Tag: geometry

Posted in Science & Nature

Pi

Posted on by Jinavie

Pi (π) a mathematical constant that is defined as the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.14159, but since it is an irrational number (cannot be expressed as a ratio), the decimal places go on and on with no repeating segments. The history of pi extends back to almost 5000 years ago, as it plays such a crucial role in geometry, such as finding the area of a circle (A = π ²). It is not an understatement to say that pi is among the top five most important numbers discovered in history (0, 1, i and e being the others).

The interesting thing about pi is that it is an irrational number. As mentioned above, this means that pi has an infinite number of non-repeating decimal places, with numbers appearing in random sequence. For example, pi to a 30 decimal places is 3.141592653589793238462643383279… Because of this feature, pi contains all possible sequences and combinations of numbers at a certain point. The corollary to this fact is, if pi is converted into binary code (a number system of only 0 and 1, used by computers to encode information), somewhere in that infinite string of digits is every combination of digits, letters and symbols imaginable. The name of every person you will ever love. The date, time and manner of your death. Answers to all the great questions of the universe. All of this is encoded in one letter: π.

That, is the power of infinity.

Posted in Science & Nature

Cryptography: Pigpen Cipher

Posted on by Jinavie

Another well-known substitution cipher is the “pigpen cipher” or “Freemason’s cipher”. As the name suggests, it was often used by Freemasons to encrypt their messages. However, as time has passed, it has become so well-known that it is not a very secure cipher at all.

The pigpen cipher does not substitute the letter for another letter, but instead uses a symbol that is derived from a grid-shaped key. The key is made of two 3×3 grids (#)(one without dots, one with dots) and two 2×2 grids (X)(one without dots, one with dots). The letters are filled in systematically so that each shape represents a certain letter (e.g. v=s, >=t, <=u, ^=v)

The cipher has many variations that attempt to throw off an attacker by rearranging the order of the grids or the letters. Thus, even if a cunning attacker picks up on the fact that the cipher is a pigpen cipher, they may use the wrong key and get a completely wrong message. Nonetheless, it is a useful skill to recognise the unique symbols of the pigpen cipher as it is a popular cipher used commonly in puzzles.

As with any substitution ciphers, frequency analysis and pattern recognition is the key to cracking the pigpen cipher.

Posted in Science & Nature

Euclidean Geometry

Posted on by Jinavie

In 300BC, a Greek mathematician called Euclid wrote a series of texts called Elements. The Elements was a textbook that outlined many principles of mathematics (especially geometry) and it would become one of the most influential works in the history of mathematics. It is composed of a series of axioms (the axiomatic approach) from which many deductions and theorems can be made. Although many of these axioms sound extremely simple and like common sense, the implications are staggering.

The following is Euclid’s Five Postulates of Plane Geometry:

  1. Two points determine a line. 
  2. Any line segment can be extended in a straight line as far as desired, in either direction. 
  3. Given any length and any point, a circle can be drawn having the length as radius and that point as centre. 
  4. All right angles are congruent (can be superimposed). 
  5. Parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side, if extended far enough. 

Using these postulates, mathematicians are able to deduce more advanced theories. For example, the Elements also describes the famous Pythagorean theorem, which states that “in any right triangle, the area of the square of the hypotenuse (the diagonal) is equal to the sum of the areas of the squares of the other two sides” (a² + b² = c²).

Thanks to Euclid’s works, we are now able to accurately model and measure the three-dimensional space around us. Not only did Euclid set the foundations for mathematics, his works were also instrumental in the development of logic and modern science.

Posted in Life & Happiness

Matchsticks

Posted on by Jinavie
  1. There are 6 matchsticks. Make 4 identical, equilateral triangles.
    The hint is that you must think differently to everyone else. If you think like everyone else, you will never find the solution.
  2. There are 6 matchsticks. Make 6 identical, equilateral triangles.
    The hint this time is quite the opposite the first puzzle: think like everyone else.
  3. There are 6 matchsticks. Make 8 identical, equilateral triangles.
    The hint for this puzzle is that you must reflect on yourself.

Answers after the break.

(Source: http://fc05.deviantart.net/fs50/i/2009/288/5/4/Matchsticks_by_oh_yesh.jpg)

Continue reading “Matchsticks” →

Posted in Philosophy

Lines

Posted on by Jinavie

There are two lines.
If they are parallel, they have many things in common but will never meet.
Sadly, all other pairs would still meet just once and then go their own ways for infinity.

One solution to this problem is making one line bend at an angle to make it continue in the same trajectory as the other line.
In life, there are moments when you must bend your line. Those who do not know how to bend will tread alone, ad infinitum.

image

Posted in Science & Nature

A Simple Task

Posted on by Jinavie

A plague struck the ancient Greek island of Delos. As the disease ravaged the island, the people went to the oracle at Apollo’s temple for help. This is what the oracle said:

Double the volume of the cube-shaped altar in Apollo’s temple

People considered this a simple task and made a new altar where each side was double the original length. However, instead of disappearing, the plague worsened and people were confused.

Reason being, given that the length of one side of a cube is a, the volume is a³; if one side is 2a, the volume becomes 8a³, or eight times the original volume. Therefore, to double the volume of a cube, the number ³√2 is required. The problem is, whether ³√2 can be found using only compass and straightedge construction (where only the two tools are used to solve a geometric problem).

This problem, also known as the Doubling the cube problem, is one of three geometric problems known to be unsolvable by compass and straightedge construction. In other words, without the help of other mathematical methods, the answer cannot be found.
However, the solution to the above story is very simple.

Find a new god.