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Offlineisaacein
exp(ix) = cosx + isinx


Registered: 05/21/08
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The Amazing Math Thread
    #11281383 - 10/20/09 01:40 AM (8 years, 1 hour ago)

Have some amazing mathematics you'd like to share? Share it here! Nice pictures are most welcome.

Recall that the prime numbers are the positive integers which have exactly two distinct divisors. They are

2,3,5,7,11,13,17,19,23,29,31,37,...


Goldbach wrote to Euler in 1742 to let him know that every even integer greater than 2 "appeared" to be the sum of two primes. (For example, 38=7+31=11+17.) In over 250 years, nobody has yet managed to show that this is true, nor has anyone found a counterexample. We have checked that there are no counterexamples up to 12 x 1017 (i.e. 1,200,000,000,000,000,000).

Here is a plot of the number of ways of writing each integer less than a million as a sum of two primes.



A beautiful thing isn't it?
By looking at the picture above you see that there appears to be many ways of writing large integers as sums of two primes. The number of representations seems to be steadily increasing, yet we are unable to show that it is always at least one.


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Edited by isaacein (10/20/09 11:13 PM)


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Invisiblecortex
[ H ] ψ = [ E ] ψ
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Re: Goldbach's Conjecture [Re: isaacein]
    #11285422 - 10/20/09 06:10 PM (7 years, 11 months ago)

Reminds me of "number spirals".

Arranging all integers so all the perfect squares line up on the right...



...and zooming way out...



..then highlighting only the primes....



...you can see the primes cluster along certain curves...



There is all kind of mystery surrounding the way primes fit into our number system.

This page has a lot more about the "number spiral", I thought it was pretty cool.


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InvisibleDiploidM
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Registered: 01/09/03
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Re: Goldbach's Conjecture [Re: isaacein]
    #11286098 - 10/20/09 07:58 PM (7 years, 11 months ago)




The Barnsley Fern is another mysterious and beautiful result in number theory that hints at deep mathematical underpinnings in biology.

Here's the algorithm:

http://en.wikipedia.org/wiki/Barnsley%27s_fern#Example:_a_fractal_.22fern.22

The first point drawn is at the origin (x0 = 0, y0 = 0) and then the new points are iteratively computed by randomly applying one of the following four coordinate transformations[3][4]:

1.

    xn + 1 = 0

    yn + 1 = 0.16 yn.

This coordinate transformation is chosen 1% of the time and maps any point to a point in the line segment shown in green in the figure.

2.

    xn + 1 = 0.2 xn − 0.26 yn

    yn + 1 = 0.23 xn + 0.22 yn + 1.6.

This coordinate transformation is chosen 7% of the time and maps any point inside the black rectangle to a point inside the red rectangle in the figure.

3.

    xn + 1 = −0.15 xn + 0.28 yn

    yn + 1 = 0.26 xn + 0.24 yn + 0.44.

This coordinate transformation is chosen 7% of the time and maps any point inside the black rectangle to a point inside the dark blue rectangle in the figure.

4.

    xn + 1 = 0.85 xn + 0.04 yn

    yn + 1 = −0.04 xn + 0.85 yn + 1.6.

This coordinate transformation is chosen 85% of the time and maps any point inside the black rectangle to a point inside the light blue rectangle in the figure.

The first coordinate transformation draws the stem. The second draws the bottom frond on the left. The third draws the bottom frond on the right. The fourth generates successive copies of the stem and bottom fronds to make the complete fern. The recursive nature of the IFS guarantees that the whole is a larger replica of each frond. Note: The fern is within the range -2.1818 ≤ x ≤ 2.6556 and 0 ≤ y ≤ 9.95851.


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2) You can't have an abortion no matter how much you don't want a child.
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4) We need a smaller, less-intrusive government.


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Offlineisaacein
exp(ix) = cosx + isinx


Registered: 05/21/08
Posts: 1,141
Last seen: 6 years, 7 months
Re: Goldbach's Conjecture [Re: cortex]
    #11287873 - 10/20/09 11:20 PM (7 years, 11 months ago)

Cortez : that is indeed a nice discovery, due to the mathematician and physicist Stanislav Ulam. He was once sitting in a boring lecture and discovered this by playing around on a piece of paper. The original spiral (which is square) is below.



Diploid : that's awesome!

Here is a nice contour plot of the Riemann Zeta Function. You can see the simple pole at Formula: 0, the "trivial" zeroes at Formula: 1 and the "nontrivial" zeroes on the vertical line Formula: 2



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Invisiblecortex
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Re: Goldbach's Conjecture [Re: isaacein]
    #11288381 - 10/21/09 12:08 AM (7 years, 11 months ago)

I wish I could come up with shit like that when I was in a boring lecture (god knows I have enough of them...)

BTW: nice thread title change!


