Math 307 - Spring 2024

Thursday, May 16, 2024

The notebook 20240516_A2P1_pure_functions.nb in our shared Dropbox directory Dropbox\307_Files\2024 contains what we discussed today. This file relates to Problem 1 on Assignment 2. In this file, I demonstrated how Mathematica Plot[] can be unreliable to give complete information about a function that we are exploring.
Before discussing problems posted in Assignment 2, we discussed a highly useful concept in Mathematica: the concept of pure function. In the code in 20240516_A2P1_pure_functions.nb, pay attention to occurrences of pure functions in action to create many points on a graph of a function.

Pure Function. In earlier problems, we often assigned a name to a function for further use. However, this approach is not always practical. Mathematica allows us to use functions as if they were named without actually naming them. For example, the square function is not explicitly named in Mathematica, but we can calculate the square of 7 by writing the following code: (#^2)&[7]. In this case, we use (#^2)& as a reference to the square function. Instead of explicitly naming the argument (independent variable), we use # which is called a slot in Mathematica. It acts as a placeholder for the argument that will be passed to the function. The symbol & is called function operator in Mathematica. It indicates the end of a pure function definition. For example, in the pure function (#^2)& the expression (#^2) defines the function, and ampersand & marks the end of the function.

The full potential of a pure function is realized when combined with the Map[] command. For instance, Map[(#^2)&,{a,b,c}] yields {a^2,b^2,c^2}. Thus, Map[] applies the specified function to each member of the given list. An alternative shortcut form for Map[] is /@. Using this shortcut, the previous example becomes (#^2)&/@{a,b,c}. This approach is extremely beneficial when dealing with complex functions and large lists containing hundreds of entries.

Monday, May 13, 2024

Today I created the notebook 20240513_Toroid. You can find it in our shared Dropbox directory \307_Files\2024\. In this notebook I explain how to create cylinders, twisted cylinders, vases, twisted vases, toroid, and twisted toroid.
Below I present two animations which show two ways to construct the one-third-twisted triangular toroid.

In this item, the one-third-twisted triangular toroid is constructed by a rotating equilateral triangle. The triangle is rotated by the full circle, that is the angle $2\pi$.

Place the cursor over the image to start the animation.

The code that is used for the above picture (not the animation) is below:


Show[ParametricPlot3D[
  3 {Cos[s], Sin[s], 0} +
   NgonCos[t, 3] (Cos[s/3] {Cos[s], Sin[s], 0} + Sin[s/3] {0, 0, 1}) +
    NgonSin[t,3] (-Sin[s/3] {Cos[s], Sin[s], 0} + Cos[s/3] {0, 0, 1}), {t, 0,
   3}, {s, 0, 2 Pi}, PlotPoints -> {51, 201},
  PlotStyle -> {Opacity[0.9]}, Mesh -> False, Exclusions -> None,
  PlotRange -> {{-4, 4}, {-4, 4}, {-1.1, 1.1}}],
 ParametricPlot3D[
  3 {Cos[s], Sin[s], 0} +
   NgonCos[0, 3] (Cos[s/3] {Cos[s], Sin[s], 0} + Sin[s/3] {0, 0, 1}) +
    NgonSin[0,
     3] (-Sin[s/3] {Cos[s], Sin[s], 0} + Cos[s/3] {0, 0, 1}), {s, 0,
   2 Pi}, PlotPoints -> {201},
  PlotStyle -> {RGBColor[0, 0, 0.75], Thickness[0.006]},
  Exclusions -> None],
 ParametricPlot3D[
  3 {Cos[s], Sin[s], 0} +
   NgonCos[1, 3] (Cos[s/3] {Cos[s], Sin[s], 0} + Sin[s/3] {0, 0, 1}) +
    NgonSin[1, 3] (-Sin[s/3] {Cos[s], Sin[s], 0} + Cos[s/3] {0, 0, 1}), {s, 0,
   2 Pi}, PlotPoints -> {201},
  PlotStyle -> {RGBColor[0.95, 0, 0], Thickness[0.006]},
  Exclusions -> None],
 ParametricPlot3D[
  3 {Cos[s], Sin[s], 0} +
   NgonCos[2, 3] (Cos[s/3] {Cos[s], Sin[s], 0} + Sin[s/3] {0, 0, 1}) +
    NgonSin[2, 3] (-Sin[s/3] {Cos[s], Sin[s], 0} + Cos[s/3] {0, 0, 1}), {s, 0,
   2 Pi}, PlotPoints -> {201},
  PlotStyle -> {RGBColor[0, 0.85, 0], Thickness[0.006]},
  Exclusions -> None], ImagePadding -> {{0, 0}, {0, 0}},
 Boxed -> True, BoxStyle -> {Opacity[0]}, Axes -> False,
 ImageSize -> 1000, PlotRange -> {{-4, 4}, {-4, 4}, {-1.1, 1.1}}]

