Tag Archives: mathematics

A math operation table for kids

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I had such a thing in Greece when I was a kid. By pulling the i-th and j-th band on a plastic board their product was being revealed. That's how I had learned the multiplication table very quickly.

I wanted to buy something like this for my little one, but I couldn't find anything similar. All math tables work now with buttons. So I decided to construct one by myself.

At first I designed it with xfig: the prototype is table and the full elements are board and leisten. Then I printed them on 300gr carton and cut them accordingly; the board:

20160130_212859and the bands:

20160130_212923which were also cut into 20 elements:

20160131_114614Then I put it all toghether:

  1. bottom carton
  2. temlate carton, where I wrote the numbers but after putting it inside
  3. horizontal bands
  4. vertical bands
  5. cover carton with 100 quadratic holes
  6. stickers (s. below)

I fixed the band sliding by attaching small white stickers on the boundary between the bands (width = 1.4mm). On the right hand side, I fixed paper pieces with paper glue on the front side and post-it glue on the back side, so that I can pull them off and change the number board inside the construction (e.g. this one is for addition, I can change the template inside for sutraction or multiplication). Here the board with all squares closed:

20160201_121539Here with all bands open:

20160201_121722

And here some operations:

20160201_12160720160201_121628

 

Quiz: How many 12-tone rows do exist?

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The composer Arnold Schönberg introduced in the 3rd decade of the 20th century the composition with 12-tone rows. Many composers after him used the 12-tone system, each one in his own way: Anton Webern, Alban Berg, Igor Stravinsky are some of the most prominent.

We will denote a 12-tone row as a permutation of the set S={0,...9,A,B} (we will use A for 10 and B for 11). But each 12-tone row has 12 transpositions (S+i, i=0,...11) and 4 forms: the original form, the inversion (the permutation with the complementary intervals), the retrograde form, and the retrograde inversion. Each form has 12 transpositions, thus we have 48 permutations. For example, the main row that uses Berg in opera Lulu is: 4 8 9 6 B 1 A 0 3 2 7 5. All 12-tone-permutations of it are listed in the following table:

Prime Retrograde Inversion Retrograde Inversion
04527968BA31 13AB86972540 087A5364129B B9214635A780
15638A790B42 24B097A83651 198B647523A0 0A325746B891
26749B8A1053 3501A8B94762 2A90758634B1 1B43685709A2
3785A09B2164 4612B90A5873 3BA186974502 205479681AB3
4896B1A03275 57230A1B6984 40B297A85613 31658A792B04
59A702B14386 68341B207A95 5103A8B96724 42769B8A3015
6AB813025497 794520318BA6 6214B90A7835 5387A09B4126
7B09241365A8 8A56314290B7 73250A1B8946 6498B1A05237
801A352476B9 9B674253A108 84361B209A57 75A902B16348
912B4635870A A0785364B219 95472031AB68 86BA13027459
A2305746981B B1896475032A A6583142B079 970B2413856A
B3416857A920 029A7586143B B7694253018A A8103524967B

Thus we can code all these 48 permutations as (0 4 5 2 7 9 6 8 B A 3 1) and denote the prime forms with Pi (where i is the first element of the Pi-form), the retrogrades of  Pi as Ri, the inversion of Pi as Ii and the retrograde inversions of Pi as RIi, where i=0,...11. Thus all 48 permutations can be coded as one form of the prime one. This is an information compression. These 48 permutations can be used in a 12-tone work, that is based on a 12-tone row (say P4).

But not all rows have 48 permutations. For some of them the retrograde form has the same interval sequence as the prime one. And for some, the retrograde inversion has the same interval sequence as the prime one. These rows have only 24 permutations.

In some musicological books I read, that all possible rows are 12!. This is not correct, because —as we saw above— we can reduce 48 permutations (or 24, in the case of the symmetrical rows) into just 1 row. And here comes the first question: how many rows are there? If we reduce them sufficiently, then we could index them, which would be very convienient for the musicological reference to the rows.

The second question is a bit more difficult. Some composers use the permutations in rotated form. Thus we could reduce the indexing further. E.g. we could denote (4 5 2 7 9 6 8 B A 3 1 0) as the 1st rotation of (0 4 5 2 7 9 6 8 B A 3 1) and not as a different row. But there are rows, that have symmetries with reference to rotation. If we respect rotation as well, how many rows do we have?

A combinatorics quiz

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Let's consider the connexion between two persons from the mothers' side and count the possibilities to express it. E.g., if the connexion is only 1 generation, then there is only 1 possibility to express it:

  1. the mother.

