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2Jun/132

Carnival of Mathematics #99

Welcome to Carnival of Mathematics #99. Wikipedia provides some nice trivia about the number 99.

99 is the ninth repdigit, a palindromic number and a Kaprekar number. It is the sum of divisors of the first eleven positive integers.

99 is the sum of the cubes of three consecutive integers:
99 = 2^3 + 3^3 + 4^3

And, I personally like that 99 is the difference of two squares: 99=10^2-1^2.

Now, onto the carnival articles.

John Cook shares Recognizing numbers. For Python users, SymPy is a symbolic math package that "takes a floating point number and tries to simplify it: as a fraction with a small denominator, square root of a small integer, an expression involving famous constants, etc."

Mike Thayer, in Algebra and Geometry, asks this question: "I teach algebra 1, to 9th and 10th graders, mainly. I also teach geometry to the same age group. I'm wondering the following: Why is it that the conversations in geometry are so much more interesting, generally?"

Peter Rowlett takes a break from PhD preparation to explore Ox Block probabilities. "I'm not blogging much in the run up to my PhD thesis deadline, but my curiosity got the better of me with this one. Having seen (via Twitter) that it was being played at a Maths Jam, I bought an old game called Ox Blocks, which offers “Noughts and Crosses[/Tic Tac Toe] with a novel twist”. Here, I investigate the probabilities of rolling an unusual die."

Thomas Woolley writes Egg shells to turtle shells. "No matter how you initially orient the gömböc it will always wobble and rotate itself to finish standing upright. Importantly, the gömböc is made of only one material, so its density is uniform. Mathematically, the gömböc is known as a mono-monostatic body. This simply means that it has exactly one stable and one unstable equilibrium point."

Tony, a university maths professor in London, in My favorite equation considers whether there's a more interesting formula than Euler's formula. "So McKay's formula may not be as immediately beautiful as Euler's, but it has something of the same spirit (and perhaps even importance). It demonstrates a very deep connection between group theory and modular forms; it's mysterious and hard to understand, and it's inspiring important mathematics. And it says a lot about the serendipity which lies behind insights even in a subject as apparently logical and rigorous as mathematics."

Simon Gladman wonders what pendulum waves might sound like in The Sweet Sound of Pendulum Waves - in Glorious Stereo! "I had a little play with Pendulum Waves the other day and since then I've been wondering what sort of sound they would make if I played a tone as each pendulum reached its apex."

Have you ever wondered Why are determinants defined the weird way they are? If you've ever wondered why, whether or not you've studied linear algebra, you might enjoy this article. It'll give you some great material for your next party conversation!

Yao-Hong Kok is a Master's student studying control theory. Math, Control Theory and Two Issues invites interested parties into a discussion. "Control theory is one of those fields that requires a lot of mathematics. I have been in the field for roughly 2 years now and I have realized that they are 2 big issues within control theory, namely: (i) identity of a control engineer/theorist, and (ii) stagnation of fundamental theory advancements. In this post, I would like to relate mathematics to the above issues and perhaps generate some discussions."

Maria Droujkova shares Math dreams meeting May 20, 2013. "Curriculum developers' elephant in the room is a simple question: "Who wants that stuff, anyway?" We decided to ask parents what do they want for their kids, in math. Deep is the chasm between what parents want, and what existing curricula provide..."

Shecky Riemann, inspired by Martin Gardner's passing to start his blog, writes Remembering... Gardner three years after his death. "Not to take anything away from our Veterans, but this is a math blog, and I'll use the opportunity of Memorial Day to once again remember Martin Gardner, whose death just over 3 years ago inspired me to start this endeavor (with no idea it would still be up-and-running 3 years later!!)."

Herminio Lopez examines an interesting puzzle in A black (and red) hole. "Thanks to a prize consisting on the proceeds of a football match, we learn about some numbers that attract the others, which can't escape from them. Mathematical sequences which lead to mathematical black holes."

In Demystifying the Möbius, Burkard and Marty take readers on a nice journey through the many twists and turns that one can take with these paper treats.

Predicting Sums is a fun article at Grey Matters. It shows a nice math trick one can perform with a little knowledge of digital roots (aka nine's complements).

Math Munch is a great blog for children of all ages that describes itself as "A Weekly Digest of the Mathematical Internet." Their latest edition is Solitons, Contours, and Thinking Sdrawkcab. Check it out if you've not yet discovered this blog.

