# exploremaths

### Enigma Machine (By Michelle So)

#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)

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)

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)

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

### Number Systems (by Sandeep Darapuneni)

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

### Artwork Ideas (by Sandeep Darapuneni)

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

### Video composition ideas (Michelle So, Hiya Ganju, Sophia Kim and Joyce Liu)

#exploremaths

After brainstorming through several concepts, our group decided to use the mathematical concept found in the novel “The Da Vinci Code”, written by Dan Brown. The novel inspired us as it contains the mathematical concepts of encryption and the golden ratio which we had discussed during one of our lessons in class. One of the most interesting topics we have learnt in class is encryption as codes and methods of cracking them are highly fascinating. In addition, we found that because this mathematical concept appealed to all the members of our group, meaning the entire class would engage with our video composition.