Showing posts with label math stuff. Show all posts
Showing posts with label math stuff. Show all posts

Tesseract - A Four Dimensional Cube

In Marvel movie The Avengers, it is stated that the Tesseract contains an unimaginable power and can be used as weapon of mass destruction if the power can be unlocked. Well, the tesseract that i'm refering to is not in the movie but rather the one in field of geometry.

Tesseract from the movie The Avengers
According to wikipedia, a tesseract is a four dimensional analog of the cube and sometimes is called 8-cell, regular octahoron or cubic prism. Tesseract is a four dimensional hypercube, (cube with dimension higher than 3) and first proposed by Charles Howard Hinton, a British mathematician in the 19th century. 

A tesseract image in Schlegel diagram.
A tesseract can be unfolded into 8 cubes in 3D space, the same as a cube unfolded 6 squares in 2D surface. The image below shows how a tesseract can be build from 1 dimensional space until the first cube projection in 4 dimensional space.

How to draw a tesseract from a single point ie one dimension.

There are other types of tesseract for example truncated tesseract and rectified tesseract in which they have different Schlafli Symbol and Schlegel diagram.  The existence of higher dimension is by far an abstract concept and none have observed these higher dimensions. Nonetheless, the mathematics suggests the possibility and therefore could exist in this vast universe.


p/s: I read somewhere in which lower dimensional object couldn't possibly see the higher dimension but the higher dimensional object can definitely see the lower one. Not sure how this implicates the existence of higher deity though. 

AES Encryption


Ok guys, its been awhile since I last posted anything here but to avoid any nagging from admin I decided to up the ante for my next post. Here for all who is interested about encryption and computer security, a brief explanation about one of the basic encryption algorithm that all of them whom call themselves |-|4xx0|2 (if you don't what is this go learn about 1337). AES is simple in terms of implementation but complex in terms of production. Both simplicity and complexity is one the requirements for the AES when it was first proposed. I have to warn you th at this kind of stuff is not for the weak hearted and mind (seriously). This is a very complex thing but I know some people would have no trouble understanding.

The Advanced Encryption Standard (AES) is an encryption algorithm for securing sensitive but unclassified material by U.S. Government agencies and, as a likely consequence, may eventually become the de facto encryption standard for commercial transactions in the private sector. (Encryption for the US military and other classified communications is handled by separate, secret algorithms.)
In January of 1997, a process was initiated by the National Institute of Standards and Technology (NIST), a unit of the U.S. Commerce Department, to find a more robust replacement for the Data Encryption Standard (DES) and to a lesser degree Triple DES. The specification called for a symmetric algorithm (same key for encryption and decryption) using block encryption (see block cipher) of 128 bits in size, supporting key sizes of 128, 192 and 256 bits, as a minimum. The algorithm was required to be royalty-free for use worldwide and offer security of a sufficient level to protect data for the next 20 to 30 years. It was to be easy to implement in hardware and software, as well as in restricted environments (for example, in a smart card) and offer good defenses against various attack techniques.
The entire selection process was fully open to public scrutiny and comment, it being decided that full visibility would ensure the best possible analysis of the designs. In 1998, the NIST selected 15 candidates for the AES, which were then subject to preliminary analysis by the world cryptographic community, including the National Security Agency. On the basis of this, in August 1999, NIST selected five algorithms for more extensive analysis. These were:
  • MARS, submitted by a large team from IBM Research
  • RC6, submitted by RSA Security
  • Rijndael, submitted by two Belgian cryptographers, Joan Daemen and Vincent Rijmen
  • Serpent, submitted by Ross Andersen, Eli Biham and Lars Knudsen
  • Twofish, submitted by a large team of researchers including Counterpane's respected cryptographer, Bruce Schneier
Implementations of all of the above were tested extensively in ANSI C and Javalanguages for speed and reliability in such measures as encryption and decryption speeds, key and algorithm set-up time and resistance to various attacks, both in hardware- and software-centric systems. Once again, detailed analysis was provided by the global cryptographic community (including some teams trying to break their own submissions). The end result was that on October 2, 2000, NIST announced thatRijndael had been selected as the proposed standard. On December 6, 2001, the Secretary of Commerce officially approved Federal Information Processing Standard (FIPS) 197, which specifies that all sensitive, unclassified documents will use Rijndael as the Advanced Encryption Standard.
The Advanced Encryption Standard (Rijndael)
The block cipher Rijndael is designed to use only simple whole-byte operations. Also, it provides extra flexibility over that required of an AES candidate, in that both the key size and the block size may be chosen to be any of 128, 192, or 256 bits. (During an early stage of the AES process, a draft version of the requirements would have required each algorithm to have three versions, with both the key and block sizes equal to each of 128, 192, and 256 bits. This was later changed to make the three required versions have those three key sizes, but only a block size of 128 bits, which is more easily accomodated by many types of block cipher design.)
The original description of Rijndael is available at: http://www.esat.kuleuven.ac.be/~rijmen/rijndael/.

