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In the PNG picture below I explain the geometry that is behind the definition of
NgonCos[] and NgonSin[]
which I defined today in class. In the PNG picture below I used the function
Mod[t,1]. It is just the function t - Floor[t].
- 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\2025\ under the name 20250220_More_plots.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 Saturday.
- 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.
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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.
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To appreciate the Beauty of Trigonometry, it is essential to understand the key feature of the trigonometric functions Cos[t] and Sin[t]:
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The trigonometric functions Cos[t] and Sin[t] provide a parametrization of the unit circle. But that is just the beginning—most importantly, a deep understanding of the sphere and torus is impossible without Cos[t] and Sin[t].
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The animation below is my attempt to illustrate how the parametrization of the unit circle works.
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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.
- The principle 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, if these functions define the coordinates of a moving point, then as the independent variable t varies, the point traces a curve in the plane.
Place the cursor over the image to start the animation.
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Today in class, a student ask about the difference between the commands Graphics[ ] on one side and Plot[ ]; ParametricPlot[ ] on the other side. The answer that I gave in class is in the notebook 20250214_More_plots.nb in our shared Dropbox directory Dropbox\307_Files\2025.
While answering this question, I started playing with the graph of the sine function. I moved it around, placed it climbing along the \(y\)-axis, coloring each point differently. After the class, I continued playing with this idea and I made two animations that are shown below.
The code for the animations is in the same file. The code is not documented, so it might be cryptic. I might documented, time permitting. You can try asking LLM models whether they understand my code.
Place the cursor over the image to start the animation.
Place the cursor over the image to start the animation.
Place the cursor over the image to start the animation.
- It is important in life to orient oneself in the world of COLORS. To help with that, I wrote a webpage: Color Cube. And all illustrations on the Color Cube page are created in Mathematica.
- Being able to navigate the world of C O L O R S is important. To help with this, I created a page: Color Cube. All illustrations on the Color Cube page were generated using Mathematica.
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To explore graphics Mathematica's graphics functions, we studied the notebook that I wrote: TheBeautyOfTrigonometry.nb.
- You can download this notebook here.
- You can also find it in our shared Dropbox directory Dropbox\307_Files\2025.
- I also created a pdf printout of this notebook. It is a long pdf document; you can read it when you don't have access to Mathematica.
- Another way how you can use the pdf file from the preceding item is to take a snippet from a command that you do not understand, paste the snippet in DeepSeek and ask for an explanation. You can accompany the snippet with a prompt like this: "Can you please explain the Wolfram Mathematica command that I gave you in the attached snippet?"
- ChatGPT does not allow free accounts to upload images. But, when you are working on a Mathematica notebook, you can copy a Mathematica command, and paste it into ChatGPT as text and it will explain the command for you. Remember, in ChatGPT, to start a new line in a prompt, you press Shift+Enter.
- You can also try Copilot. I did not explore its strengths and weaknesses.
- Share your experiences in Discussions on Canvas.
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On Tuesday, we emphasised the important role played by brackets in Mathematica. Pay attention to the parenthesis ( ), the square brackets [ ], the curly brackets, or braces: { }, the double square brackets [[ ]], and finally, the stared parenthasis (* *). To illustrate the double square brackets [[ ]], I talked about the Hall of Fame of Numbers. The Hall of Fame of Numbers is the set consisting from the five numbers: the number one, \(1\), the number zero, \(0\), the number pi, \(\pi\), the number \(e\), and the imaginary unit \(i\). In Mathematica notation, I write this set as a list and name it HFM:
HFN = {1, 0, Pi, E, I}With this notation, I illustrate the use of the double square brackets. To extract the third entry in the preceding list I write:
HFN[[3]]
Next, I will write Mathematica code for a famous mathematical equation and invite you to recognize it:
HFN[[4]]^(HFN[[5]]*HFN[[3]]) + HFN[[1]] == HFN[[2]]
Which famous equality is hidden in Mathematica code above? -
I was curious whether ChatGPT will recognize what is hidden in the Mathematica code above. My prompt and ChatGPT's response is below.
- The Mathematica part of the class will start on Monday, February 10, 2025 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. 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.
- In the study of the 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.
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As we begin the Mathematica portion of this class, a natural question arises: Why learn Mathematica? This is not an easy question to answer, but below are some key aspects that make Mathematica 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 abstract mathematical thinking and structures.
- Mathematica offers powerful visualization tools, allowing you to create compelling graphics that enhance the understanding of mathematical concepts and effectively illustrate abstract mathematical results, fostering a varied and engaging mathematical learning experience.
- 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.
- Mathematica makes solving mathematical problems more efficient. How? For problems involving symbolic manipulation, we can perform symbolic manipulations in Mathematica exactly. For problems in which symbolic manipulations are inefficient, numerical approximations can be performed. Additionally, powerful graphical visualizations help guide the problem-solving process, offering deeper insights.
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I have enjoyed the interaction between my mind and computers since I first started using them in the early 1990s. Recently, I came across a powerful metaphor for this relationship, expressed by Steve Jobs. In an old interview, he illustrated the usefulness of computers with a compelling analogy: "For me, a computer has always been a bicycle for the mind." That perfectly describes Mathematica and other modern tools like ChatGPT. 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.
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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.
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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.
