Here is a set of 5 photographs: click on them to enlarge.

pix 1: December 2004 (courtesy Binayak)

A panoramic view of the East bank of the river Nile. In fact, the views are about the same for both the banks. There are vegetation and cultivation within a width of about 1Km from the edge of the river on both banks. Almost the entire population of Egypt lives on that narrow fertile strip of land. Beyond that, there are only hilly desert areas. That is the general nature of Egypt’s landscape. It is very true that without Nile the whole of Egypt would have been a lifeless desert.

pix 2: December 2004 (courtesy Binayak)

A small jetty used by the Egyptian farmers on the river bank. One can take note of the types of vegetation that grow there. The date palm trees are easily recognizable.

pix 3: December 2004 (courtesy Binayak)

Desert has encroached almost up to the edge of the river here. A microwave radio communications tower is visible in the background.

pix 4: December 2004 (courtesy Binayak)

Another view of the East bank of Nile.

pix 5: December 2004 (courtesy Binayak)

A highway and a railroad run along almost the entire length of the Nile connecting all the important cities of Egypt. He we can see train on the railroad.

2. For the scientific minded:

This month’s topic: Arithmetic of remainders (continued).

Some more advanced theorems with proofs are shown in Plates 1 to 4 below.

Plate 1: Theorem 7

Plate 2: Theorem 8

Plate 3: Theorem 9

Plate 4: Theorem 10

3. For “pop” music lovers:

“Rivers of Babylon” was written and recorded in 1972 by Brent Dowe and Trevor McNaughton of the group "Melodians." It appearded in the sountracks of the 1972 movie “The Harder They Come.” However the song got more publicity after the German group "Boney M" re-recorded it in 1978.

Here is a set of 4 photographs: click on them to enlarge.

Pix 1: December 2004 (courtesy Binayak)

An external view of a Nile luxury cruise ship at Luxor. The passenger cabins are organized in 3 tires and the upper deck is an open one. The ships have 3-star to 5-star ratings depending on the facilities provided.

Pix 2: December 2004 (courtesy Binayak)

A swimming pool on the upper deck of one of the cruise ships. Another ship is seen in the background.

Pix 3: December 2004 (courtesy Binayak)

A luxury ship sailing up the Nile on its way to the town of Edfu from Luxor.

Pix 3: December 2004 (courtesy Binayak)

The luxury ships docking at the town of Edfu after arriving in the early hours of morning.

2. For thescientific minded.

This month’s topic: Arithmetic of remainders (continued).

The concept described last month is also applicable for negative integers. Plate 1 shows a modulo 7 organization for such integers. Compare it with Plate 2 of July issue.

Plate1: Modulo 7 organization for negative integers

Some basic theorems relating to the Arithmetic of remainders with proofs are shown in Plates 2, 3 and 4 below.

Plate2: Theorems 1 and 2

Plate3: Theorems 3 and 4

Plate4: Theorems 5 and 6

3. For “pop” music lovers:

Que Sera Sera (Whatever Will Be Will Be) was first published in 1956. It is a song written by Jay Livingston and Ray Evans. Since then it has been performed by several singers and still remains popular.

Here is a set of 4 photographs: click on them to enlarge.

Pix1: December 2004 (courtesy Binayak)

A panoramic view of the general surroundings of the Valley of the Kings from Hatshepsut’s temple. The Valley of the Kings is located on the west bank of Nile, opposite to the city of Luxor. This is a hilly area in the desert where Pharaohs were buried in concealed tombs from the 16th century to 11th century B.C.

Pix2: December 2004 (courtesy Binayak)

A view inside the Valley of the Kings with signboards showing locations of the excavated tombs of different Pharaohs.

Pix3: December 2004 (courtesy Binayak)

Another view inside the Valley of the Kings showing visitors trying to read signboards in order to locate the tombs they want to visit.

Pix4: December 2004 (courtesy Binayak)

An ancient temple like structure at the Valley of the Kings.