However, the variations of Rijndael which act on larger block sizes apparently will not be included in the actual standard, on the basis that the cryptanalytic study of Rijndael during the standards process primarily focused on the version with the 128-bit block size.
Rijndael is a relatively simple cipher in many respects.
Rijndael has a variable number of rounds. Not counting an extra round performed at the end of encipherment with one step omitted, the number of rounds in Rijndael is:
  • 9 if both the block and the key are 128 bits long.
  • 11 if either the block or the key is 192 bits long, and neither of them is longer than that.
  • 13 if either the block or the key is 256 bits long.
To encipher a block of data in Rijndael, you first perform an Add Round Key step (XORing a subkey with the block) by itself, the regular rounds noted above, and as already noted, the final round with the Mix Column step, as described below, omitted.

Now that you know a bit about this type of encryption, here is an example of what the encryption looks like kinda. You can download a pdf from:

Female Scientists Who Changed The World

Marie Curie
Together with her husband, they performed ground breaking and risky procedures in their laboratory. On the heels of the discovery of radiation by Henri Becquerel in 1896, she developed ways to separate radium from radiation leading to many current practices, including chemotherapy. Later in life, she became the Director of the Curie Laboratory in the Radium Institute of the University of Paris. For her work, she was awarded various prizes, including the Nobel Prize for Physics in 1903 and again for chemistry in 1911.

Ada Byron
Also known as Lady Lovelace, she was the daughter of world renowned poet, Lord Byron. Although she strived to be an analyst and metaphysician, her father’s gift of imagination and creativity led her to see the then Analytical Engine as something far more profound. After her translation of the engines creator’s work took on a life of its own, she suggested a plan for how the engine could calculate Bernoulli numbers. This plan, developed in 1843, would then be regarded as the first ever computer program.

Rosalind Franklin
Born in Great Britain in 1920, Rosalind decided to be a scientist at the age of 15. However, she hit a speed bump when her father refused to pay for higher education, believing women were unfit for it. With some prodding from family members, Rosalind was finally able to go where she was able to assist in making amazing breakthroughs in DNA research. She was able to adjust x-ray equipment to produce a fine beam and extract DNA fibers like never before. Unfortunately, her life was cut short at the age 37 by ovarian cancer. But her name and work still live on the Rosalind Franklin University of Medicine and Science in Illinois.

Lisa Meitner
Born in 1878 in Austria, she was the third of eight children. After obtaining a doctorate from the University of Vienna, she went on to study with noted chemist Otto Hahn who would later go on to win the Nobel Prize for Chemistry in 1944 for their collaborated work. It led to the discovery of a new element protactinium, the Auger effect, and eventually discovered the process of nuclear fission with her nephew. Being of Jewish descent, her work was interrupted by the Nazi invasion, but she survived to continue her work.

Hypatia of Alexandria
Born at approximately 360 A.D. in Alexandria, Egypt, she is considered the first female mathematician. Along with her father, they collaborated together on impressive works such as “Arithmetica,” “Almagest,” and “Elements.” She is also believed to have written “The Astronomical Canon.” Hypatia also did work in philosophy, accounting, and astronomy. She is even credited with the charting of celestial bodies, the invention of the hydrometer, and more.

Dorothy Hodgkin
Born in Cairo in 1910, this British scientist is noted with discovering protein crystallography, which is a method of x-rays that can determine the arrangement of atoms. She became interested in chemistry and in crystals at about the age of ten and was encouraged by a friend of her parents who gave her chemicals and helped her analyze them. When returning to Oxford in 1934, she would continue the work that earned her a Nobel Prize in Chemistry in 1964. Her work was also vital in the studies of insulin and penicillin.