If the connexion is 2 generations, then there are 2 possibilities to express it:

  1. the mother of the mother
  2. the grandmother.

If the connexion is 3 generations, then there are 4 possibilities to express it:

  1. the mother of the mother of the mother
  2. the grandmother of the mother
  3. the mother of the grandmother
  4. the great-grandmother.

If the connexion is 4 generations, then there are 7 possibilities to express it:

  1. the mother of the mother of the mother of the mother
  2. the mother of the mother of the grandmother
  3. the mother of the grandmother of the mother
  4. the grandmother of the mother of the mother
  5. the grandmother of the grandmother
  6. the mother of the great-grandmother
  7. the great-grandmother of the mother.

Now, if the connexion is 100 generations, how many possibilities are there to express it by using only the words "mother", "grandmother", "great-grandmother", "the" and "of"?

Calculate the same for 1000 generations. Can you have the answer in less than 10 seconds? (10'' of computer calculations)

Here is my answer.

Update:

Damn it! I have just guessed an algorithm, that can calculate this in linear time and in less that 1 sec! Can you guess it too? ;o)

Here is my second answer.

Project Euler

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Project Euler is a project that offers mathematical problems that should be solved by computer programming. The problems are easy to understand and a theoretical solution is always obvious. But... a solutions is regarded as acceptable, if the algorithm that is applied lasts at most 1 minute. And of course the problems have such parameters that make brute force inefficient. So the ambitious solver has to invent good ideas or look for some in mathematical books, that solve the problem quickly.

Solving or just dealing with a problem makes much fun, because one experiences the power of mathematics and recalls old or learns knew knowledge.

Recommended to everyone who loves mathematics and programming!

Here is the webpage.

An Application of the Fibonacci Sequence in Music

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Every mathematician and almost every composer knows the Fibonacci sequence:

\begin{array}{ccl}F_0 &=& 0\\F_1 &=& 1\\F_n &=& F_{n-1}+F_{n-2},\ \ \ n>1\end{array}

The first 11 members of this sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

This sequence is one of the most used sequences in music. Of course I could not be an exception as a composer. My particular use of the sequence is that I used it modulo 12 (beginning from 1), that is the rests of the division of the  Fibonacci numbers by 12. The period of this is 24:1

i Fi Fi mod(12)
1 1 1
2 1 1
3 2 2
4 3 3
5 5 5
6 8 8
7 13 1
8 21 9
9 34 10
10 55 7
11 89 5
12 144 0
13 233 5
14 377 5
15 610 10
16 987 3
17 1597 1
18 2584 4
19 4181 5
20 6765 9
21 10946 2
21 17711 11
23 28657 1
24 46368 0

Translated to pitches, it results for gis as 0:

Fibonacci Row with 24-tones

I discovered this —mathematically trivial— result in 1995. Here some observations I did at that time:

  • The sequence can be divided in two halves: the second equals the first stretched at 1 perfect fourth and transposed 4 half-tone higher
    (e.g. a-ais-b equates to cis-fis-b).
  • Notes 24,1-6 form the complementary set of 12-18 (also symmetrical property).

[For more interesting relations, see the linked image at the end.]

According to the 2nd observation I was high motivated to build a 12-tone row with notes 24,1-6 for the 1st half and notes 18-12 for the second half in the appropriate transposition:
12-tone row based on the above fibonacci 24-tone row"

I have used this 12-tone row for my early Fugue for Orchestra and Composition I for piano. Then I used the whole 24-tone row for Transpositionen (ensemble) and Konzert (2nd movement, piano). I have kept the sketches with some observations that I did to this before composing the 2nd movement of Konzert here (image size 11MB).

  1. Every sequence of the Fibonacci sequence residua is periodical. ↩

A Rotation in R3

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In the Summer term of 2009 I worked at the Technical University of Berlin as a Tutor for the lecture "Mathematics for Physicists". In the 5th exercise sheet there was an interesting exercise (3b):

Let

X = \left(\begin{array}{ccc}<br /> 0&-1&0\\<br /> 1&0&0\\<br /> 0&0&0<br /> \end{array}\right)

Show that etX is a Rotation at angle t and give the rotation axis.

I had given two solutions for this exercise. A non intuitive but elegant one and a technical one. I still like this approach, that's why I publish it here (in German).

For the first solution I was inspired by the idea, that multiplication with eti for real t is a rotation by angle t, and that i is represented by the matrix X but without the 3rd row and column in the set of all matrices of the form

\left(\begin{array}{cc}<br /> a&-b\\<br /> b&a<br /> \end{array}\right)


with real a and b, which is isomorphic to the set of complex numbers (see the above link about this isomorphism).

