# Month: February 2015

### Tessellation and Symmetry (By Joyce Liu)

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#exploremaths

Today we worked with symmetry, and how it is incorporated
from regular shapes to beautiful drawings. We know of the many types of
symmetry; Rotational, Reflectional, Scale and Translation Symmetry. Tessellation
and symmetry is all around us. Like the golden section, it is in nature, manmade
objects, everywhere. Tessellation does not only consist of a single shape
conjoined together to form a pattern, it can use more than one shape, for
example in the design of a soccer ball. These types of tessellations are called
semi-regular tessellations. We also see regular tessellations (single shapes)
in many aspects of nature, for example, the seeds in a sunflower and the hexagons
in beehives. The more we discover about maths, the more I realise that maths and nature is a perfect pair.

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### The Use of Maths in Music and Billiards (By Joyce Liu)

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#exploremaths

In class we studied some videos which described the way mathematics
is incorporated into the modern world – how math can be used in music and a game
of billiards.

It is interesting to see how music, something that we
constantly hear is mostly derived from pitch frequencies that make up geometric
series. It makes me surprised that it is possible for all the different types
of instruments and their beautiful pieces to be created from maths. Music and math
is not commonly associated with each other and to me I find it very fascinating
that they are a pair that must go together. Although this interests me, I have
very little knowledge about music (as I don’t play any instruments or sing) and
so therefore it makes me slightly confused. Even when I face a problem like
this, it doesn’t make too much difference and I am able to understand the
meaning of the videos.

In another video, we watched how the game billiards is
played using mathematical strategies. For my first impression, I was rather
confused as it appeared complicated by the heavy reliance of angles for the
strategies. I was also unsure of what was explained at first since I did not
understand how to play that game, but soon it was explained and I began to
understand. When I understood, the way the game was played made it seem like a
very hard game of techniques and mathematical strategies. I was fascinated to
see the way that the table was created – with all the markings that aid in the
strategies of a player.

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### Interesting Tessellations (Alex Ho)

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In the lesson, we applied rotational symmetry on our own
free-hand designed polygons, where the more points we had, the more exciting
the activity became. Although only practising with rotational symmetry, we
learnt of the others, including reflectional (also known as line symmetry),
scale symmetry and translational symmetry (better known as tessellation).

When thinking of an example of tessellation, the most common
thought would be of beehives and their hexagonal shapes. Research of other
tessellations has made me think of how blindly I played without noticing it.

Tessellation can be found within games. A chess board
consisting of 64 squares, with 2 alternating colours (usually a dark colour and
a light colour) feature a tessellation of squares, unlike a beehive, with its
hexagons and relatively same colours.

Throughout the lesson, we were tasked to form a 10cm by 10cm
square, then creating smaller ones inside at the half and the quarters (on the
right hand side of the half line), creating an interesting looking spiral
effect. But when finishing the squares as far as the pencil allows, the half
way lines (when joined) surprisingly form their own golden spirals.

The completed shape with the spirals joined, although what
started off as something simple, turned out to produce something complex.

#exploremaths

### The Magic of Symmetry (Terrence Wong)

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The Magic of Symmetry

Today in our class, we studied the different types of
symmetry and how we can apply it through our irregular drawings. These
different types of symmetry included: rotational symmetry, reflectional
symmetry, scale symmetry and translational symmetry which is better known as a tessellation.

The most common type of tessellation which can be found in
nature is the beehive. But as an additional piece of work, we were asked to
find another example. My research has led to the tessellation of the pineapple.
The outer skin of the pineapple is constructed by pentagonal shapes. Very much
like the beehive which is constructed of the shape hexagon. This lesson has
made me realise how mathematics is all around us, especially in places as
normal as in nature. Many times we overlook the simplicity of nature and don’t
realise how maths can tie in with all of this. An example was shown in class. A
simple 10 by 10 square can be dissected into quarters and each time, a scale of
the same shape appeared and was symmetrical to the previous shape. As you
complete the shape, you can connect the halfway points to the centre and get a
golden spiral. This just shows how simple something can be but yet consists of
something so complex. This lesson has made me really think about what is around
us and how ‘beautiful’ it is.

#exploremaths

### The Golden Ratio (by Erik Willison)

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#exploremaths
The golden ratio is found throughout architecture and nature. Examples of this are the Notre-Dame Cathedral and the Great pyramids. In nature we can see it in sunflowers, plants with spiral leaves and even in spiral galaxies.