Sophie Germain
She was born in France in 1776 to an era of revolution. Thirteen years later, her country would undergo its own revolution, and due to her confinement at home, she became interested in mathematics. Although her parents objected to a female learning mathematics, Sophie continued to read on the subject and teach herself. Unable to attend institutions of higher education, Sophie found ways to study by obtaining lecture notes, using a pseudonym to write to scientists, and even attending academic functions. Her work ultimately led to the theory of elasticity and noted advances in number theory.

Barbara McClintock
Born in 1902, this American was one of the world’s most noted cytogeneticists, a geneticist who studies the structure and function of the cell with particular interest in chromosomes. She attended Cornell University in 1927 where she received a PhD in botany. However, after an undergraduate course in genetics, she found herself fascinated by the subject. Barbara devoted her studies to seeing how chromosomes change during the reproduction process. It was during this work that she discovered that genes could move between and on chromosomes, leading to many other important discoveries. For this, she was awarded the Nobel Prize in Physiology or Medicine in 1983.

Rachel Carson
She was a writer, scientist, and ecologist born in Pennsylvania in 1907. After obtaining a MA in Zoology from Hon Hopkins University, she was hired by the U.S. Bureau of Fisheries to write radio scripts during the depression and feature articles on natural history for the “Baltimore Sun.” After becoming concerned by the use of synthetic chemical pesticides after World War II, Carson changed her focus in order to warn the public about the long term effects of misusing pesticides. One of the very first environmentalists, she changed the world by inspiring a host of present and future ones.

**Materials taken from here. Continue reading to see the remaining six who's still alive.

Brain Teaser

Below are a few questions that you can use to test your friend's intelligence.


1. A bear walks south for one kilometer, then it walks west for one kilometer, then it walks north for one kilometer and ends up at the same point from which it started. What color was the bear?


2. A chicken farmer also has some cows for a total of 30 animals, and the animals have 74 legs in all. How many chickens does the farmer have?


3. Three students checked into a hotel and paid the clerk $30 for a room ($10 each). When the hotel manager returned, he noticed that the clerk had incorrectly charged $30 instead of $25 for the room. The manager told the clerk to return $5 to the students. The clerk, knowing that the students would not be able to divide $5 evenly, decided to keep $2 and to give them only $3.
The students were very happy because they paid only $27 for the room ($9 each). However, if they paid $27 and the clerk kept $2, that adds up to $29. What happened to the other Dollar?


4. You start walking north and you walk in a straight line for two kilometers. When you look at the map, you discover that you actually walked one kilometer north and one kilometer south. How is this possible?

5. At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?


6. The grandson is about as many days old as the son is in weeks. The grandson is approximately as many months old as the father is in years. The ages of the grandson, the son, and the father add up to 120 years. What are their ages in years?

7. You have to measure exactly 4 liters of water, but you only have a 3-liter bottle and a 5-liter bottle. How do you do it?

8. The government pays farmers a specific fee for each row of four trees that they plant. An enterprising, but dishonest farmer found a way of planting five rows of four trees using only ten trees. How did he do it?

See all the answers and also some other question here.

General Theory of Relativity Lecture.

Hi. I found this video on Youtube where you can expand your knowledge by watching it. It's more than 1 hour long, so make sure you have a fast internet connection.


The video is the courtesy from Stanford University and the one who gave the lecture was Leonard Susskind. You can browse the youtube for more videos and lectures from him. The General Theory of Relativity has 12 lectures featuring him. Make sure you learn something while watching the video!

A Peek Into The History of Mathematics

This will just be a quick update. I still am reading a few materials so that i can understand them before i share it with you guys. 

Below are something that we don't know about mathematics.



Pi is mathematical symbol that gives the value approximately 3.14159. The value for pi is actually the ratio between the circumference and the radius in Euclidean Space. 


The most ancient mathematical texts available are Plimpton 322, which is believed to come from Babylonian Mathematics, 1900 years before century


The natural logarithm e is called Euler's number, after the Swiss mathematician Leonhard Euler


The tangent function (tan) was found by Islamic scholar, Muhammad Ibn Musa Al Khawarizmi


The term "angle" was a Latin word meaning the corner

I think that is it for my short update. Right now I am also searching for the right material or topic to be discuss. So if you have something valuable, do contact me. Thank you