Math movie: How to play a Rubik’s Cube like a piano

By Murray Bourne, 09 Dec 2015

We use group theory every day, without ever thinking about it. Here's a simple example.

The integers form a group under the operation of addition.

4 + − 4 = 0

In this case, 0 is the identity and − 4 is the inverse. This example aslo demonstrates closure (4 and − 4 are integers, and when we add them, we also get an integer, 0.)

Another property of a group is that it is associative. That is, if we have more than one operation, it doesn't matter what order we do it in. For example:

(4 + − 4) + 3 = 4 + (− 4 + 3) = 3

Similarly, the positive rational numbers form a group under multiplication. If we start with the number 7, we can find its inverse, 1/7 and the identity is 1. Thus we have:

1/7 xx 7 = 1

Similarly, each of 1/7, 7 and 1 are rational numbers, so the group is closed.

Many kinds of groups

Group theory covers many different areas, including permutations, matrices, transformations, topology and so on.

Applications include number theory, chemistry, physics, mechanics and music.

Group theory and music

Notes on a piano are periodic. That is, the 12 notes starting from the lower end of the piano are repeated 7 times to form the rest of the keyboard.

A, A#, B, C, C#, D, D#, E, F, F#, G, G#

A lot of music makes use of what is called the cycle of 5ths, where the chord structure follows notes which are spaced 5 notes apart, like this:

B E A D G C F A# D# G# C# F#

This forms a cyclic group structure which can also be periodic.

And this brings me to today's math movie. It's a TED-Ed offering by Michael Staff and outlines the connections between Rubik's Cube and music.

See the 1 Comment below.

One Comment on “Math movie: How to play a Rubik’s Cube like a piano”

  1. Stepanov Sergey says:

    Computer based maths module in Elementary music education
    Math & Music
    It is well known that the difficulties in the perception of any information, including musical one, cause strain of the main functional systems in the child’s organism. The innovative technology " Reflection " gives us some explanation in this direction and raises the topic on the necessity of applying of Digital Key in Elementary Music Education.
    The physiological substantiation on the application of the digital system for coding and decoding of a melody is the following : children begin their contact with digits already in preschool age, when they are taught to count, and this system is learned by children quite firmly, since it is often used in their daily life. But the generally accepted music grammar is new for them and ,naturally, requires some additional period of time to be acquired by children. It is for that reason that in the initial period of musical teaching ,children inevitably spend a lot of time and efforts to read a melody written down in music signs. Naturally, it slows down rate of training ,causes psychoemotional discomfort, lowers the child’s interest to music .Therefore, in the initial stage of teaching, besides work with generally accepted music grammar, will be useful to replace it with a digital system for a certain period. This does not mean that we want to do without standard music grammar, but at the initial stages of musical education, the system of digital coding and decoding of music sounds is undoubtedly useful, as it speeds up teaching of children.
    With this purpose ought to study the load appearing in the child's psychic sphere, the experimental researches are to be performed, namely : ENG , EMG , EEG – tests. The method of ENG and EMG joins the visual analyzer with neuro-motor function of the hands and explains ,on the scientific point of view, the ratio between the load on muscles of eyes and muscles of hands. One can see the interdependence between the perception of information ( Readout Algorithm ) and the synapse transmission ( Action Potential ) and also it proves the possibility of development of muscular fatigue in hands depending on the quantity of eyeballs’ fluctuations. EEG - test allows us to make up the diagrams of dynamics of the proceeding neurophysiological processes, and it offers an opportunity to investigate the functional activity of neurons during practical study on two different modes ( Comparing Methods ). Realization of the described investigations will allow us to approach closer to understanding of more subtle mechanisms of child’s mental activity and to detect physiological factors that promote to improvement of quality, speed and efficiency in Elementary music education.
    In practice, using the generally accepted music grammar, the child connects the definition of the location of the melody to the pitch i.e., to the system of dimension , which is written down in the form of an expanded construction , both on x – the horizontal and on y - the vertical. By reading the music information ,the direction of eyeballs’ movements is spasmodic, and it has a multistep combination both on y – the vertical and on x - the horizontal often with a return of eyesight to the starting point of support. For an integration, synthesis and the modification of the complex pattern of the received information the structures of the central nervous system require an additional period of time. It is a neurophysiological process proceeding in an interval of time between the moment of perception of the music information from the sheet and the moment of the hands’ response on the keyboard of an instrument. A great number of irregular nervous impulses are transferred to the central nervous system per unit of time and , as the consequence of this , the fatigue of hand muscles is considerably increased (Berosov, Korovkin1990). An amplitude of muscle tension is directly dependent on the frequency of innervations, where each subsequent nervous impulse coincides with the phase of increased excitability of the muscle ( Green, Stout,Taylor 1990 ).At the level of the synaptic terminal we can see untimely synthesis of the neurotransmitter, deep and stable depolarization of the postsynaptic membrane and ,as a result, the convulsive reflexes are thus formed. An important neurophysiological moment has been marked : within a short time interval the contracture , that is , constantly high muscular tension is formed, which in turn , is harmfully reflected on the content and character of the melody.
    In practice , using the method of the digital key, the child connects the definition of digital melody to the system of dimension which is written down in the form of an integral construction both on x – the horizontal and on y – the vertical . Reading the digital information the trajectory of eyeballs’ movements on y– the vertical is projected to the exact determinant ( digit ,sign, symbol ), the trajectory of eyeballs’ movements on x – the horizontal is projected in one direction, forward. In the given system of dimension the integration of the digital information proceeds instantly, its realization on an instrument proceeds in reflexive time - ratio. The paradoxical phenomenon is revealed : the time interval between the moment of perception of the digital information and the moment of the hands’ response on the keyboard of an instrument, is contracted to the minimum .We achieve a reduction of load on hand muscles at the expense of decreasing of an amplitude between muscle tension and the resulting movement and, consequently , the time intervals between effort and accuracy of pressing of a key are considerably shortened. At the level of the synaptic terminal we can see an allocation of neurotransmitter directly proportionally to the frequency of generated impulses by neurons and , as a result , the coordinated reflexes are thus formed. An important neurophysiological moment has been marked : the reciprocal muscular innervations is formed , that is, the rational distribution of the manual technique on the keyboard of an instrument, which in turn, is considerably reflected on the content and character of the melody. Grain of truth lies in the fact that at the expense of perception of melody by means of digits its realization becomes faster and easier, which in turn , is positively reflected on the psychoemotional status of the child and enables him to dynamically realize the potential music abilities in psychosomatic action as a result .
    Kids of new generation should have different methods of training. They are capable to perceive information faster, with cross-modal processing, activating all senses at once : visual perception, audio analyzers, neuromotor functions.
    NeuromusicLab Reflection Ukraine

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