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InvisibleMinstrel
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Re: Goldbach's Conjecture [Re: isaacein]
    #11291151 - 10/21/09 12:46 PM (7 years, 11 months ago)

Quote:

isaacein said:
Cortez : that is indeed a nice discovery, due to the mathematician and physicist Stanislav Ulam. He was once sitting in a boring lecture and discovered this by playing around on a piece of paper.




The same Stanislav Ulam who helped Teller design the hydrogen bomb?  :eek:


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Offlinedkamp18
The Real World


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Re: Goldbach's Conjecture [Re: Minstrel]
    #11291801 - 10/21/09 02:37 PM (7 years, 11 months ago)

pascals triangle



etc.

inherent symmetry and elegance


Edited by dkamp18 (10/21/09 02:40 PM)


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Invisiblecortex
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Re: Goldbach's Conjecture [Re: Minstrel]
    #11292987 - 10/21/09 05:38 PM (7 years, 11 months ago)

Quote:

Minstrel said:
The same Stanislav Ulam who helped Teller design the hydrogen bomb?  :eek:




I could be wrong, but I don't think there are many well known physicists named Stanislav Ulam!


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OfflineHighTek
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Re: Goldbach's Conjecture [Re: cortex]
    #11293464 - 10/21/09 06:48 PM (7 years, 11 months ago)

Latin Squares, Magic Squares and Magic Circles are a pretty awesome concept in the realm of mathematics.

Latin Squares are when you have a n x n matrix with no repeating numbers in the rows and columns. Sudoku are Latin Squares. But they have have properties that extend beyond sudoku.

Magic Squares and Cirlces are interesting but hard to explain. I saw a presentation on them this past weekend and need to look into it more.


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OfflineAnnomM
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Re: The Amazing Math Thread [Re: isaacein]
    #11298507 - 10/22/09 11:53 AM (7 years, 11 months ago)

Great thread!

Formula: 0

:drooling:


Edited by Annom (10/22/09 11:54 AM)


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Offlineisaacein
exp(ix) = cosx + isinx


Registered: 05/21/08
Posts: 1,141
Last seen: 6 years, 7 months
Re: The Amazing Math Thread [Re: isaacein]
    #11306344 - 10/23/09 03:49 PM (7 years, 11 months ago)

Ford Cicles : above every rational number p/q, a circle of radius 1/(2q^2) is placed tangent to the real line. The result is an infinite number of circles which do not intersect but each of which is tangent to infinitely many others.





The total area of the Ford circles between 0 and 1 is



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Offlineisaacein
exp(ix) = cosx + isinx


Registered: 05/21/08
Posts: 1,141
Last seen: 6 years, 7 months
Re: The Amazing Math Thread [Re: Annom]
    #11306495 - 10/23/09 04:11 PM (7 years, 11 months ago)

Quote:

Annom said:
Great thread!

Formula: 0

:drooling:




Indeed it is hard to avoid mentioning Euler's formula in a thread about beautiful mathematics. But here is a nice generalization.

A Lie group, essentially speaking, is a group in the abstract sense which is also a "smooth" topological space (it's more complicated than that, but you get the idea). The real line or the unit circle in the complex plane are basic examples of commutative Lie groups. The first examples of non-commutative Lie groups are the matrix groups (or, equivalently the automorphism groups of topological vector spaces); the group of n x n invertible matrices over the reals for example (the "General Linear Group") and its subgroups (examples of which are : the subgroup of matrices having determinant 1, called the "Special Linear Group"; or the subgroup of rotation (orthogonal) matrices, called the "Special Orthogonal Group").

If M is a Lie group, then the tangent space to the identity element of M is called the Lie Algebra of M, denoted m. It is typically a vector space over the reals or the complex numbers.

If the sphere below were a Lie group, and the point its identity element, then the plane would be the Lie Algebra. (Note incidentally that the sphere cannot be made into a Lie group.)


For instance the Lie Algebra of the unit circle is the set of purely imaginary numbers - the identity element is 1, and the tangent space is a vertical line. What is nice is that Euler's formula generalizes to all Lie groups. Take an element of the Lie algebra, take its exponential and it will map you back into the Lie group. (Example : for any imaginary number y, e^y lies on the unit circle. Euler's formula is the specific case where the Lie group is taken to be the circle group.)

Here are more examples :

The Lie algebra of the General Linear Group is the set of all n x n matrices, and indeed if A is any n x n matrix then e^A is an invertible matrix.

The Lie algebra of the Special Linear Group is the set of all n x n matrices having zero trace, and indeed if A is any n x n matrix having zero trace then e^A is an invertible matrix having determinant 1.

The Lie algebra of the Special Orthogonal Group is the set of all n x n skew-symmetric matrices (matrices such that A^T = -A, where A^T is the transpose), and indeed if A is any n x n skew-symmetric matrix then e^A is a rotation matrix.


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