The Aperidical is another of my favorite blogs. They describe themselves as "a meeting-place for people who already know they like maths and would like to know more. It was begun by Katie Steckles, Christian Perfect and Peter Rowlett as a shared blogging outlet and grew out of our desire to have a place on the web where we could keep up to date with what’s going on elsewhere, and to share the mathematical things we do." You might also recognize Aperiodical as the stewards of this Math Carnival. Christian authored this fun piece, Integer sequence review: A000959.

If you've ever wondered what math and the meaning of life were related, check out 42 at Calculus Humor. This article deserves to go viral. Really.

Finally, I'll share one of my own favorite recent articles, Ken Fan: Inspired by Math #29. It's a podcast interview where Ken and I had a nice informal chat without much preparation before-hand.

The next carnival, #100, will be posted in July and hosted by Richard at Simple City. More information about the Carnival of Mathematics, and a submission link, is posted at the Aperiodical site.

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  1. Hi there;
    I am a civil engineer. Since I have retired I am studying the matters that amused me at Faculty. So, I have developed the theory ‘The mathematics of evolution’ (We could call it of ‘The basic equation of Quantum Physics’) through which we can to accompany the natural evolutions of all natural phenomena systems. I have many articles to prove it. Here I am showing an article about the classical musicians.
    Thank You. Cheers;
    Hélio Barnabé Caramuru – helio.barnabe@gmail.com

    Are the Musical genius connected with the intimate nature of the universe?

    Hélio Barnabé Caramuru

    Engineer – Independent researcher – not Scholar Scientist.

    Summary

    The musical genius Johann Sebastian Bach (1685-1750) proposed the chromatic musical scale ‘spicy’, consisting of notes, the #, re, mi, fa, f #, sun, sun #, la, la #, you of, as we know and use, ending the polyphonic music. The names of basic notes were proposed by the genius of Guido Arezzo in the eleventh century, based on the first syllables of the verses of the letter of the ‘Hymn to St. John the Baptist.’
    The notes of the scale ‘spicy’, with frequencies whose measurements were obtained in the laboratory and coincide with the values calculated by the theory ‘Mathematics of Evolution’, will be considered here.
    We will also make a small foray into the field of fiction, implying that the music could be heard through the ‘interpretation’ of the results obtained in the evolution of natural phenomena. Disclose a program in ‘Basic’ through which the results were obtained in this article and may provide input to a researcher in the field of music.

    Involvement of mathematical frequencies. Presentation of the Basic Equation of the Theory ‘The mathematics of evolution”

    The simplified equation for this case is:

    Hn = Hn, 0 (1 – B / C), derived from the fundamental equation of the theory given below. Hn +1.0 represents the frequency of any note of the scale ‘spicy’ and Hn, 0, the frequency of the note immediately below the same scale.
    The fundamental equation of the theory is expressed by:

    n B ^ i
    Hn,i = ( )—————- [(C – B) ^ (n – i)]
    i C ^ (n – 1)

    Or on a form more elegant:

    Hn,i = k * ( B ^ i) * [(C – B) ^ (n – i)] / [C ^ (n – 1)]

    In doing if i = 0 [condition typical for the phenomenon of ‘decay’ (radioactive isotopes, for example)], we obtain:

    C – B
    a) Hn,0 = C [(————-) ^ n], and considering the amount of:
    C

    C – B
    b) Hn +1,0 = C [(————–) ^ C (n + i)], then we have to respect
    C

    b / a, the value:

    Hn+1,0
    ————- = (1 – B / C), as we wanted, and taking the values:
    Hn, 0

    B = 523.2511 – 493.8833 = 29.3678 Hz and,

    C = 523, 2511 Hz [(which is the value of ‘potential’ of the scale: H0,0 = C (in this case)], we have to (1 – B / C), the value of 0.9438744.

    The value of n corresponds to the periods in which there are ‘decays’ (change of notes of the scale).
    The notes and their frequencies measured in the laboratory or calculated are as follows (in descending scale):

    do si blá lá bsol sol bfá fá
    523.251 493.883 466.164 440.000 415.305 391.996 369.994 349.228

    mi bré ré bdo do
    329.628 311.127 293.665 277.183 261.626

    Examples of calculations:

    523.251 x 0.9438744 = 493.88322 Hz; 493.88322 x 0.9438744 =466.16372 Hz, etc..

