- The construction of the Cesaro fractal is somewhat similar to the construction of the von Koch curve in the file 20200505_A2P5.nb. The main difference is that the Cesaro fractal depends on an angle $\alpha \in [0,\pi/2].$ So, in the construction in the first step towards the Cesaro fractal we need to use the trigonometry to figure out the location of the points pC, pD, pE. To help you out I wrote a notebook in GoodNotes on my iPad. Please pay attention, that the construction of the Cesaro curve involves a variable $\alpha \in [0,\pi/2].$ So, all the formulas, in addition to two points, or a list of points must involve this variable $\alpha \in [0,\pi/2].$ The reward for this work is that in the end we can animate the Cesaro fractals in Manipulate[]
- Today I did live coding of the Koch curve in the file 20200505_A2P5.nb which is in 307_Files\2020. I felt that the construction went quite smoothly today. In the past I constructed the Koch curve in vonKoch_curve.nb in the same directory. In the same directory there is a more complicated fractal called Pythagora's Tree: PythagorasTree.nb. All these files can serve you as "guides" for constructions of fractals in Problem 5.
- The first step towards a construction of the von Koch curve I did in a GoodNotes notebook.
- In the first part of Problem 5 you are asked to create a function that would produce iterations of the quadratic type 2 curve. In the first picture below I show the 0th iteration in blue and the 1st iteration in red. I emphasize the points that are used. You do not need to do this on your plots. In the second picture below I show the 1st iteration in blue and the 2nd iteration in red. The large picture is the fourth iteration of the quadratic type 2 curve.
- In the second part of Problem 5 you are ask you to create a function that would produce iterations of the Cesaro fractal which depends on angle $\alpha$. The four pictures below show the 0th, 1st, 2nd and the 3rd iteration of the Cesaro fractal with the small angle $\alpha = \pi/16$.
- Below is an animated gif file that cycles through the 1st, 2nd, 3rd, 4th, 5th and the 6th iteration of the Cesaro fractal and, within each of the iterations cycles through all the angles starting from $\pi$, proceeding towards $0$ and then back to $\pi$ in steps of $\pi/50$. The animation starts by the first iteration and cycles through angles from $\pi$ to $0$ and back to $\pi$. This is repeated for 2nd, 3rd, 4th 5th and 6th iteration. For each iteration there are 101 pictures.
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Place the cursor over the image to start the animation.
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Today we worked on Problem 4 on modified Assignment 2. The notes are in the notebook started yesterday 20200504_A2P4.nb. As always, this notebook is in the folder 307_Files\2020 in Dropbox. After class I added more explorations related to Goldbach partitions of even positive integers. An interesting function to look at is the count of all Goldbach partitions of a specific even positive integer; call this count the Goldbach length of a positive even number. Here are the first Goldbach lengths:
{18, 2}, {20,2}, {22, 3}, {24, 3}, {26, 3}, {28, 2},
{30, 3}, {32, 2}, {34, 4}, {36, 4}, {38, 2}, {40, 3}}
Below are Goldbach lengths of 500000 positive even integers, my recreation of the picture from Wikipedia.
- Today we further worked on Problem 3 on Assignment 2. The notes are in the notebook started yesterday 20200430_A2P3.nb. As always, this notebook is in the folder 307_Files\2020 in Dropbox.
- To answer your question about geometric probabilities I wrote these notes in OpenBoard freeware white-board.
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Today we further worked on the involute of the unit circle. Let us celebrate it by the animation below. We also started working on Problem 3 on Assignment 2. The notes of today's explorations are in the notebook 20200430_A2P3.nb. As always, this notebook is in the folder 307_Files\2020 in Dropbox.
Place the cursor over the image to start the animation.
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Please ask questions whenever something is not clear.
Mathematics is a field in which annihilation of confusion is possible. Let us annihilate confusion in mathematics together. Let us bring out the clarity and the beauty of mathematics.