Firstly architecture, The Notre-Dame Cathedral was built way back in 1163 and was designed by some random guy that was of great importance to somebody; anyway, this spectacular work of art has golden rectangles scattered throughout it. the three primary examples are:

1. The ratio of the height of the first floor to the height of the second floor (starting from the top of the first floor) is the golden ratio, or pretty close to it (outlined in red)

2. The height of the second floor to the height of the third floor is a golden ratio (outlined in blue)

3. The ratio of the length of the top left box including the space between it and the top right one to the length of the right one by itself is another golden ratio.

My second example of the golden spiral in architecture is the Great Pyramids of Giza. Each of these pyramids use the golden section. The diagram below is so good that it is basically self-explanatory, but for those who are lost I shall explain. Inside this pyramid you can see a triangle reaching from the centre of one edge to the centre of the base and up to the very top of the pyramid. The Egyptians built this pyramid with dimensions that just so happen to be able to be simplified into the diagram below where Phi is the hypotenuse. The dimensions that the Egyptians used were:

– A base of 230.4m

– An estimated original height of 146.5m

The ratio between these two numbers is 0.636. Now I know that 0.636 is not the golden ratio and may seem quite far off 0.618etc.; but the height only needed to be 0.0376m taller and the ratio between these dimensions would fit the golden ratio to those three significant figures.

Onto nature. Nature proves to be mathematical in every sense and likes to use the golden spiral.

Starting off with the sunflower; the seeds of a sunflower grow in a particular pattern although they may seem to appear scattered randomly. A sunflower needs to decide how far it should turn around before it grows a new seed from its centre, the angle that it turns will determine how much of the flower’s circular centre will be filled with seeds, and obviously the aim is to optimise this number. Through micro-evolution, or something like that, the sunflower came to a decision: it will have each new seed that grows from its centre 137.5 degrees  (or close enough to that) appart. This value seems random but it is infact the corresponding angle to the golden ratio, also know as the golden angle. and that is where the sunflower uses golden mathematics, optimizing its seed growth.

Secondly, in nature, we have plants with spiral leaves. These types of plants simply use the golden angle to grow their leaves so that no new leaves block out the sun of exiusting leaves on the plant, also this leaf formation helps dew and rain that fall on the plant flow directly towards its roots.

Finally, on the biggest scale in the universe, we have galaxies. A spiral galaxy, like our own, has arms that are meant to wrap around it as the galaxy turns from the centre. For some unknow reason, the arms of a spiral galaxy seem to defy this, and with that newtonian physics, as the stars on the extremities of the arms move at a higher velocity; thus maintaining the golden spiral in our galaxy.

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### Reflection on Mathematical Ideas (Josh Luong)

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The Relation Between Music and Maths

The concept learnt after watching the various videos in
class have extended my knowledge on how much closer music and maths really is.
Being made to do mathematics all my life, it was definitely interesting seeing
that simple fractions and equations could be so similar to the octaves of notes
or triads played. After learning to play my own instrument, I can’t promise
that every time I play a note, several equations will be visualized and used to
my advantage like the genius musician Beethoven most likely could but it’s definitely
something that I can think about in the future.

The Philosophy of Pythagoras

only seen Pythagoras as a mathematician who had been someone who has given the
rule to find sides in a right angled triangle which has only made me grieve
over my maths homework but after viewing the video I had come to understand
that he was not only someone who had discovered maths formulas but his philosophy
tells that everything in the world is  made
of numbers which is not entirely true. It was interesting seeing a cartoon representation
of Pythagoras having to secretly discover these new concepts of mathematics alongside
the Pythagoreans which only makes me appreciate that we get to relearn and
maybe improve on his findings from thousands of years ago. #exploremaths

### Music and Maths: How does it Connect?

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Through this lesson, we viewed
three mathematically entertaining videos which connected music and maths. We
explored the concept of the Pentagram in which we learnt that you could keep
adding a pentagram in the middle of the previous one to form an infinite
pattern. This allowed us to acknowledge the fact that your mind is the only
method of thinking infinitely. Something that surprised me this lesson, was the
fact of how mathematics is so connected with music. We saw that calculating
wavelengths of musical notes, you could make a sound which was pleasant to the
ear. This is called consonance. Apart from the pentagram and how music is made
with maths, we learnt that Pythagoras’, an ancient Greek Philosopher, was the
father of maths and music. Although often arguing with Democritus about what
the world revolved around, Pythagoras was the main reason we have our music
today, not through only musical practices, but from a mathematical viewpoint,
i.e. Pythagoras discovered an octave, which is 8 keys, had a frequency of which
equalled 2:1.

The questions I have that are
left unanswered from today’s lesson, reflecting on the basis of maths and
music, is how can you construct a piece or musical score using maths. How can
you make a song which combines both influences of maths and music into one
thing?

#exploremaths