2. For the scientific minded.

This month’s topic: Arithmetic of remainders.

Brief history:

This concept was invented by German mathematician Karl Friedrich Gauss and appeared in his thesis Disquisitiones Arithmeticae submitted to the French Academy in 1801. Gauss was just 24 years old then.

Review of concept:

This is a way of grouping the vast range of integers (from zero to an infinitely big one) into groups. Any such group known as a module consists of a known collection of integers - those that are less than a predetermined integer n. Any integer, whether big or small, can be shown to be modulo or mod n equivalent (congruent is the term used by Gauss) to some integer in the module. Plates 1, 2 explain the concept. Each column in Plate 2 is a module and the integers 3, 10 and 7 are all equivalent to 3 (mod 7). Plate 3 shows a modulo 10 organization that we are most familiar with. Plate 4 shows the concept from another angle for example: the integer 13 is equivalent to 1 (mod 3) as well as 1 (mod 4).

Plate1: Defining Modulo n numbers

Plate2: Modulo 7 organization

Plate3: Modulo 10 organization

Plate4: Different modulo representations

3. For “pop” music lovers:

“Limbo Rock” was recorded in 1962 by the American singer-songwriter Chubby Checker (real name Ernest Evans). It reached no. 2 in the national charts 1962 and was the last of his top ten hits.

Internal view of the Karnak temple at Luxor. The structures on the left and right symbolize the West and East bank of river Nile and the empty central area the Nile itself. The two huge statues, one on each side in the foreground belongs to the great Pharaoh Ramesses II.

Pix2: December 2004 (courtesy Binayak)

A statue of Pharaoh Ramesses II with his queen Nefertari at his feet at the Karnak temple. According to the customs of those days, the queen was to be shown in diminished size when accompanying the Pharaoh. He ruled Egypt from 1278 BC to 1213 BC, known to be the longest reign by any Egyptian Pharaoh. Undoubtedly, he was also one of the most powerful Pharaohs and won several military campaigns against Syria, Nubia and Libya. Apart for this, he built several cities, temples and monuments throughout Egypt.

Pix3: December 2004 (courtesy Binayak)

Two sets of massive stone pillars decorated with Hieroglyphics support the two structures of the Karnak Temple.

Pix4: December 2004 (courtesy Binayak)

Sunken relief on the walls of the Karnak temple depicting coronation of a Pharaoh by Egyptian gods. Above, Hieroglyphics describe the scene.

Pix5: December 2004 (courtesy Binayak)

The Obelisk of some Pharaoh at the Karnak Temple. Most Pharaohs erected one such structure inscribing all his achievements on it.

2. For the scientific minded:

This month’s topic: More properties of Fibonacci numbers

We conclude discussion on Fibonacci’s and related work in this issue. Plates 1 to 4 describe some additional properties of Fibonacci numbers without proof.

Plate1: Property 1 of Fibonacci numbers

Plate2: Property 2 of Fibonacci numbers

Plate3: Property 3 of Fibonacci numbers

Plate4: Property 4 of Fibonacci numbers

3. For “pop” music lovers:

"Sway" is the English version of the 1953 song “Quien sera?” by the Mexican composer Pablo Beltran Ruiz. In 1954, the song was translated in English by Norman Gimbel and recorded by Dean Martin. Since then, this song was re-recorded by many artists. One of the popular versions is the one by Michael Buble recorded in 2003.

Here is a set of 5 photographs: click on them to enlarge.