- Today we studied a geometric construction related to Problem 2 on Assignment 2. The notes of today's explorations are in the notebook 20200428_A2P2.nb. As always, this notebook is in the folder 307_Files\2020 in Dropbox. In the notebook 20200428_A2P2.nb we presented the construction of the involute of the unit circle. The involute of the unit circle is a particular curve related to the unit circle. Its non-mathematical definition is as follows. Let a spool of string has a shape of the unit circle. The endpoint of this string, as it unwraps from this spool, draws an involute of the unit circle. I explained the vector geometry of the unwrapping process in this drawing. This vector geometry is implemented in Mathematica in 20200428_A2P2.nb.
- Another curve that you are asked to explore in Problem 2 on Assignment 2 is a cardioid. Below I will describe how to obtain a parametrization of a cardioid.
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Here is a derivation of the equation of a cardioid.
Click through several helpful images.
- The origin $O = (0,0)$ of the coordinate system is the gray point. Denote by $C$ the black point in the picture.
- Both circles in the picture are unit circles, the gray one centered at $O,$ the black one centered at $C.$
- The angle at the gray point marked by the blue arc between the green and the blue point measures $t$ radians. That is, the length of this blue arc is $t.$
- There are two other angles in the picture which also measure $t$ radians: the angle marked by the blue arc between the blue and the red point and the angle marked by the black arc between the blue point and the purple point.
- The green point is $(1,0),$ the blue point is $(\cos t, \sin t),$ the black point $C$ is $(2\cos t, 2\sin t),$ the purple point is $(2\cos t-1, 2\sin t).$
- The goal is to determine the coordinates of the red point.
- Denote by $\mathbf{a}$ the unit vector whose tail is the black point and whose head is the red point. Based on the previous items we have \begin{align*} \mathbf{a} & = \bigl(\cos( \pi+2 t ) \bigr) \mathbf{i} + \bigl(\sin( \pi+2 t ) \bigr) \mathbf{j} \\ & = -\bigl(\cos(2 t ) \bigr) \mathbf{i} - \bigl(\sin(2 t ) \bigr) \mathbf{j}. \end{align*}
- Denote by $\mathbf{c}$ the position vector of the black point. Then the position vector of the red point is \[ \mathbf{c} + \mathbf{a} = \bigl( 2 \cos t - \cos(2 t ) \bigr) \mathbf{i} + \bigl(2 \sin t - \sin(2 t ) \bigr) \mathbf{j}. \]
- Thus the parametric equations for this cardioid are \begin{align*} x & = 2 \cos t - \cos(2 t ) \\ y & = 2 \sin t - \sin(2 t ) \end{align*}
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In the Cardioid part of Problem 2 on Assignment 2 you need to reproduce two pictures from Wikipidia's
Cardioid page
I made an animation that unifies these two pictures. You should be able to produce something like this. If not, you can present one animation and one picture.
Place the cursor over the image to start the animation.
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Once you have figured out how to consruct a cardioid, you should be able to consturct Generalized Cardioids. In this part of Problem 2 you need to produce generalized cardioids as shown in the animations below. These animated gif files are large. On a slow internet connection it takes a while for them to load.
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Below is the cardioid generated by a wheel with radius 1/2 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 1/3 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 2 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 3/2 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 1/2 rolling on a circle with radius 1.
- Today finished Problem 1 onAssignment 2. The notes of our explorations are in the notebook 20200423_A2P1.nb which is, as always, in the folder 307_Files\2020 in Dropbox.
- The next task is to understand how to derive a parametrization of the cycloid and a parametrization of the cardioid and construct animations related to these parameterizations.
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Wikipedia's definition of a
Cycoid is: A cycloid
is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. One could demonstrate a cycloid in a dark room by attaching a small light to a rolling wheel and taking a long exposure photography of the motion. This is what American photographer Berenice Abbott did to create her photograph Cycloid:
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In this item I will illustrate the derivation of the parametric equations for the cycloid which you need to implement in the animations in Mathematica.
