I am a master by research student in statistics. I worked on research projects in both math (recursive relations) and statistics (robust statistics and model selection). The topic I want to use for SOME2 is in the area of complex analysis which is not related to my research but is still super fun. I will try to review the content with someone in the area. However, I am here mainly to find a suitable content producer to make a video with them.
Quick Summary
The XY-plane allows us to visualize the roots of a quadratic equation (y = ax^2 + bx + c) and how they behave as we change the coefficients a, b, and c. However, that's only true for the real roots of the equation. The imaginary roots are completely invisible in the XY-plane! That's also true for cubic equations and higher degree polynomials ... i is always invisible! But not anymore ... today we will extend our vision to the complex plain and shine the spotlights on our cute little friend i. So come on everyone let's put on our 3D i glasses and admire the beauty of math.
Target medium
The plan is to make a video on Youtube and publish Geogebra files. The video will explain the concepts and present interesting observations and results. The Geogebra files will allow the audience to play with the coefficients of polynomials and visualize the effect of that on their extended 3D plots. The focus will be on polynomials of degrees 2, 3, and 4.
I am looking for a collaborator to help me make the video. I can make the graphs that we will use in the video by GeoGebra. I also have a plan for the mathematical content that I want to include in the video (I am open to suggestions though). However, we will discuss and agree together on the visual content and effects that are most appropriate for the video.
More details
The main idea in the video is to extend the 2D XY-plane to a 3D graph that allows us to visualize the complex roots of a polynomial. I will illustrate this with a simple example ... f(X) = X^2 or Y = X^2
If X is complex, then we can represent it by X = (X_real + X_imag i). Similarly, we can represent Y by Y = (Y_real + Y_imag i) If we substitute in f(X) we get:
f(X_real + X_imag i) = (X_real + X_imag i)^2 = (X_real^2 - X_imag^2) + (2 X_real X_imag) i
Therefore, the real part of Y is Y_real = X_real^2 - X_imag^2, and the imaginary part is Y_imag i = 2 X_real X_imag i.
To visualize both inputs (X_real, X_imag) and both outputs (Y_real, Y_imag) we would need a 4D graph. This can be done by using time as the fourth dimension. However, luckily we don't need the fourth dimension. We can use 2 axes for X_real and X_imag and plot Y_real in the third axes but only when Y_imag = 0. This means that if f(X_real + X_imag i) is not real then it will not appear on the 3D graph. Thankfully though, this 3D graph is enough to visualize all the roots and how they behave when we change the coefficients of the polynomial.
To plot our simple example Y = X^2, we need to set Y_imag = 0 and therefore Y_imag = 2 X_real X_imag = 0. There are 2 solutions for this equation:
X_imag = 0 -> Y_real = X_real^2 this will plot a parabola on the X_real axes facing up
X_real = 0 -> Y_real = - X_imag^2 this will plot a parabola on the X_imag axes facing down
see the demonstration below:
https://www.geogebra.org/3d/t8guepyq
When we go to polynomials of higher degrees we start seeing more interesting visualizations. For instance, the function f(X) = X^3 - X has the following graph:
https://www.geogebra.org/3d/r4f8qaac
Ok enough with the spoilers already ... if you are interested please contact me and let's show the world what i looks like (;
Contact details
ibrahim.inj@gmail.com
Additional context
I should note that this form of visualizing complex roots is not a new idea. The standards of novelty in the SOME2 reads as follows:
"It doesn't necessarily have to be an original idea or original topic, but it should offer someone an experience they might otherwise not have by searching around online. Some of the greatest value comes from covering common topics in better ways. Other times there's value in surfacing otherwise obscure ideas which more people should know about"
There are some articles online that explain this form of visualization especially for the quadratic formula. Geogebra wrote an article about this: https://www.geogebra.org/m/U2HRUfDr
The aim in this project is to provide visual illustrations for higher degree polynomials and provide visual interactive tools that help the audience understand the topic better.