Between two (or more) worlds

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In my life I had --and still have-- the possibility to live between two or more worlds in basically two dimensions: Country-Dimension (Greece/Germany) and professional one (Music/Mathematics, where in Music there are some orthogonal subdimensions as well: Interpretational/Compositiona/Scientific). All these were of great importance to my personality, as it concerns my way of thinking and of understanding people.

Just in the last years I have realised this hybrid aspect in me. This was not accomplished by just observing my past works and recollecting my past thoughts, or by watching other people who belong to only one discipline. One important factor that contributed to this direction was the behaviour of these people towards me. The awkwardness with which musicians see the mathematical side of me, or the mathematicians see my musical qualifications made me realise that for many people --not for all-- this is enough to consider me as a foreign body, an alien. It is the feeling of many immigrants, which they have after returning to their countries for vacations: because of their hybrid mentality they are considered as foreigners everywhere. But --the question is-- where do they feel at home?

For me, this question has a simple answer: feeling at home gives the sense of sitting comfortable. By having experience in different --if not opposite-- fields I see continuously improvement possibilities  in them. That is, I never feel comfortable because I have always something to improve, to do. And I love this!

Παιχνίδι με επίπεδους γράφους

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Ένα πολύ ωραίο παιχνίδι σκέψης είναι το παιχνίδι επιπεδότητας(;) που μπορεί κανείς να παίξει στην ιστοσελίδα της έδρας «Αλγόριθμοι και Πολυπλοκότητα» του Ινστιτούτου Πληροφορικής του Χούμπολτ (χρειάζεται να έχετε ενεργοποιημένη την Java). Λαμβάνετε έναν επίπεδο γράφο και πρέπει να τοποθετήσετε τις κορυφές του γράφου έτσι ώστε να μην τέμνεται καμία ακμή.

Αν κατεβάσετε το παιχνίδι, το τρέχετε με την εντολή "java -jar planar.jar".

Καλό παιχνίδι!

ΥΓ. Η λύση του παραπάνω είναι η εξής:

ΥΓ. Το κατάφερα ως το 40. Έχω βρει μια ενδιαφέρουσα στρατηγική. Αλλά πιστεύω πως οποιοσδήποτε το παίξει θα καταλήξει στην ίδια στρατηγική. Βρείτε την μόνοι σας!

Νέα ιστοσελίδα

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Σήμερα εγκαινίασα την νέα μου επαγγελματική ιστοσελίδα. Μαθήματα μαθηματικών και μουσικής στο Βερολίνο: Unterricht per Klick. Απλά πράματα: μπαίνεις, βλέπεις, παραγγέλνεις, και... (ελπίζουμε) πληρώνεις. Για κάθε μάθημα θα δίνω και λογαριασμούς με αριθμό εφορίας. Βέβαια, θα πρέπει να τρέξω να βάλω και χαρτάκια (διαφημίσεις) σε σχολεία και σχολές, αυτό δεν το γλιτώνω. Αλλά τώρα θα έχω και την ιστοσελίδα ως εφεδρεία.

Γράψτε τα σχόλιά σας, προτείνετε ιδέες εύρεσης μαθητών. Και κάντε και καμιά διαφήμιση αν ξέρετε κανέναν στο Βερολίνο. Άντε μπράβο!

Zeugnis

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Πήγα την Πέμπτη στο Adlershof και πήρα επιτέλους το πτυχίο μου... Μαζί μ' αυτό και ένα πιστοποιητικό που μου δίνει το δικαίωμα να φέρω τον ακαδημαϊκό τίτλο Dipl.-Math. (Diplom-Mathematiker). Απρίλιος 2003-Οκτώβριος 2008. Πέντε χρόνια διαβάσματος, απόκτησης ενός τεράστιου φάσματος γνώσεων, επένδυση χρόνου, κόπου, χρημάτων (αν δεν σπούδαζα θα δούλευα). Άξιζε πάντως. Ο δε συγκεκριμένος τίτλος έχει πολύ μεγάλη αξία στη Γερμανία. Μάλιστα είμαι από τους τελευταίους τυχερούς που τον φέρουν: Με στόχο την εναρμόνιση με τις αγγλοσαξωνικές χώρες και αγορές, προωθείται τώρα ο τίτλος Bachelor-Mathematiker...

Και --γι' αυτούς που ενδιαφέρονται-- το πτυχίο σε αριθμούς βρίσκεται στην σχετική σελίδα.