Let $t\in [0,\pi].$ The position vector of the gray center of the circle in the above pictures is \[ t \, \mathbf{i} + \mathbf{j}. \] If the unit circle was positioned at the origin, then the position vector of the red point on the unit circle would be \[ \bigl(\cos(-t -\pi/2) \bigr) \, \mathbf{i} + \bigl(\sin(-t -\pi/2) \bigr) \mathbf{j} = - (\sin t ) \, \mathbf{i} - (\cos t ) \mathbf{j}. \] Consequently, the position vector of the red point on the unit circle in the above pictures is \[ \bigl( t - \sin t \bigr) \, \mathbf{i} + \bigl(1 - \cos t \bigr) \mathbf{j}. \] In Mathematica notation the vector $\mathbf{i}$ is the pair {1,0} and the vector $\mathbf{j}$ is the pair {0,1}. Thus, the parametric equations of the cycloid in Mathematica notation is(t - Sin[t]) {1,0} + (1 - Cos[t]) {0,1} = {t - Sin[t] , 1 - Cos[t] } The following animation might help you to understand the above derivation.Place the cursor over the image to start the animation.
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Problem 2 consists of two parts: cycloids and cardioids. In the first part I ask you to reproduce the following animations.
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.
- Today continued our investigations related to Problem 1 onAssignment 2. The notes of today's explorations are in the notebook started yesterday 20200423_A2P1.nb and which is, as always, in the folder 307_Files\2020 in Dropbox.
- Today we started with Assignment 2. We did talk about Problem 1. The notes of our explorations are in 20200423_A2P1.nb which is as always in the folder 307_Files\2020 in Dropbox.
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On your assignment notebooks:
- Assignment 1 is due on Sunday, April 26, 2020 at 23:59.
- Your assignment should be submitted as one notebook. This notebook should be named Yourlastname_A1.nb. The notebook Yourlastname_A1.nb should be placed in the Dropbox folder 307_Yourlastname which you shared with me. After I am done with grading, I will put the notebook named Yourlastname_A1g.nb in the same folder. This notebook will contain my comments on your work and the numerical grade for your assignment.
- Your assignment notebooks should be organized neatly. A notebook should start with a title cell. Assigned problems should be presented as sections. The structure of your assignment notebook should follow the structure of my notebook in which I presented the problems. However, your assignment notebook should not contain any of my text and calculations that I presented while introducing problems. Your notebook should contain only your work.
- Your work on each problem should contain a sufficient amount of text (placed in text cells) so that I can make sense of what is being presented. If I ask a specific question in a problem, then you should answer that question in English. Your answer should be followed by a justification.
- When your notebook is almost ready for submission do the following test. Save the notebook and close Mathematica. Start Mathematica and open your assignment notebook. Start a stopwatch on your phone or use online stopwatch in 307_Files\Assignments. Click on , then click on . The keyboard shortcut for this operation is .
- Make sure that your notebook evaluates completely and properly. Make sure that your notebook evaluates completely in under 90 seconds.
- At this point, and always, before saving your notebook delete all output deleted: Click on , then click on . The keyboard shortcut for this operation is . (Alt-ce el)
- In 307_Files\2020 I created the notebook TheBeautyOf_Octagonal_Trigonometry.nb in this file I use the parametrization of the octagon that we obtained earlier to explore the relevant parts of TheBeautyOfTrigonometry_small.nb. Since the functions that parametrize octagon satisfy all the properties of "funny" cosine and sine, all the explorations in TheBeautyOfTrigonometry_small.nb make sense for these functions. In particular, in this file you can find parametric plot of a "octagonal" spiral, "octagonal" sphere, a "octagonal" torus.
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Today we finished explorations related to Problem 4 on Assignment 1. The explorations are in the file 20200421_A1P4.nb in folder 307_Files\2020. This file should provide you with sufficient guidance for what to do in Problem 4. In our in class explorations the final product was a special circle, so I created the function
CirCircle[{pA1_,pA2_},{pB1_,pB2_},{pC1_,pC2_}] which is a function of three points. In Problem 4, the final product is a formula for a parabola. Therefore, in addtion to three points, it should be a function of an indepndent variable, say t. At the end of 20200421_A1P4.nb I will add the code that should help you to get the final Manipulation correctly.
- In 307_Files\2020 I created the notebook TheBeautyOf_Pentagonal_Trigonometry.nb in this file I use the parametrization of the pentagon that we obtained earler to explore the relevant parts of TheBeautyOfTrigonometry_small.nb. In particular, in this file you can find parametric plot of a "pentagonal" spiral, "pentagonal" sphere, a "pentagonal" torus. However, several parts of TheBeautyOfTrigonometry_small.nb do not make sense since the functions which parametrize pentagon do not satisfy the definition of "funny" trigonometric functions; these functions are not shifts of each other. In fact, I shoule heave explored TheBeautyOfTrigonometry_small.nb for octagon. Looking forward to that.
