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Algebra plus Geometry equal Calculus: A Detailed Explanation to Understand Three-Feature Equations on 4D Movements Applications for AI

In my current studies on Artificial Inteligence (AI) I have come across several equations with three features; then, 4D surfaces, that describe common phenomena in real life. Those who are familiar with AI development, computer science, or engineering may know already that its basis is in linear algebra and geometry which are the basis of calculus and means that life is a symphony of numbers.


Now, how to understand or try to visualize something that is beyond our comprehension? A 4D surface, three features and one variable, it can't be drawn as four axes are needed; however, recalling the course calculus III from my time as a pregrade student, a way to go around this restriction is to draw the level-curves which assume one of the features to be fixed at some value, tipically zero.


Therefore, where should one start? a good beginning is what I call "the Bible" of calculus (now of AI), the book Calculus by Strang where in the page 397 you will find the first interesting 4D equation:


At first sight, a cosine equation is a wave line on the xy plane and a wave surface in the xyz coordinate system, something as seen below:


Consequently, a 4D coordinate system XYZT would be the surface in 3D reshaping as T changes value; thus, Z. The following gif leaves it clearer for the reader:



cosine equation, cosine wave, python, plotly, jupyter notebook
z=cos(x-y-t) - Source: Own elaboration

Why does the cosine behave like that? Remember that inside the cosine the features x, y, and t compound the following equation: "x- y - t"; hence, the cosine will have a maximum value as that equation equals zero. If one fixes t=0; then, the wave surface will have a maximum value at the plane x=y as shown below:


cosine equation, cosine wave, python, plotly, jupyter notebook, complex planes
z=cos(x-y) - Source: Own elaboration

As one can see in the top image, the plane x=y is exactly at the maximum value of the cosine -pay attention to the scale of x and y axes as it is not in the middle, reason why the graph seems to be not in such plane-, the bottom one is the same surface but for the conjugate or x=-y -not important for this article-. Therefore, when t comes into the equation the plane shifts by t giving the sensation of a moving wave.


For more details on computations check the code on Google Colab here.


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