Strange Property of Mobius Strip ( Trivia # 25 )

My first post from IIT Kanpur ... I thought that after coming to IIT K my dream MATHHUB will come to an end .. but i was wrong .. Now I realize that it is just a begining .. The trivia which i am going to share with u today is something which i didn't knew ... Keshav (the senior dude whose lappy i am using to post this trivia) told me and after verifying it experimentally i was surprised ...
U can consider it as the continuation of my previous post of Mobius Strip ...
Draw a line in the middle of the mobius strip. Now cut along this line. Now the surprising part is that instead of getting two separate strips, the Möbius strip will become one long strip. This long strip has four half-twists in it.

Make a Mobius strip and try this out ... itz amazing .. !

MOBIUS STRIP ( 1 TURN )

DIVIDED FROM MIDDLE

GOT ANOTHER MOBIUS STRIP ( 4 TURNS )

Number Relationships ( Trivia # 24 )

Here are some beautiful Number Relationships.
Notice the consecutive exponents :



Now, taken one place further, we get :

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.

Good Primes ( Trivia # 21 )

Erdös and Strauss defined a good prime as a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes.

p (n)^2 > p (n − i) * p (n + i)

for all values of i from 1 to n − 1.

The infinite sequence of good primes starts :
5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149 ........