In my current studies on ** Artificial Inteligence (AI)** I have come across several

**with three features; then,**

*equations***, that describe common phenomena in real life. Those who are familiar with**

*4D surfaces***,**

*AI development***, or**

*computer science***may know already that its basis is in**

*engineering***and**

*linear algebra***which are the basis of**

*geometry***and means that life is a symphony of numbers.**

*calculus*Now, how to understand or try to ** visualize** something that is beyond our comprehension? A 4D surface, three features and one variable, it

**as four axes are needed; however, recalling the course**

*can't be drawn***from my time as a pregrade student, a way to go around this restriction is to draw the**

__calculus III__**which assume**

__level-curves__**of the features to be**

*one***at some value, tipically**

*fixed***.**

*zero*Therefore, where should one start? a good beginning is what I call ** "the Bible"** of calculus (now of AI), the

**Calculus by**

__book__**where in the**

__Strang__**you will find the first interesting 4D equation:**

*page 397*At first sight, a cosine equation is a ** wave line** on the xy plane and a

**in the xyz coordinate system, something as seen below:**

*wave surface*Consequently, a 4D coordinate system ** XYZT** would be the surface in

**value; thus, Z. The following gif leaves it clearer for the reader:**

*3D reshaping as T changes*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

**value as that**

*maximum***. If one fixes**

*equation equals zero***; then, the wave surface will have a maximum value at the**

*t=0***as shown below:**

*plane x=y*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

**or**

*conjugate***-not important for this article-. Therefore, when**

*x=-y***comes**

*t***the equation the**

*into***giving the**

*plane shifts by t***of a**

*sensation***wave.**

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

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