About the author
I am a master by research student in statistics. I worked on research projects in both math (recursive relations) and statistics (robust statistics and model selection). The topic I want to use for SOME2 is in the area of complex analysis which is not related to my research but is still super fun. I will try to review the content with someone in the area. However, I am here mainly to find a suitable content producer to make a video with them.
Quick Summary
The XY-plane allows us to visualize the roots of a quadratic equation (y = ax^2 + bx + c) and how they behave as we change the coefficients a, b, and c. However, that's only true for the real roots of the equation. The imaginary roots are completely invisible in the XY-plane! That's also true for cubic equations and higher degree polynomials ... i is always invisible! But not anymore ... today we will extend our vision to the complex plain and shine the spotlights on our cute little friend i. So come on everyone let's put on our 3D i glasses and admire the beauty of math.
Target medium
The plan is to make a video on Youtube and publish Geogebra files. The video will explain the concepts and present interesting observations and results. The Geogebra files will allow the audience to play with the coefficients of polynomials and visualize the effect of that on their extended 3D plots. The focus will be on polynomials of degrees 2, 3, and 4.
I am looking for a collaborator to help me make the video. I can make the graphs that we will use in the video by GeoGebra. I also have a plan for the mathematical content that I want to include in the video (I am open to suggestions though). However, we will discuss and agree together on the visual content and effects that are most appropriate for the video.
More details
The main idea in the video is to extend the 2D XY-plane to a 3D graph that allows us to visualize the complex roots of a polynomial. I will illustrate this with a simple example ... f(X) = X^2 or Y = X^2
If X is complex, then we can represent it by X = (X_real + X_imag i). Similarly, we can represent Y by Y = (Y_real + Y_imag i) If we substitute in f(X) we get: f(X_real + X_imag i) = (X_real + X_imag i)^2 = (X_real^2 - X_imag^2) + (2 X_real X_imag) i Therefore, the real part of Y is Y_real = X_real^2 - X_imag^2, and the imaginary part is Y_imag i = 2 X_real X_imag i. To visualize both inputs (X_real, X_imag) and both outputs (Y_real, Y_imag) we would need a 4D graph. This can be done by using time as the fourth dimension. However, luckily we don't need the fourth dimension. We can use 2 axes for X_real and X_imag and plot Y_real in the third axes but only when Y_imag = 0. This means that if f(X_real + X_imag i) is not real then it will not appear on the 3D graph. Thankfully though, this 3D graph is enough to visualize all the roots and how they behave when we change the coefficients of the polynomial.
To plot our simple example Y = X^2, we need to set Y_imag = 0 and therefore Y_imag = 2 X_real X_imag = 0. There are 2 solutions for this equation: X_imag = 0 -> Y_real = X_real^2 this will plot a parabola on the X_real axes facing up X_real = 0 -> Y_real = - X_imag^2 this will plot a parabola on the X_imag axes facing down see the demonstration below: https://www.geogebra.org/3d/t8guepyq
When we go to polynomials of higher degrees we start seeing more interesting visualizations. For instance, the function f(X) = X^3 - X has the following graph: https://www.geogebra.org/3d/r4f8qaac
Ok enough with the spoilers already ... if you are interested please contact me and let's show the world what i looks like (;
Contact details
ibrahim.inj@gmail.com
Additional context
I should note that this form of visualizing complex roots is not a new idea. The standards of novelty in the SOME2 reads as follows: "It doesn't necessarily have to be an original idea or original topic, but it should offer someone an experience they might otherwise not have by searching around online. Some of the greatest value comes from covering common topics in better ways. Other times there's value in surfacing otherwise obscure ideas which more people should know about"
There are some articles online that explain this form of visualization especially for the quadratic formula. Geogebra wrote an article about this: https://www.geogebra.org/m/U2HRUfDr The aim in this project is to provide visual illustrations for higher degree polynomials and provide visual interactive tools that help the audience understand the topic better.