    Conclusion

    The coincidence of the choice of notes of the chromatic scale, whose frequencies are in agreement with the ‘Calculations of Nature’, leaves us perplexed. What we count as a musical genius is a seer? He is connected with the essence of nature with the objects of the divine?
    With the discovery of the theory ‘The Mathematics of Evolution’ we are certain that nature performs all its disciplined developments within mathematics. The numbers are used by nature to rule over everything, including us. Why do we like music? Because it is drawn up in numbers, such as ourselves? We are brothers in music? We are her children? It is made of the same matter that made us too? All that is beautiful, perfect is framed in numbers, so it pleases us, because it is part of our nature.
    The ‘Chromatic Musical Scale Well-Tempered Bach’ is built within the ‘rules of mathematics’. Will the music genius uses the same rules? Or the music of genius existed potentially in ‘air’ and they were taken? Produced and harvested two words are synonymous for geniuses? We can mathematically write the songs? We believe that those who please us, that are in line with our own nature, can be harvested, can be produced .Michelangelo said that he carved the statues were there, inside the stone, ready, he just found them. We can say the same music?
    Today, with the knowledge of the theory ‘The Mathematics of Evolution’, someone dares to produce mathematically music? The equation is here (…). For the fundamental equation of Hn,i that am gave up, and calculate the ‘decay’ Scale of Bach, the notes are calculated in parallel families Hn,i derived. How will be hearing the sequencing of the full set of lecture notes prepared?
    Taking Cartesian axis, where we consider the horizontal periods n and in the vertical axis the frequencies Hn,i and in plotting all the calculated points of frequencies of scale ‘Well Tempered’, and linking them, are faced with beautiful curves. . . But that’s not all! . . .

    Suggestion

    From C and B of fundamental equation given above calculates a multitude of families Hn,i sound. . . It is the symphony of the universe? Who dares to generate (on the computer polyphonic / example – ‘Microdigital / TK 3000 / / 2e) all notes curves Hn,i? We suggest the study of music (preferably classical), with the establishment of relations of the frequencies of your notes to check the validity of this article, in fact, is only speculation on the subject of the frequencies of musical notes and their combinations.
    We remember the musician Vincenzo Galilei (father of the renowned scientist Galileo Galilei), among other qualities, he was also a research scientist who wanted to study music from the point of view of mathematics (frequencies). For those who want to ‘venture’ to see the combinations of the frequencies of musical notes, we give here a program through which you can see, from a frequency and a variable C B, like the one shown in the text, ‘arrangement ‘of frequencies.

    Program ‘Basic’ used in this Article.

    10 REM Hn, i – Fundamental Equation of ‘The mathematics of evolution’
    20 REM Chromatic Scale
    30 REM BASIC PROGRAM
    40 FOR N = 0 TO X: REM X is at the discretion of the composer – Recommendation: X = 20 to try
    45 PRINT ;N ;”/”;
    50 FOR I = 0 TO N
    100 B = 29.3678 C = 523.2511: REM start with the composer Criteria frequency C = 523.2511 – can be any frequency
    110 LET F = 1
    120 FOR J = 1 TO N
    130 LET F = F * J
    140 NEXT J
    200 LET G = 1
    210 FOR K = 1 TO N-I
    220 LET G = G * K
    230 NEXT K
    300 LET H = 1
    310 FOR L = 1 TO I
    320 LET H = H * L
    330 NEXT L
    500 BX = F / (G * H)
    510 HB = ((B ^ I) / (C ^ (N-1 )))*(( C-B) ^ (N-I))
    550 HBC = BX * HB
    560 PRINT HBC ;”,”;
    600 NEXT I
    700 NEXT N

    Note – This program was developed through a micro-computer. . .can run on a computer program ‘windows’ below the year 1995, however, need a helper program (application) ‘Basic’.
    – We recommend that initially the value of C is one of the frequencies of the notes ‘do’ and B as indicated as 28.3678

    Engenheiro Hélio Barnabé Caramuru

    helio.barnabe@terra.com.br / helio.barnabe@gmail.com / helio_barnabe@hotmail.com

    tels. (55-11) – 3251.4338 / 3251.4338

    São Paulo, 05 de março de 2008

  2. Every odd number is the difference of two squares. Actually, every odd number is the difference of two consecutive squares, since (n+1)^2 – n^2 = 2n + 1. In our case, 99 = 50^2 – 49^2.

    So how many other pairs of numbers a and b are there such that 99 = a^2 – b^2?


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