## Exercises

 Nr. Exercise Level Topic Solution 1 Symmetric Patterns * - ** Symmetry, Trigonometry Partly 2 Strange Message * Decimal Numbers, Cryptography No 3 Error Propagation ** Infinitesimals, Chaos Yes 4 The Experiment * Linear Regression, Chaos Partly 5 Laser * Logarithm, Chaos Yes 6 Capacity * Algebra, Sine Yes 7 Black Hole * Decimal Numbers No 8 Law Of Reflection ** Vectors Yes 9 Averaging *** Statistics Open

The exercises focus on mathematical aspects of the FroZenLight application and can be used for example in modern math classes. To avoid a permanent replication of the geometry topic it has been left away. Exercises with a chaos topic treat the strong dependency on initial uncertainties. No solution has been provided for exercises that require simple experimentation.

### Exercise 1: Symmetric Patterns (* - **)

For certain positions of the light source and rotation angles of the circles symmetric reflection patterns become visible. What values have to be put into the fields of the Variables window (Menu: Settings -> Variables) in order to create the following reflection patterns:
```  a) Grid size: 2x2, radius: 50%.
__
a1) Short line:   ___     a2) Long line: _____   a3) Square: |__|

a4) Set the rotation angle to 0. What pattern can be made if the
angle between initial and reflected ray is always pi/4.
__
a5) Twist:  |><|     a6) Triangle:   \/

b) Grid size: 10x10, radius: 50%.
__    __
b1) Zigzag:  .../\  /\  ...  b2) Computer clock: ... _/  \__/  \__ ...
\/  \/
```
Set the precision to 125 (Menu: Settings->Precision) and use the high precision calculator (Menu: Util -> Calculator). For some patterns you will need to use a root finding method (see help of calculator).

### Exercise 2: Strange Message (*)

A message originating from an unknown source in deep space has been intercepted by an outpost of our solar system. Unfortunately it consists only of long and short numbers and some words. What could be the meaning of this strange message?
```  ***************************************************************
Grid size: 9 x 9
Circle radius: 1/2Y
Light source:
distance: 1.54
height: 0.70571088063360923332860364417861977154869948908523
74798222078686824186052124368550006595229047749691
05293063116438101415369883837581909253296372920655
76639999740730094523127719560634740148985483692437
69770452792949829093403043752518795412066191036428
357041284579547686472463946261997324192764

angle: -0.07298088214969363282560511856346091343178262849927
19711221455504896668592433328244821104186749015523
68316830370548255517572487122148999591071597649954
15308000211333176996871974473886024239262905766106
22750556084256136049249968070575354199179873443329
674116820532614225353375845840786175204681 * pi
***************************************************************
```
A digit Y in the denominator defining the radius got lost due to much noise in the receiver caused by a heavy super novae outburst. The symbol table (Menu: Util->Character Map) that has been found recently in an alien ship wreck from mars might be of some help to you.

Make sure that the precision is set to 500 (Menu: Settings->Precision ->500) and open the high precision calculator (Menu: Util->Calculator). Also open the variables window (Menu: Settings->Variables) and switch off truncation at the options menu of this window.
1. Find the hidden meaning of this message. Use all digits of the radius value shown by the calculator.
2. How many digits of the radius value can be removed until the secret meaning gets lost.
3. How likely is it that somebody finds out the meaning of the message if he has no hint for the radius. Does a computer of unlimited speed change the situation?

### Exercise 3: Error Propagation (**)

Let Δ ϕ be the positive angle error of the ray, l the length, n the normalized direction vector and r the circle radius.
1. Show that the following upper bound for the error propagation can be obtained by geometric analysis: Here the indexes 1 and 2 refer to the incoming and outgoing ray which intersect at the reflection point. This formula can be chained to obtain an upper bound for Δ ϕn / Δ ϕ1 = (Δ ϕn / Δ ϕn-1) (Δ ϕn-1 / Δ ϕn-2) ... (Δ ϕ2 / Δ ϕ1).

Hint: Assume the initial ray to be horizontal and remove the uncertainty of height by assuming a somewhat bigger angle error Δ ϕ1. Overestimate the angle error Δ ϕ2 of the reflected ray so that the uncertainty of the position of the reflection point is subsumed.

2. By using the expression for the angle error Δ ϕ2 show that the following lower bound for the error propagation holds: 3. Derive from the previous inequality by using r = a rmax the following inequality: Plot a graph that shows the reduction of accurate decimal places implied by this inequality.

Hint: Replace li with the smallest and the sqrt(1-n_i*n_i+1) with the largest possible value.

### Exercise 4: The Experiment (*)

Select a 20 x 20 lattice and set the radius to r = 0.5 * rmax (50%). Make sure that the precision is set to 500 (Menu: Settings->Precision ->500) and open the variables window (Menu: Settings->Variables) and switch on truncation at the options menu of this window. Rotate the lattice a bit and collect data by doing the following steps multiple times:

1. Put the light source somewhere between the circles.
2. Remove the last decimal place from the height number and press enter.
3. Proceed only if you haven't been observing any change with respect to the path of the light beam. Otherwise go back to 1.
4. Read off the number of reflections and write it down (x-value).
5. Repeat step 2. as long as the final light beam does not change.
6. Read off the length of the height number and write it down (y-value).
What kind of data did you collect here? How can it be visualized in a simple way?

Teacher: This exercise is well suited for group work where every group is doing the experiment for a different radius.

### Exercise 5: Laser (*)

Is it possible to display all light patterns of the application with a physical model consisting of circular shaped mirrors and a laser? Use the last inequality of exercise 3 and assume that the circle radius is given by r = rmax/2 and the initial angle uncertainty is Δ ϕ1 = 0.001 rad.

Hint: If the error is about Δ ϕn = 1 rad than the reflections have become random. Patterns with more than n-1 reflection points are therefore impossible. Determine n-1.

### Exercise 6: Capacity (*)

Take a look at the light patterns containing a message (e.g. Poe,Goethe,Newton,PI,...) and the symbol table (Menu: Util->Character Map).
1. Assume that the size of the circle grid is s x s. Determine the maximum number of character symbols that can be displayed. The end of line (eol) and begin of line (bol) symbols are no character symbols.
2. Give an estimation for the minimal rotation of the circle lattice that is required to make a message pattern possible. The circle lattice is always contained in a square of side length 2.
3. Show that the formula for the minimal rotation depends only on a = r / rmax.

### Exercise 7: Black Hole (*)

Is it possible to cast a ray on the circles so that it will never leave the circle lattice? Select the lowest precision (Menu: Settings->Precision) and a 2 x 2 grid size (Menu: Settings->Grid Size) for this exercise. Open the Variables window (Menu: Settings->Variables) and try to increase the number of reflections by adding more and more digits to the height.

### Exercise 8: Law of Reflection (**)

A ray r1 with gradient m1 touches the bound of a circular shaped mirror C at point P and gets reflected into a ray r2 with gradient m2. Let m be the gradient of the line connecting the center of C and the reflection point P.
1. Show that the following connection between m, m1 and m2 holds: 2. Let m' and m'' be the two mathematically possible solutions for m. How are these two gradients related? Determine the result of the product m' * m'' ?

### Exercise 9: Averaging (***)

Let r be the radius and a the lattice spacing of a circle grid of unbounded size.
1. How large is the average reflection count within a square obtained by connecting the centers of four neighboring circles?
2. How large is the mean distance between the nth (n>0) reflection point of a light ray and its light source?

Last edited: 30.06.2014