Golden Rectangle ( Trivia # 23 )

A golden rectangle is one whose side lengths are in the golden ratio, 1: φ (one-to-phi) or 1 : ( 1 + √5 )/2 or approximately 1 : 1.618 .
A distinctive feature of the golden rectangle is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, in which case we will get golden rectangle again and again with decreasing size .


Ratio of new sides =

1 / (φ-1) = (φ+1) / (φ+1)(φ-1) = (φ+1) / (φ^2 -1)

Now we all know φ^2 = φ+1

Also 1/φ = φ - 1 ...... property of phi

(φ+1) / (φ^2 -1) = (φ+1) /φ = 1 + 1/φ = 1 + φ - 1 = φ : 1

This is equal to the ratio of sides of the original rectangle . Hence we proved the fact that when a square section is removed, the remainder is another golden rectangle. This property makes it unique .

Boy's surface ( Trivia # 22 )

Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 . He discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3-space .


To make a Boy's surface:
1. Start with a sphere. Remove a cap.
2. Attach one end of each of three strips to alternate sixths of the edge left by removing the cap.
3. Bend each strip and attach the other end of each strip to the sixth opposite the first end, so that the inside of the sphere at one end is connected to the outside at the other. Make the strips skirt the middle rather than go through it.
4. Join the loose edges of the strips. The joins intersect the strips.

Boy's surface can be used in sphere eversion, as a half-way model. A half-way model is an immersion of the sphere with the property that a rotation interchanges inside and outside, and so can be employed to evert (turn inside-out) a sphere. Boy's surface is the beginning of a sequence of half-way models with higher symmetry first proposed by George Francis.

If Rubik’s Cube Was a Ball ? ( Trivia # 20 )

If you took the classic Rubik’s Cube mechanical puzzle and changed it into the shape of a ball, you’d get something that looked much like this.

Königsberg Bridge Problem ( Trivia # 16 )

The Konigsberg Bridge problem is perhaps the best known example in the graph theory. It was a long-standing problem until solved by Leonhard euler in 1736, by means of a graph. The problem is depicted in the pictures given below.

Fig. 1

Fig. 2

Two islands , C and D, formed by the Pregel River in Konigsberg were connected to each other and to the banks A and B with seven bridges, as shown. The problem was to start at any of the four land areas of the city , A , B , C , or D, walk over each of the seven bridges exactly once, and return to the starting point ( without swimming across the river, of course :D ).

U can now see this situation by means of a graph, as shown in the figure below. The vertices represent the land areas and the edges represent the bridges.

Fig. 3

Before dealing with this problem let us generalize this problem .... as proposed by the Euler Graphs
def : If some closed path in a graph passes through all the vertices , then this path is called an Euler line and the graph an Euler graph . Thus an Euler Graph has no isolated vertex and all vertices form an interconnected network ...
Another important property of an Euler Graph is that all the vertices are of even degree ?? Puzzled ....
Since it is an Euler Graph it contains an Euler line(the closed path). In tracing this line we observe that every time it meets a vertex 'x' two lines are created .. one "entering" that vertex and another "leaving" the vertex . This is not only true for all intermediate vertices but also for the terminal vertex (because we "entered" and "exited" this vertex at the begining and end) . This proves that the degree (no. of lines radiating from a vertex) of every vertex is even .

An Euler Graph

Thus for finding the solution to the Konigsberg bridge we need to check whether its graphical representation is an Euler Graph .... U can check urself that its vertices are not having even degree .Thus it is not possible to walk over each of the seven bridges exactly once and return to the starting point . Hence the Konigsberg Bridge problem possesses no solution .

Another variation of this kind of problem is mentioned below ...... u might have tried it when u were a kid ...
Without lifting the pencil draw this figure without passing through the same line or curve twice ..


I made another version of this Konigsberg Bridge .... try this too ... and post ur solution in comments ...

Mahato Ka Bridge :)

Swastika ( Trivia # 10 )

The swastika is an equilateral cross with its arms bent at right angles , in either right-facing (卐) form or its mirrored left-facing (卍) form. Archaeological evidence of swastika-shaped ornaments dates from the Neolithic period. It occurs mainly in the modern day culture of India, sometimes as a geometrical motif and sometimes as a religious symbol. It remains widely used in Eastern religions / Dharmic religion such as Hinduism, Buddhism and Jainism. Though once commonly used all over much of the world, its iconic usage in Nazi Germany has stigmitized the symbol in the Western world. It's usage has been outlawed in Germany.

Geometrically, the swastika can be regarded as the area inside of an irregular icosagon or 20-sided polygon. The proportions of were fixed based on a 5x5 diagonal grid.
Characteristic is the 90° rotational symmetry and chirality, hence the absence of reflectional symmetry, and the existence of two versions of swastikas that are each other's mirror image.

Indeed Swastika is one of the most popular shapes in the world !!

A right-facing swastika might be described as "clockwise"

The swastika was used as an official emblem of the Nazi Party, a use sometimes continued by modern Neo-Nazis.

The swastika in the decorative Hindu form.

Möbius strip ( Trivia # 9 )

The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It is also a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

A model can easily be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a loop. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it has "handedness" (as in right-handed or left-handed).

A Möbius strip made with a piece of paper and tape. If an ant were to crawl along the length of this strip, it would return to its starting point having traversed both sides of the strip, without ever crossing an edge.

The Universal Recycling Symbol is a form of Möbius strip.

Lute of Pythagoras ( Trivia # 4 )

In my earlier posts i told you about the Pythagoras Tree . Now i am going to tell u something about the LUTE ....
The lute of Pythagoras is a geometric form made of pentagons with inscribed pentagrams the sides of the pentagrams are the sides of the smaller pentagons. The sides of the lute are based on the number phi, which is the golden ratio and an irrational number like pi. If you measure any line in the lute and also a similar line just before it , the ratio of both comes out to be phi.

Lute of Pythagoras

Pythagoras Tree (Trivia # 2)

The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. From the left each image is an iteration of this function growing exponentially more complex. The top is a 45,45,90 triangle and the bottom is a 30, 60, 90 triangle resulting in a lopsided tree.

Pythagoras tree

If we add a third dimension by making the squares into cubes and exploring the geometry in 3 dimensional space we get a Dragon Curve. The cubes were rendered 95% transparent so you can see the geometry inside of it.

Pythagoras Dragon Curve