Pix1: December 2004 (courtesy Binayak)

Modern settlements on the West bank of the River Nile in Egypt viewed from the outskirts of the city of Luxor located on the East bank. Luxor is the center of the ancient Egyptian civilization, being the capital of the “New Kingdom” and the city of the great Egyptian god Amon-Ra. Luxor was known as “Waset” and “Taipet” in the ancient times and re-named “Thebes” by the Greek and Roman occupiers later on. Luxor stands amongst the ruins of two ancient Egyptian temples: Karnak and Luxor dedicated to Amon-Ra. In the hilly desert areas on the West bank of Nile, directly opposite to Luxor, is the Valley of Kings and Queens. This is the burial place for the pharaohs, and their spouses. Ancient Egyptians always used to consider the East bank of Nile to be the land for the living and the West bank the land for the dead.

Pix2: December 2004 (courtesy Binayak)

A panoramic view of the Valley of Kings.

Pix3: December 2004 (courtesy Binayak)

The remains of the burial temple of pharaoh Amenhotep III on the West bank that was destroyed by Nile floods within 200 years of its construction. This temple was the largest in Luxor and Egypt but virtually non-existent today. The two colossal statues, later re-named the Colossi of Memnon by the Greek, are those of the pharaoh himself, guarding the temple gates. Amenhotep III was the ninth pharaoh of the eighteenth dynasty and ruled from 1388 B.C. to 1351 B.C.

Pix4: December 2004 (coutrtesy Binayak)

The restored burial temple of queen and pharaoh Hatshepsut. She was the fifth pharaoh of the eighteenth dynasty reigning from 1479 B.C. to 1458 B.C., being one of the most successful amongst all pharaos, male or female. She assumed the position of pharaoh while acting as the regent for her infant son Thutmose III and always used to dress like a man.

Pix5: December 2004 (courtesy Binayak)

An Egyptian bloom on the banks of Nile.

2. For the scientific minded:

This month’s topic: Geometrical shapes incorporating the Golden Ratio.

1. Brief History:

The Golden Ratio has fascinated intellectuals from the ancient to modern times. These include Phidias, Pythagorus, Plato, Euclid, Fibonacci, Leonardo da Vinci, Luca Pacioli, Johannes Kepler and Roger Penrose.

2. The “Golden” geometrical shapes:

(a) The right-angled triangle:

Plate1 shows a right-angled triangle ABC with the base and perpendicular are in the ratio a : b = the Golden Ratio “phi” .

Plate1: The golden right-angled triangle

(b) The isosceles triangle:

Plate2 shows two isosceles triangles ABC and PQR. For any one of them the ratio of the longer side(s) is to the shorter side(s) is a : b = “phi”.

Plate2: The golden isosceles triangles

(c) The rectangle: Plate3 shows a rectangle ABCD with side ratio of a : b = “phi”.

Plate3: The golden rectangle

(d) The regular pentagon with diagonals:

Plate4: The golden pentagon and pentagram

Plate4 shows a regular pentagon ABCDE with all its diagonals drawn. It contains several large, medium and small triangles of type1 and type2 as shown in Plate2. For example, ADC and ATP are a large and a small triangle of type1. Similarly, AEB and APB are a large and a small triangle of type2. Again, DET is a medium triangle of type1.

The diagonals and the sides of the regular pentagon are in the ratio a : b = “phi”. In addition, the diagonals divide each other in the same ratio. For example, AS is to DS is “phi”, DS is to TS is “phi” and AD is to AS is also “phi”.

The diagonals minus the sides (like a five pointed star) is also an ancient mystical symbol known as the pentagram or the pentangle.

The intersection points of the diagonals create another regular pentagon PQRST which will also show all the above properties if its diagonals are drawn.

3. Analysis:

We’ll look into the proofs of the statements made above.

Plate5: Proof of the golden isosceles triangles

(a) The proof for the right-angled triangle is given in Plate1 in accordance to the definition of “phi”.

(b) The proofs for the isosceles triangles type1 and type2 are given in Plate5, again in accordance to another definition of “phi”. It can be observed that in this plate, ABC (and BCD) is the type1 triangle and ABD is the type2 triangle. Therefore the same proof works for both. In addition to this, it can be seen that the definition of the Golden Ratio can be extended to: "phi" = a : b = (a + b) : a = b : (a - b). It can also be noted that the numbers (a - b), b, a, (a + b) form a Fibonacci sequence if the first two are Fibonacci numbers.