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Here are few remarks, in writing, about Problem 2 on Assignment 1.
- In several parts of this problem you have to define two kinds of functions, one recursively defined and one defined by a closed form expression. Please make sure that you distinguish which one is which. The structure how we define these different kinds is completely different. Do not mix them up. Also, we do not use "dynamic programming" (memory) with functions defined by closed form expressions.
- In each item in the problem, two kinds of functions from the previous item should give the same output. Although, this is a math class, I am not asking you to prove this. However, please make sure that you test that the functions give the same output on thousands of positive integers. I showed how to do this during one of the classes.
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Here are few remarks, in writing, about Problem 3 on Assignment 1.
- This problem has two quite distinct parts. In Part A, for a fixed positive odd integer, call it $m$, we explore values that the expression under investigation takes for different permutations of coefficients in numerators. Then we explore small integers $m = 5, 7, 9,$ (I am not sure if $11$ is doable or not in BH 215. Please try.) I almost finished the explorations for Sin[x]. You need to complete the investigations for Cos[x] and summarize results of your investigations in English, addressing all questions that I asked.
- Part B is inspired with a special permutation from Part A. That is the end of the connection between two pars. In Part B, we explore special sums of the squares of the function Csc[x]. I did most of the explorations for Csc[x]. Explore both odd and even fractions of $\pi.$ You should also explore the analogus sums for Sec[x]. Summarize your findings in English and answer all my questions clearly. Support your answers with Mathematica calculations. .
- Very important tool in Mathematica that we encountered discussing Problem 3 on Assignment 1 and on this assignment is the concept of Pure Function. For exapmple, this is the square function as a pure function: (#^2)&[x]. The output is x^2.
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Here are few remarks, in writing, about Problem 1 on Assignment 1.
- Your "funny" trigonometric functions must be defined for all real numbers.
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It is clear from the descriptions and graphs of "funny" trigonometric functions what should be their values at any real number. Use tables with rational numbers to check whether your definition gives the values that are indicated in the description. For example, for LiCos[x] the table of values
Table[{x,LiCos[x]},{x,-1,1,1/2}]
should give the following table
{{-1,0},{-1/2,1/2},{0,1},{1/2,1/2},{1,0}}.
However, to make sure that your functions is correct you should generate larger tables. - When exploring "funny" unit circles make sure that you understand the meaning of parametric equations as illustrated in the post on April 9, 2020 below. Sometimes Mathematica's graph of a unit circle generated by ParametricPlot[] can be misleading.
- When exploring TheBeautyOfTrigonometry_small.nb make sure that you make all the necessary adjustment for domains of independent variables. The domains in TheBeautyOfTrigonometry_small.nb are all related to Pi. It is clear that Pi has nothing to do with LiSin[] nor with TrapSin[]. To be able to make all the adjustments you need to understand the role of each of the independent variable.
- Since your formulas for "funny" trigonometric functions are somewhat complicated Mathematica gets confused and thinks that some of these functions are discontinuous. We, humans, know that the funny trig functions that we use to explore the beauty of funny trigonometry are continuous. To communicate this simple fact to Mathematica use the option Exclusions->None.
- Some three-dimensional graphs in your explorations of the Beauty of Funny Trigonometry might not look perfect. If the edges are not sharp you can play with the option PlotPoints->{61,31}
- Some manipulations in the your explorations of the Beauty of Funny Trigonometry can be slow. If that happens, then make sure that all your formulas and domains are correct and try adjusting PlotPoints. If this does not work leave your code in your homework notebook but comment out the slow code by bounding it with the stared parenthesis (* *).
- We further discussed Problem 1 and started the discussion of Problem 2 on Assignment 1 today.