The above code uses the functions that we defined earlier:


Clear[NgonCos, NgonSin];
NgonCos[t_, nn_] := (1 - (t - Floor[t])) Cos[(2/nn) \[Pi] Floor[t]] + (t -
      Floor[t]) Cos[(2/nn) \[Pi] (1 + Floor[t])];
NgonSin[t_, nn_] := (1 - (t - Floor[t])) Sin[(2/nn) \[Pi] Floor[t]] + (t -
      Floor[t]) Sin[(2/nn) \[Pi] (1 + Floor[t])];

In this item, the one-third-twisted triangular toroid is constructed by a rotating line segment. The line segment is rotated by the three full circles, that is by the angle $6\pi$.

Place the cursor over the image to start the animation.

The code that is used for the above picture (not the animation) is below:


Show[ParametricPlot3D[
  3 {Cos[s], Sin[s], 0} +
   NgonCos[t, 3] (Cos[s/3] {Cos[s], Sin[s], 0} + Sin[s/3] {0, 0, 1}) +
    NgonSin[t, 3] (-Sin[s/3] {Cos[s], Sin[s], 0} + Cos[s/3] {0, 0, 1}), {t, 0,
   1}, {s, 0, 6 Pi}, PlotPoints -> {51, 201},
  PlotStyle -> {Opacity[0.9]}, Mesh -> False, Exclusions -> None,
  PlotRange -> {{-4, 4}, {-4, 4}, {-1.1, 1.1}}],
 ParametricPlot3D[
  3 {Cos[s], Sin[s], 0} +
   NgonCos[0, 3] (Cos[s/3] {Cos[s], Sin[s], 0} + Sin[s/3] {0, 0, 1}) +
    NgonSin[0, 3] (-Sin[s/3] {Cos[s], Sin[s], 0} + Cos[s/3] {0, 0, 1}), {s, 0,
   6 Pi}, PlotPoints -> {201},
  PlotStyle -> {RGBColor[0, 0, 0.75], Thickness[0.006]},
  Exclusions -> None], ImagePadding -> {{0, 0}, {0, 0}},
 Boxed -> True, BoxStyle -> {Opacity[0]}, Axes -> False,
 ImageSize -> 1000, PlotRange -> {{-4, 4}, {-4, 4}, {-1.1, 1.1}}]

The above code uses the functions that we defined earlier:


Clear[NgonCos, NgonSin];
NgonCos[t_, nn_] := (1 - (t - Floor[t])) Cos[(2/nn) \[Pi] Floor[t]] + (t -
      Floor[t]) Cos[(2/nn) \[Pi] (1 + Floor[t])];
NgonSin[t_, nn_] := (1 - (t - Floor[t])) Sin[(2/nn) \[Pi] Floor[t]] + (t -
      Floor[t]) Sin[(2/nn) \[Pi] (1 + Floor[t])];

Saturday, May 11, 2024

On Friday, we introduced functions inspired by the trigonometric functions cosine and sine, but designed to draw regular ngons, for example equilateral triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, hendecagon, dodecagon, and so on. The notebook that I created today you can find in our shared Dropbox directory \307_Files\2024\ under the name 20240510_A1P2.nb.
The most important feature of the functions Cos[t] and Sin[t] is that they provide a parametrization of the unit circle. An animation illustrating this feature was posted on Thursday.
The principle presented in the animation posted on Thursday is the general principle of parametric equations of curves in the plane. Given two functions (one green and one blue) of the same independent variable t, taken as the coordinates of a moving point, then, as the independent variable t changes, the point does trace a curve in the plane.
In the PNG picture below I explain the geometry that is behind the definition of NgonCos[] and NgonSin[] which I defined in class. In the PNG picture below I used the function Mod[t,1]. In class I used the equivalent expression t-Floor[t]. In fact you can check in Mathematica that
```
FullSimplify[Mod[t, 1] == t - Floor[t]]
```
The result of the preceding command is True, meaning that Mathematica recognizing the above identity as correct.

The formulas derived in the preceding item, implemented in Mathematica are as follows:


Clear[NgonCos, NgonSin];
NgonCos[t_, nn_] := (1 - (t - Floor[t])) Cos[(2/nn) \[Pi] Floor[t]] + (t -
      Floor[t]) Cos[(2/nn) \[Pi] (1 + Floor[t])];
NgonSin[t_, nn_] := (1 - (t - Floor[t])) Sin[(2/nn) \[Pi] Floor[t]] + (t -
      Floor[t]) Sin[(2/nn) \[Pi] (1 + Floor[t])];

The final product of this work are the graphs of the first nine regular $n$-gons inscribed in the unit circel: the equilateral triangle, the square, and the pentagon, the hexagon, the heptagon, the octagon, the nonagon, the decagon, the hendecagon, the dodecagon, see the Regular Polygon webpage at Wolfram MathWorld.

