM.png

A More Mathematical Explanation

Note: understanding of this explanation requires: *Geometry and Algebra

The 3D Pythagorean Tree is a manipulation of the Pythagorean Theorem of UNIQ3c94a3ad377d94ca-math-00 [...]

The 3D Pythagorean Tree is a manipulation of the Pythagorean Theorem of a^2+b^2=c^2 into the third dimension.


In the standard two dimensional tree the two smaller squares made from the legs of the triangle, when added together equal the square made by the hypotenuse in area. In the 3D Tree the sum of the surface areas of the smaller cubes is equal to that of the larger one. This is because of the Pythagorean Theorem itself.


Image:Screen shot 2013-06-11 at 2.04.16 PM.png

Image:GPythag3d.png


Lengths- The lengths are just the 2D distance

Area- The areas are the lengths squared (squared and then doubled for the second set of cubes)

Surface Area- The areas times 6

Volumes- The lengths cubed are the volumes


The Surface Area Relationship

The main focus here is the relationship between the cube connected to the plane that represents the hypotenuse and the other two smaller cubes. (Insert Graph)

The equation for the surface area of a cube is, if the variable a is the leg length, 6a^2. Looking at just one face and one iteration, the lengths follow the Pythagorean Theorem:

a^2+b^2=c^2

The variable b represents the length of the other leg. The variable c represents the length of the hypotenuse. For an example, we will used 1 as the length of legs a and b to find the length of the hypotenuse c.

(1)^2+(1)^2=c^2

2=c^2 \sqrt{2}=c

When we have all three side lengths, we can use the surface area equations to find a relationship.

(6*1)^2+(6*1)^2=(6\sqrt2)^2

36+36= 6\sqrt{2}^2

72=72

Because the two sides of the equation are equal, we can conclude that the sum of the surface areas for the cubes formed by the legs is congruent the surface area of the cube created by the hypotenuse.

Now we will rewrite this equation using the variables a and b for the legs and c for the hypotenuse. The importance of the pythagorean theorem will become more apparent as the equation progresses.

a^2+b^2=c^2

The equation for surface area is 6a^2, with a being the length of the segment used. Now we will plug in a, and b into the surface area relationship that we think is true. We are going to apply the Pythagorean theorem into the surface area equation.

6(a)^2+6(b)^2

We can then factor out 6 from this equation.

6(a^2+b^2)

Because we know that a^2+b^2=c^2, we can say that:

6(a^2+b^2)=6(c^2)

C represents the surface area of the cube created by the hypotenuse. The simplified version of this equation is:

a^2+b^2=c^2

This is the pythagorean theorem. This proves that the surface area relationship is hinged on the pythagorean theorem. We can further simplify this equation because we know that a=b because the right triangle is isosceles.

(*when a=b)

 2a^2=c^2

Then we can simplify it even further by taking the square root of the equation.

\sqrt{2}*{a}=c


The Edge Length Relationship

The ratio between the the edge of the base cube and the edge of an immediately following cube is \sqrt2 :1. The equation for this is as follows \sqrt2 (a)= c This theorem is derived from the pythagorean so it applies to all cubes in an isosceles 3D pythagorean tree. a^2+b^2=c^2 (*When a=b)

a^2+a^2=c^2

2a^2=c^2

\sqrt(2)*(a)=c

This equation holds true for both all isosceles pythagorean trees both 3D and 2D The edge or side of the side of the base square or cube is equal to the length of the edge or side of the subsequent square or cube multiplied by \sqrt(2).

The Volume Relationship

The volume of the original cube has a special relation to the sum of the two congruent cubes that follow it. The sum of the two smaller cubes multiplied by \sqrt(2) is equal to the volume of the larger cube they are immediately built upon. This too can be derived from the Pythagorean Theorem.

 a^2+b^2=c^2 *when a=b

 (2)(a^2)=c^2 raise to 1.5 power, so that a and c are in the 3rd power, thus creating the 3rd dimension and volume

 [2(a^2)]^{1.5}=(c^2)^{1.5}

2^{1.5}(a^3)=c^3

 2^{1.5}* a^3 is the sum of the volumes of the smaller cubes or just  
[\sqrt2 (a)]^3 (this comes from the edge length relation)

 c^3 is the volume of the larger cube

(\sqrt2)(a)=c all to the 1/3 power or cubed root

The Wrap-Up

If we look at the three equations, we notice that the outcome comes out the same for each:

(\sqrt2)(a)=c

This is the simplified version, in terms of c, of the pythagorean theorem when Leg a is congruent to Leg b. In conclusion, all of the relationships in an isosceles right triangle 3-D Pythagorean tree are connected to the Pythagorean theorem. Therefore, the Pythagorean theorem provides the template



Why It's Interesting

This is interesting because we have discovered that the Pythagorean theorem is the foundation for all of the relationships that we have decided to explain. The Surface Area, Volume, and Edge Length relationships all have the same outcome when their equations are simplified. The outcome, for an isosceles right triangle (a=b), is always:

\sqrt(2)*(a)=c

How the Main Image Relates

The main image is a direct representation of a 3-D Pythagorean Tree. It also has accurate decreases in depth.

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