(c) The proof for the rectangle can easily be derived from that of the right-angled triangle.

(d) Calculating angles as shown in Plate4, one can show, for example, BE is parallel to CD. Then it becomes easy to find all the large and small type1 and type2 triangles and prove many of the statements. To prove the rest, it can be noticed that ACD is a type1 triangle with CS bisecting angle ACD. Therefore, according to Plate5, AS and DS are in the ratio “phi”. Again, DET is a type1 triangle with ES bisecting angle DET. Then, again DS and TS are also in the ratio “phi”. Lastly, ABCS is a parallelogram and AS = BC = b. So AD is to AS is a : b = “phi”.

3. For (pop) music lovers:

“Jambalaya” was first released in 1952 by the American country music singer Hank Williams. Later, it was recreated by many others including the American sister and brother group Carpenters. The song, released in 1974 by Carpenters, was one of the biggest hits for the group. “Jambalaya”, a Louisiana dish of Spanish, French and Cajun influence is made from chicken, sausage, rice, celery, onion and green ball pepper.

Here is a set of 5 photographs: click on them to enlarge.

Pix1: December 1995

This is not Tajmahal, but a memorial dedicated to another great lady: Queen Victoria, the Queen of England from 1837 to 1901. It is the Victoria Memorial in Calcutta, India. Calcutta was the capital of British ruled India from 1757 to 1911 before New Delhi replaced it. Today Calcutta is the fourth important city in India. The monument, designed by Sir William Emerson, was built between 1906 and 1921. The architecture clearly displays a fusion between Western and Indian (muslim) styles.

Pix2: December 1995

The St. Paul’s Cathedral, Calcutta, India, initiated by Bishop Wilson, was built in Gothic Revival style between 1839 and 1847. It is located close to the Victoria Memorial.

Pix3: February 2008 (courtesy Binayak)

The City of San Francisco is one of the oldest in the state of California and the US itself. It is located amongst hilly areas at the tip of the San Francisco peninsula extending into the Pacific Ocean. Established as a fort by the Spanish in 1776, the city grew during the times of the California Gold Rush in 1848. It was devastated by earthquake and fire in 1906 but quickly re-built. Today it is an important international financial, transportation and cultural center apart from being a popular tourist destination.

Pix4: February 2008 (courtesy Binayak)

Another view of the City of San Francisco showing city folks settling down for a game of chess using temporary dining places on the wide walkways. These belong to the several fast food joints along the road.

Pix5: February 2008 (courtesy Binayak)

Many buildings in San Francisco are old-styled (Greco-Roman) like this one. This shows the city’s ancient cultural heritage.

2. For the scientific minded:

This month’s topic: The Golden Ratio or Divine Proportion

1. Brief History:

Though the modern history of Golden Ratio starts from Luca Pacioli’s (a catholic priest and a mathematician) definition of the constant in his book Divina Proportione published in 1509, it seems to be known since around 500 BC, being in the knowledge of several scholars including Phidias, Plato, Euclid, Fibonacci and Leonardo da Vinci. This constant (1.618, correct to 3 decimal places) is actually a ratio or proportion that looks aesthetically perfect when used in architecture and paintings. Several proportions in the structures shown in Pix 1, 2 and 5 above are likely to incorporate the constant. According to Leonardo da Vinci’s study, a lot of proportions in the human body are approximations of this constant. For example, the average length ratio of the human forearm (upto the fingertips) is to the upper arm is 1.6. The same holds for the ratio of the lower section to the upper section of the human leg. Hence, the alternate name Divine Proportion comes into picture.

