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.

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).

*************************************************************** 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.

- Find the hidden meaning of this message. Use all digits of the radius value shown by the calculator.
- How many digits of the radius value can be removed until the secret meaning gets lost.
- 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?

- 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.

- By using the expression for the angle error Δ ϕ2
show that the following lower bound for the error propagation holds:

- 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.

- Put the light source somewhere between the circles.
- Remove the last decimal place from the height number and press enter.
- Proceed only if you haven't been observing any change with respect to the path of the light beam. Otherwise go back to 1.
- Read off the number of reflections and write it down (x-value).
- Repeat step 2. as long as the final light beam does not change.
- Read off the length of the height number and write it down (y-value).

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

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.

- 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.
- 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.
- Show that the formula for the minimal rotation depends only on a = r / rmax.

- Show that the following connection between m, m1 and m2 holds:

- 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'' ?

- How large is the average reflection count within a square obtained by connecting the centers of four neighboring circles?
- 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