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For all of the proofs, click the link below for each proof and art piece:
Finding The Square Roots: Faces
Mathematics can be beautiful. The following proofing is expressed as an art piece for my latest EP: Teacher Training.
The subsequent geometric proof details how the artwork was created using the free geogebra app. While the color base is red, blue, and yellow; the shapes are derived from manipulating the values of the radius of a circle along with the altering of the squared function. First, the proof will be explored. Then, the final picture will be shown.
I hope you enjoy...
For Math remediation and guidance, check out these short papers:
Interactive Proof:
Infinite Geometric Series: Concentric Circles and the Squared Function
Infinite Geometric Series: Concentric Circles and the Squared Function
This infinite geometric series are derived from geometric proofs applied to the coordinate plane. The main concept of this series is to find the intersect points of the parent functions: y=x, y=x^2, y=x^(1/2), y=n^x, and y=1/x; and the circle, y^2+x^2=r; with the value of r as an element in the set of positive real numbers.
This first exploration involves the intersection of the functions y=x^2, y=-x^2, x=y^2, and x=-y^2; and y^2+x^2=r; with the value of r as an element in the set of positive real numbers.
Protocol for intersect points for a value of r.
1. Draw the unit circle: y^2+x^2=1
2. Draw a square with the intersect points of
the unit circle and the x and y axis.
3. Draw triangles at the intersect points of
the unit circle and the functions:
y=x^2, y=-x^2, x=y^2, and x=-y^2
This is a division of the unit circle based on the intersection of the functions y=x^2, y=-x^2, x=y^2, and x=-y^2; and the unit circle.
The series become an infinite series once the value of r is derived from the set of positive real numbers. At certain values of r, the intersect points between the functions and the unit circle invert. For instance at r=2^(1/2), the intersection of the squared functions and the circle create an equivalent square to resultant square of the intersection of the x and y axis and the circle at r=2^(1/2). This is due to the functions relationship with the points (1,1), (1,-1), (-1,1), and (-1,-1); which is where the squared functions themselves intersect. This reduces the number of intersect points of the circle at r=2^(1/2) and the squared functions from 8 total to 4 total.
Protocol for intersect points for a value of r.
1. Draw the unit circle: y^2+x^2=2
2. Draw a square with the intersect points of
the unit circle and the x and y axis.
3. Draw triangles at the intersect points of
the unit circle and the functions:
y=x^2, y=-x^2, x=y^2, and x=-y^2
The circle, y^2+x^2=2, can act as the base of this particular series. Meaning, at r=2^(1/2) is the only point where the circle is divided evenly through the axis and squared functions
For each circle in the series, the intersection of the squared functions and respective circle can be elements of a sub set of the set of the intersect points and resultant divisions of the circle when r is an element of the set of positive real numbers. The two sub groups are when r>2^(1/2) and when r<2^(1/2). The protocol for each individual proof occurs when an element is selected from the set of positive real numbers. This selection of the element does not effect the iterations of the parent function, y=x^2.
How does this proof relate to the artwork?
How does this proof relate to the artwork?
If there are concentric circles created through selecting elements from the set of positive real numbers, then one can create the divisions of the circles based on the intersection at the altered squared functions. This process creates the artwork below
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