2. The definition and value of the Golden Ratio:

The Golden Ratio is defined as (a + b) : a = a : b. If this relation is satisfied by two positive numbers a and b, then a : b is the Golden Ratio. It will be evident that this relationship cannot be satisfied for rational values of both a and b. Consequently, this ratio works out to be an irrational number (like “pi” = 3.141592654… or “e” = 2.718281828…) with a value of 1.618033988… and is denoted by the Greek letter (lowercase) “phi”. The inverse as well as the conjugate of Golden Ratio is also an irrational number with a value of 0.618033988… and is denoted by the Greek letter (uppercase) “phi”.

3. Analysis:

We now proceed to show why the ratio of two succesive terms of a Fibonacci sequence tend to the Golden Ratio or its inverse as the terms become large, as shown in Plates 3 and 4 of the March 2009 issue. We observed in Plate 2 of March 2009 issue that in a Fibonacci sequence, any term is the sum of the previous two terms. Using this relationship, we show in Plates 1 and 2 that the ratio of a term to the next term, for large terms, can be expressed as a continued fraction. Let b and a be two successive terms then the next term will be a+b. If these are very large terms, then the continued fraction representations of b : a and a : (a + b) will practically have no difference. Then by the definition above, b:a is the inverse of the Golden Ratio. This is shown in Plate 3 below.

Plate1: Ratio of successive terms in continuation for a Fibonacci sequence

Plate2: Ratio of successive large terms in a Fibonacci sequence

In Plates 3 and 4 below we proceed to find the conjugate surd form of the Golden Ratio and its inverse along with their decimal values.

Plate3: The Golden Ratio and its inverse

Plate4: The conjugate surd form of the Golden Ratio and its inverse

Finally, using the relationships and results obtained in Plates 2 to 4 we write the Golden Ratio and its inverse in the infinite continued fraction form in Plate 5 and in the infinite continued square-root form in Plate 6.

Plate5: The infinite continued fraction form of the Golden Ratio and its inverse

Plate6: The infinite continued square-root form of the Golden Ratio and its inverse

3. For (pop) music lovers:

“Rasputin” was released in 1978 by the German pop group Boney M. This song was one of the best by the group. It incorporates portions of a classic Turkish folk song. The subject of the lyrics is the life of Grigori Rasputin, a friend and advisor of Tsar Nicholas II of Russia.

This is a monthly magazine for people of all ages and with all kinds of interests. The items presented here are factual and usually have some historical content. None of these material is copyrighted and can be shared freely. It is being presented in good faith and with sufficient research. A lot of care has been taken to make the material interesting but not harmful in any way. In the event that anybody finds any material objectionable, please let me know. Suggestions to improve are also most welcome. A new issue will be published towards the end of every month and the best time to visit this site is the around the second or third week of the month.

The formula derived in Plate5 (February 2009 issue) can generate only a part of all possible Pythagorean triples. Using Number Thory, the complete soution is given by:

M = s^2 - t^2; N = 2st; L = s^2 + t^2

where s, t are integers such that:

s > t > 0, s and t have no common factors and s - t is not divisible by 2.

Other Number Systems

The number systems in use today, other than the Decimal system, are the Binary, Octal and Hexadecimal systems. The humans understand and use the Decimal system whereas the computers use the others. The Decimal system uses ten as the base whereas the Binary, Octal and Hexadecimal systems use two, eight and sixteen. The two symbols in a Binary sytem are 0 and 1. The eight symbols in an Octal system are 0, 1, 2, 3, 4, 5, 6 and 7. The sixteen symbols in a Hexadecimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Binary system the one that is mostly used by computers because computers can work with and store this system naturally. The Decimal, Octal and Hexadecimal numbers can also be written using Binary symbols. Then the numbers are said to be Binary coded. Thus, just using Binary symbols 1 and 0, computers can store and work with numbers in any of the four systems.

For example, the Binary number 11110 can be written as:

(a) 36 in Octal.

(b) 011 110 in Binary Coded Octal. (c) 30 in Decimal. (d) 0011 0000 in Binary Coded Decimal.