Thursday, May 9, 2024

Today we introduced functions inspired by the trigonometric functions cosine and sine, but designed to draw regular ngons, for example equilateral triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, hendecagon, dodecagon, and so on. The notebook that I created today you can find in our shared Dropbox directory \307_Files\2024\ under the name 20240222_ngons.nb.
To appreciate the Beauty of Trigonometry, one needs to understand the most important feature:

The trigonometric functions Cos[t] and Sin[t] provide a parametrization of the unit circle.

The animation below illustrates how this parametrization is obtained.
The main components of the animation are as follows
- The black circle is the unit circle.
- The values of t $\in [0,2\pi]$ are represented by orange lengths. In each scene there are three orange lengths: the horizontal length, the vertical length and the arc length on the unit circle.
- The values of Cos[t] are green horizontal line segments.
- The values of Sin[t] are blue vertical line segments.

Place the cursor over the image to start the animation.

The principle presented above is the general principle of parametric equations of curves in the plane. Given two functions (one green and one blue) of the same independent variable t, taken as the coordinates of a moving point, then, as the independent variable t changes, the point does trace a curve in the plane.

Wednesday, May 8, 2024

In this part of the class we study computational software system Wolfram Mathematica. We will focus on symbolic and numerical calculations, visualization, and programming in Mathematica. But Mathematica is much more than that. To read more about Mathematica see An Elementary Introduction to the Wolfram Language.
The following question naturally arises as we start with the Mathematica portion of this class: Why learn about Mathematica? This is not an easy question. Here are some aspects of Mathematica that make it worth your attention.
- Mathematica is a powerful computational tool that can handle a wide range of mathematical problems, from basic algebra to advanced calculus and beyond.
- Unlike many other mathematical software systems, Mathematica excels in symbolic computation, allowing you to manipulate mathematical expressions and equations symbolically closely resembling mathematical structures and thinking.
- Mathematica provides sophisticated visualization capabilities, enabling you to create compelling graphical representations of your mathematical results.
- Mathematica is a high-level programming language that offers an extensive array of predefined functions, designed to simplify and streamline tasks that are commonly encountered in mathematical and technical computing. My impression is that this makes it more accessible and user-friendly compared to other mathematical software, enabling users to achieve complex objectives with greater ease.
- A lot of help is available on-line, for example Mathematica Stack Exchange, Wolfram Community and ChatGPT is quite good at offering help with Mathematica questions.
- Solving mathematical problems is easier using Mathematica. How so? If a problem involves symbolic manipulations, we can do them in Mathematica exactly; if that approach does not lead to a solution, we can perform numerical approximations, and to guide us toward the process, we can use graphical visualization.
I have enjoyed the interaction between my mind and computers since I first started using them in the early 1990s. Only recently, I encountered a powerful metaphor for the relationship between the mind and computers expressed by Steve Jobs. In an old interview, Steve Jobs illustrated the usefulness of computers with a compelling analogy. He said: "For me, a computer has always been a bicycle for the mind." That is what ChatGPT is. Isn't that wonderful!
See the Bicycle for the Mind video. Or, a longer version.
I am convinced that your mathematical learning experience will be enriched if you engage in mathematical explorations using technology.
I recently encountered a fascinating geometric fact online: Imagine two squares where a vertex of the larger square is positioned at the center of the smaller square. Regardless of how these two squares are positioned, the area of their intersection remains constant. My description might not fully capture the beauty of this fact — indeed, an animation is worth a thousand words. How am I to make an animation? I must use technology. Here is an animation created in Mathematica.

Place the cursor over the image to start the animation.
I should have left it as an exercise for you to figure out why what I claim in the preceding item is true. But, the solution supported by a Mathematica animation might indicate why using technology can be powerful tool in communicating Mathematics.
Place the cursor over the image to start the animation.

Monday, April 1, 2024 (updated on Sunday, May 5, 2024)

The Mathematica part of the class will start on Monday, May 6, 2024 in BH 215 at 2:00 pm.
The information sheet
We will use
To get started with Mathematica see my Mathematica page. Please watch the videos that are on my Mathematica page before the first class. Watching the movies is essential for being able to organize your homework notebooks well! I will not discuss the basics of notebook structuring in class.

Spring 2024
MATH 307: Mathematical computing
with Mathematica

Branko Ćurgus

Spring 2024 MATH 307: Mathematical computing with Mathematica

Branko Ćurgus

Spring 2024
MATH 307: Mathematical computing
with Mathematica