Month: June 2015

Enigma Machine (By Michelle So)

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#exploremaths

1. What
is the flaw in enigma?

The
flaw in the enigma machine is that the machine cannot represent the letter that
is being typed into the enigma machine. For example, if the letter ‘c’ was
typed into the machine, it cannot generate the letter ‘c’ as the encoded
letter. This made it easier for the allies to crack the code as through trial
and error, the code can be obtained when there are no letters that are repeated
as itself.

2. How
did the allies crack it and get around this sophisticated machine?

Allies cracked this machine by using a machine
known as the ‘Bomb’. This machine operates like a search machine, searching for
the code. It generates a inter-electrical circuit. In addition, as the machine
cannot generate itself in the machine, the allies slide the code until there is
no repeated letters. Furthermore, the allies were able to get around the
machine easier as the operators in the Nazi tend to use the same setting to
send the message, therefore allowing it to be easier to crack the code.

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A Fraction of Fractals (Nina Qiu)

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Snow and ice comes to mind when the word “fractal” is
mentioned. But of course, they are indeed examples of fractals. What a fractal
is, is that it’s a shape or an on-going complex pattern that is self-similar
across different scales. Some of well-known fractals include the Sierpinski
triangle, the Mandelbrot set, the Koch snowflake, the Julia set, and the
Apollonian gasket. Some examples of fractals occurring in nature are river
networks, frost crystals, coastlines, Mountain Goat horns, and patterns on tree
leaves.

One of my favorite fractals is definitely the Apollonian
gasket, named after the Greek mathematician, Apollonius of Perga. It seems
quite simple, but it’s maths. The fractal is generated from triples of circles,
where each of them is tangent to the other two. And the Apollonian network is a
graph derived from finite subsets of the Apollonian gasket.

#exploremaths

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Of Symmetry and Tessellation (Nina Qiu)

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There are four types of symmetry mentioned: rotational
(point symmetry), reflectional (line symmetry), scale, and translational (tessellation)
symmetry. If you look closely into nature, you could find all kinds of
symmetry, especially tessellation.

 

Some examples include reptile scales, spider webs,
honeycombs, and pineapples. You can identify tessellation by looking at the
shapes that assembled together to form an object’s appearance, such as a snake’s
skin. You can see the fan shaped scales overlapping each other, thus translational.
It is quite amazing how nature has a way of forming mathematical patterns that
humans cannot fathom. It gives a certain aspect towards the question: whether
or not is math invented or discovered?

#exploremaths

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Enigma Machine (by Sandeep Darapuneni)

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What is the Flaw
in Enigma?

1.   
The
Enigma Machine had one notable limitation, that a letter cannot be encrypted as
itself.

2.   
It
could also be said that how the Germans used the machine allowed the British to
crack it more easily. The Germans sent many daily messages, and would normally
end their messages with ‘Heil Hitler’, meaning that the British could eliminate
many possibilities by knowing this. 

 

What did the
allies do to crack it?

 

The British used the fact that a letter cannot be encrypted as itself and
created ‘The Bomb’, a machine that would go through all of the possibilities
until it decoded the message. This machine was used by the British to regularly
intercept Nazi messages, and it is estimated that the war would have went on
for around 3 more years if ‘The Bomb’ wasn’t created.

 

#exploremaths

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Number Systems (by Sandeep Darapuneni)

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Egyptian Number System

 

The Purpose and the Distinctive Features of this
Number System

Egyptians
used this number system like we use the Hindu-Arabic System today. But instead
of numbers, this system used everyday objects to represent the numbers.

 

Where and How this Number System is Used Today

The
Egyptian Number System is not used today.

 

Advantages and Disadvantages

This
number system made it easier for the Egyptians to carry out simple operations
such as addition and subtraction. But this system required you to draw out many
symbols, meaning that a lot of time and space would be used.

 

Hexadecimal

 

The Purpose and the Distinctive Features of this
Number System

Hexadecimal
is of base 16 and uses 0-9 for the first 10 characters and A, B, C, D, E for
the next 6 characters. Primarily, this number system was created to write
binary code shorter. For example, 1111 0011 1000 1010 in binary code can be
simply written as F38A in hexadecimal.

 

Where and How this Number System is Used Today

Computer
programmers use this language today to quickly read binary numbers.

 

Advantages and Disadvantages

Hexadecimal
is useful as it allows for the quick reading of binary. But one drawback of
hexadecimal is that there is a limit to the size of numbers that can be made,
with 65 535 being the largest number possible.

#exploremaths

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Artwork Ideas (by Sandeep Darapuneni)

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I
was flipping through my book for ideas for this assignment when I came across
the Koch Snowflake. I realised how both the Koch Snowflake and Sierpinski’s
Triangle both used triangles to form their basic outline and how combining
these 2 fractals to create ‘SierKochski’s Snow Triangle’ would be perfect for
this assignment. I don’t know the exact details of my artwork, but I am
definitely going to incorporate this into it.

 

#exploremaths

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