- Today I started coding of the Koch snowflake in the file 20210311_A3P4.nb which is in 307_Files\2021. 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, see also the corresponding pdf printout PythagorasTree.pdf in the same directory. All these files can serve as "guides" for constructions of fractals in Problem 4.
- In the first part of Problem 4 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 4 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 3 on Assignment 3. The notes are in the notebook started yesterday 20210308_A3P3.nb. As always, this notebook is in the folder 307_Files\2021 in Dropbox. After class I fixed the problem that I encountered in class. We looked at 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.
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Today we studied a geometric construction related to Problem 2 on Assignment 2. We construucted the involute of the unit circle. Let us celebrate this construction by the animation below.
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.
- In Problem 2 on Assignment 2 you are asked to explore 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 construct 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.
- Another curve you are asked to explore is the cycloid.
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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.
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On your assignment notebooks:
- Assignment 1 is due on Tuesday, February 23, 2021 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)
- At the end of the file 20210222_A1P3.nb, I added hints on how to organize your final Manipulation[] command to avoid the cases in which Problem 3 does not have a solution.
- The second part of Problem 1 requires to repeat the explorations in the notebook TheBeautyOfTrigonometry_small.nb (I created a smaller file that should be easier to adjust to new commands) 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 approprietly.
- 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 (* *).
- In 307_Files\2021 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. I also created TheBeautyOf_Octagonal_Trigonometry.nb for octagon. Check this file as well.
- We also discussed the concept of recursively defined functions which is needed for Problem 2. 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.
- In the file 20210219_P2P3A1.nb in the part that relates to Problem 3 I used the concept Pure Function. In this case I used it just as an abbreviation, but it is an important Mathematica construct. For exapmple, this is the square function as a pure function: (#^2)&[x]. The output is x^2. The Pure Function is particularly efficient when we "map" it to a list: (#^2)&/@Range[20], try this to get the list of the first twenty positive squares.
- The problem which is alike Problem 3 is as follows: Given three noncollinear points, find the center of the circle that passes through each of the given points. As I have demonstrated in 20210219_P2P3A1.nb, Mathematica easily solves this problem symbolically. Given points $A = (A_x, A_y)$, $B = (B_x, B_y)$, $C = (C_x, C_y)$, we seek the point $S = (S_x,S_y)$ (the center of the circle) which is at the same distance, say $r$, from each of the points $A$, $B$, $C$. That is, we need to solve for $S_x, S_y, r$ the following system of equations: \begin{align*} \sqrt{(A_x - S_x)^2 + (A_y-S_y)^2} & = r \\ \sqrt{(B_x - S_x)^2 + (B_y-S_y)^2} & = r \\ \sqrt{(C_x - S_x)^2 + (C_y-S_y)^2} & = r \end{align*} Mathematica is very efficient in solving this system. It finds: \begin{align*} S_x & = \frac{B_x^2 \left(C_y-A_y\right)+C_x^2 \left(A_y-B_y\right)+\left(B_y-C_y\right) \left(A_x^2+\left(A_y-B_y\right) \left(A_y-C_y\right)\right)} {2\left(C_x \left(A_y-B_y\right)+ A_x\left(B_y-C_y\right)+ B_x \left(C_y-A_y\right)\right)} \\[10pt] S_y & = \frac{ A_x\left(B_x^2-C_x^2+B_y^2-C_y^2\right) +C_x \left(A_y^2-B_y^2\right) +B_x \left(C_x^2-A_y^2+C_y^2\right) +\left(C_x-B_x\right) A_x^2 -B_x^2 C_x}{2\left(C_x \left(A_y-B_y\right) +A_x\left(B_y-C_y\right) +B_x\left(C_y-A_y\right)\right)} \\[10pt] r & =\frac{1}{2} \sqrt{ \frac{\left(\left(A_x-B_x\right)^2+\left(A_y-B_y\right)^2\right) \left(\left(A_x-C_x\right)^2+\left(A_y-C_y\right)^2\right) \left(\left(B_x-C_x\right)^2+\left(B_y-C_y\right)^2\right)} {\left(C_x \left(B_y-A_y\right)+B_x \left(A_y-C_y\right)+A_x \left(C_y-B_y\right)\right)^2}} \end{align*} With a little bit of human help these expressions can be brought to a more symmetric form: \begin{align*} S_x & =\phantom{-} \frac{\left(A_y-B_y\right) \left(A_y-C_y\right) \left(B_y-C_y\right) + A_x^2 \left(B_y-C_y\right)+C_x^2 \left(A_y-B_y\right)+B_x^2 \left(C_y-A_y\right)}{2 \bigl(\left(A_x-B_x\right)\left(B_y-C_y\right) -\left(B_x-C_x\right)\left(A_y-B_y\right)\bigr)} \\[10pt] S_y & = -\frac{\left(A_x-B_x\right) \left(A_x-C_x\right) \left(B_x-C_x\right)+A_y^2\left(B_x-C_x\right) + B_y^2 \left(C_x-A_x\right) + C_y^2 \left(A_x-B_x\right) }{2 \bigl(\left(A_x-B_x\right) \left(B_y-C_y\right)-\left(B_x-C_x\right) \left(A_y-B_y\right)\bigr)} \\[10pt] r & = \frac{\sqrt{ \left(\left(A_x-B_x\right)^2+\left(A_y-B_y\right)^2\right) \left(\left(A_x-C_x\right)^2+\left(A_y-C_y\right)^2\right) \left(\left(B_x-C_x\right)^2+\left(B_y-C_y\right)^2\right)}} {2\bigl| \left(A_x-B_x\right) \left(B_y-C_y\right)-\left(B_x-C_x\right) \left(A_y-B_y\right) \bigr|} \end{align*} This circle is known as the circumscribed circle of the triangle $ABC$.
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In 20210219_P2P3A1.nb we use the output of the Mathematica Solve[] command to create the main two functions to construct the circumscribed circle:
RadT[{Ax_,Ay_},{Bx_,By_},{Cx_,Cy_}] CenT[{Ax_,Ay_},{Bx_,By_},{Cx_,Cy_}] Each of these is a function of three points. In Problem 3, the final product is a formula for a parabola. Therefore, in addtion to three points, it should be a function of an independent variable, say t, something likePar3P[{Ax_,Ay_},{Bx_,By_},{Cx_,Cy_},t_]
- 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 principle presented above is the general principle of parametric equations of curves in the plane. Given two functions, one green and one blue, taken as the coordinates of a moving point, they do trace a curve in the plane. In class, we discussed the converse question: given a curve in the plane, for example an octagon inscribed in the unit circle, find a green and a blue function that do trace that octagon.
- We discussed Problem 1 on Assignment 1 today. The notebook 20210216_A1P1.nb contains this discussion. A typical definition of a function in Mathematica is given in the snippet posted below. 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 semicolon ;.
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Inspired by the parametrization of the unit circle presented above, today 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 which I did in class. 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.
Place the cursor over the image to start the animation.
- The Mathematica part of the class will starts on Tuesday, February 9, 2021 as an online class at 14:00. You will get a Canvas invitation for the Zoom meetings.
- In the meantime I will send you an email 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.
- 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.