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I s s u e 9
O c t o b e r
2 0 1 1
The harmony of numbers
Maths Club
Strategy Games
Sprouts
Two players start
with a few spots
drawn on a sheet of
paper. Players take
turns to draw a line
between two spots
(or from a spot to
itself) and adding a
new spot
somewhere along
the line. The idea is
to make it
impossible for the
other player to draw
a line.
The line may be
straight or curved,
but must not touch
or cross itself or
any other line.
The new spot
cannot be placed
on top of one of
the endpoints of
the new line. Thus
the new spot splits
the line into two
shorter lines.
No spot may have
more than three
lines leading to or
from it.
The last person to draw
a line is the winner.
Maths Careers website has some good background information on the connection
between maths and music, including the tale of how Pythagoras came to make the
connection.
„The story goes that one day, probably deep in thought about the nature of
triangles, the Greek philosopher was walking past a blacksmith‟s when he
heard something strange. As the blacksmith‟s hammers struck the anvil,
some of the hammers would produce harmonious sounds while others sounded
discordant. He decided to investigate and found that there was a mathematical
relationship between the weight of the hammers and the sounds they
produced. ‟ Read more here and view a Science Channel video clip here
The magical mathematics of music
In an article in Plus magazine, Jeffrey Rosenthal writes:
„The astronomer Galileo Galilei observed in 1623 that the entire universe "is
written in the language of mathematics", and indeed it is remarkable the extent to which
science and society are governed by mathematical ideas. It is perhaps even more surprising
that music, with all its passion and emotion, is also based upon mathematical relationships.
Such musical notions as octaves, chords, scales, and keys can all be demystified and
understood logically using simple mathematics.‟ Read more here and view a short video here
Mathematics and Music
In the BBC radio programme ‘In our Time‟, Melvyn Bragg and guests including
Marcus du Sautoy discuss the mathematical structures that lie within the heart of
music. The seventeenth century philosopher Gottfried Leibniz wrote: 'Music is the
pleasure the human mind experiences from counting without being aware that it is
counting'. Read more and listen to the 45 minute discussion.
The Code
In his recent BBC documentary The Code:
Shapes (part 2), Marcus du Sautoy tells us
that „numbers determine how we hear
sound‟. He uses an oscilloscope to look at frequencies of notes played an octave apart on a
piano and demonstrates that „every combination of notes used in music is defined by simple
ratios‟. When complex ratios are used, there is no common pattern, resulting in dissonant,
harsh sounds. View a clip here on
Differential Dynamics
John Whitney first described the idea of spinning shapes producing notes, using
the principle of "differential dynamics" in his book “Digital Harmony”, The first
digital Whitney Music Box was created by Jim Bumgardner, and is hosted on his
website here Inspired by Whitney‟s pre-computer and computer films, Lawrence
Ball and Michael Tusch developed a form of mathematics devised for the production of
musical, visual and audio/visual art forms, called Harmonic Mathematics. This was applied
to graphic visuals, sound timbres and melodic loops that evolved in the 80s. Read more here
Useful links
Click here to view the
Maths Item of the Month
“There is geometry in the
humming of the strings,
there is music in the
spacing of the spheres.”
Pythagoras
BC 580-500
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Pythagoras
Circa 580 - 500 BC
Pythagoras is
known as the first
pure mathematician
and is a very
important figure in
the development of
mathematics.
Relatively little is
known about his
mathematical
achievements as he
left no writings.
Pythagoras was
born on the Greek
island of Samos
around 580 BC. He
studied in Egypt
and Lebanon. Many
of the practices of
the society he
created later in Italy
can be traced to the
beliefs of the
Egyptian priests
who he studied with,
such as the codes
of secrecy, striving
for purity, and
refusal to eat beans
or to wear animal
skins as clothing.
He was captured
when he tried to
return to Greece
and continued his
studies with Magoi
priests in Babylon.
He later returned
briefly to Samos
before settling in
Crotona, Southern
Italy, where he
founded a school of
philosophy,
mathematics and
natural science.
The Pythagoreans
In the philosophical and religious school where his many followers lived and worked, the main
focus of Pythagorean thought was ethics, developed primarily within philosophy, mathematics,
music, and gymnastics. In their view everything was composed of number, and the explanation
for an object‟s existence could only be found in number. The Pythagoreans followed a code of
secrecy, so everything was passed on by word of mouth and written accounts were forbidden.
The society studied properties of numbers that are familiar to modern mathematicians, such as
even and odd numbers, prime and square numbers. The society also believed in such
numerical properties as masculine or feminine, perfect or incomplete, and beautiful or ugly.
These opposites, they believed, were found everywhere in nature, and the combination of
them brought about the harmony of the world. To the Pythagoreans, each number possessed
its own special attributes, e.g.
Number
Property of the number
1
monad (unity) generator of numbers, the number of reason
2
dyad (diversity, opinion) first true female number
3
triad (harmony = unity + diversity) first true male number
4
(justice, retribution) squaring of accounts
5
(marriage) = first female + first male
6
(creation) = first female + first male + 1 ?
10
(Universe) tetractys
The Pythagoreans lived by rules of behaviour, including when they spoke, what they wore and
what they ate. Pythagoras was the Master of the society, and the followers, both men and
women, who also lived there, were known as mathematikoi. They had no personal
possessions and were vegetarians. Although the Pythagorean community was ultraconservative, they accepted women as equals on all terms. When Pythagoras died, his wife
Teano (who had been a teacher at the school) ran the school .
It is difficult to be certain whether all the theorems attributed to Pythagoras were originally his,
or whether they came from the communal society of the Pythagoreans. However, the
Pythagoreans gave credit to the Master for the following:
The sum of the angles of a triangle is equal to two right angles.
The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is
equal to the sum of the squares on the other two sides. The Babylonians understood
this 1000 years earlier, but Pythagoras proved it.
Constructing figures of a given area and geometrical algebra. For example they solved
various equations by geometrical means.
The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely
to have been the idea of Pythagoras because it does not align with his philosophy that
all things are numbers, since number to him meant the ratio of two whole numbers.
The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It
is believed that Pythagoras knew how to construct the first three but not last two.
Pythagoras taught that Earth was a sphere in the centre of the universe, that the
planets, stars, and the universe were spherical because the sphere was the most
perfect solid figure. He also taught that the paths of the planets were circular.
Pythagoras recognized that the morning star was the same as the evening star, Venus.
Year 6
Mathematics Resources
PowerPoint presentation with notes and worksheet are
available to download on the Monthly Maths home page
Six dominoes have been placed in this grid and there is only
one way of filing the grid with more dominoes.
Try it:
We can do better than 6 dominoes. The way in which these five
dominoes have been placed means that there’s only one way of
completing the grid
Can we do
better still?
These four dominoes lead to a unique tiling.
Better still! These three dominoes lead to a unique tiling.
What is the best you can do with other grids?
Number o rows
Number of columns
1
2
3
4
5
6
7
8
2
0
1
1
1
2
2
2
3
4
0
1
6
0
2
8
0
3
3
Valhalla
Val and Hal take it in turns to place a domino on the grid. Val
places dominoes vertically and Hal places dominoes
horizontally.
You can choose to be either Val or Hal. Your opponent then
chooses whether to go first or second. The first player unable
to place a domino loses.
Can you find a winning strategy?