- The second part of Problem 1 requires to repeat the explorations in the notebook TheBeautyOfTrigonometry_small.nb with the standard trigonometric functions Cos[t] and Sin[t] replaced with a pair "funny" trigonometric functions LiCos[t], LiSin[t], or TrapCos[t], TrapSin[t], or RoCos[t], RoSin[t]. For this task, it is essential for you to pay attention to domains that you need in each plot. In TheBeautyOfTrigonometry_small.nb I deal with Cos[t] and Sin[t], so the most important domain is [0,2 Pi] and related smaller or sometimes larger domains. When you work with "funny" trigonometric functions this domain will be different. Please also adjust Ticks.
- We also discussed the concept of recursively defined functions which is needed for Problem 2. The notebook 20200414_A1P2.nb contains the relevant discussion. In this notebook I explain how to define recursive functions in Mathematica. Please notice the big distinction between recursively defined functions and functions defined by closed form expressions. In your homework please make clear which functions are recursively defined and which are defined by closed form expression. Read more about recursively defined functions on my Mathematica page.
- We discussed Problem 1 on Assignment 1 today. The notebook 20200413_A1P1.nb contains this discussion. A typical definition of a function in Mathematica is given in the snippet posted on Saturday. First Clear[ ] the name of the functions that you will use and the variables that you will use. The syntax of a function definition is that the variables in the definition on the left-hand side have underscores. separate Clear[ ] and the definitions with ;.
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Inspired by the parametrization of the unit circle presented yesterday, during the class on Friday I derived a parametrization of a regular $n$-gon inscribed in the unit circle. Here $n$ is a positive integer greater than $2.$ In class I derived the functions analogous to Cos[t] and Sin[t].
Here is a sketch of that derivation. We derived the parametrization functions to be
- 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 Wikipedia page.
- The Problem 1 on Assignment 1 is inspired by the trigonometric functions Cos[t] and Sin[t]. The most important feature of the functions Cos[t] and Sin[t] is that they provide a parametrization of the unit circle. The animation below illustrates how this parametrization is obtained.
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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 principal presented above is the general principle of parametric equations of curves in the plane. Given two functions, one green and one blue, they do trace a curve in the plane. Tomorrow I want to discuss the converse question: given a curve in the plane, for example aan octagon inscribed in the unit circle, find a green and a blue function that do form that octagon.
Place the cursor over the image to start the animation.
- Your homework for Tuesday and Wednesday is to watch the movies posted on my Mathematica page and to read six short sections after the movies.
- I updated the post of yesterday. Please read it again.
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Here I will summarize what you need to set up on your computer to start with this class:
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Open a Dropbox account. I sent you an email about this process. This email is posted as an announcement on Canvas. In your Dropbox account you should have these three folders
- \307_Yourlastname This is a folder in which you will place your assignment notebook.
- \307_Files This is a folder in which you can find a Mathematica notebooks related to the class. In this folder you can find the file 00_Read_me_307_Files.pdf which explains the structure of the folder.
- \307_Questions This is a folder in which you can put small notebooks with your questions. In this folder you can find the file 00_Read_me_307_Questions.pdf which establishes some conventions for using this folder. When you place your question in this folder it might be a good idea to post a discussion topic on Canvas and let us know what your question is.
- Establish your account at Wolfram so that you can use online Mathematica. I sent you an email with the instructions. This email is posted as an announcement on Canvas.
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Set up your personal computer for remote access to computers in BH 215 computer lab. I sent you an email with instructions on April 6. This email is posted as an announcement on Canvas. When you establish the remote access to a computer in BH 215 computer lab you will find Mathematica installed. Pay attention, we use the version 12.
- You can install your own Dropbox on a computer in BH 215. Since you do not have administrative privileges on this computer, installing Dropbox requires patience. Please let us know if you were successful in Discussions on Canvas.
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If you are not successful with the previous item, all Math 307 files you can find on K- drive on the university network. On your K-Drive you will find
K-Drive\Math\Curgus\307_Files and
K-Drive\Math\Curgus\307_Questions.
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Open a Dropbox account. I sent you an email about this process. This email is posted as an announcement on Canvas. In your Dropbox account you should have these three folders
- I sent an email to all with the instructions how to set-up a Dropbox folder for your homework assignments and how to gain access to a shared Dropbox folder with the class materials.
- Mathematica part of the class starts on Monday, April 6, 2020 as an online class at 14:00. You will get a Canvas invitation for this class.
- 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.