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Mathematical Curiosities

Kaprekar numbers 

Consider the number 7,777² = 60,481,729. If we separate this number into two parts, each four digits long, and then add them, we get 6,048 + 1,729 = 7,777, which is the same as the base we began with. 

The numbers need not be repetitive digits to be interesting. Consider the number 297:  297² = 88,209, and 88 + 209 = 297. 

Such a number is called a Kaprekar number, named after the Indian mathematician Dattaraya Ramchandra Kaprekar (1905-1986), who discovered such numbers. Some other Kaprekar numbers are 1; 9; 45; 55; 99; 703; 999; 2,223; 2,728; 4,879; 4,950; 5,050; 5,292; 7,272; 7,777; 9,999; 17,344; 22,222; 38,962; 77,778; 99,999; … ; 538,461; 857,143; …

We also have such things such Kaprekar triplets, which behave as follows: 

45³ = 91,125 with 9 + 11 + 25 = 45

Other Kaprekar triplets are 1; 8; 10; 297; 2,322.

While we are dealing with discoveries made by Kaprekar, we can admire the Kaprekar constant, 6,174. This constant arises when one takes any four-digit number and forms the largest and the smallest number from these digits, and then subtracts these two new numbers. Continuing this process will always eventually result in the number 6,174. When the number 6,174 is arrived at, we continue the process of creating the largest and the smallest number, and then taking their difference (7,614 - 1,467 = 6,174), we notice that we get back to 6,174. This is, therefore, called the Kaprekar constant. To demonstrate this with an example, we will consider the number 2,303:

  • The largest number formed with these digits is:            3,320.
  • The smallest number formed with these digits is:          0,233.
  • The difference is:       3,087.
  • The largest number formed with these digits is:            8,730.
  • The smallest number formed with these digits is:          0,378.
  • The difference is:       8,352.
  • The largest number formed with these digits is:            8,532.
  • The smallest number formed with these digits is:          2,358.
  • The difference is:       6,174.
  • The largest number formed with these digits is:            7,641.
  • The smallest number formed with these digits is:         1,467.
  • The difference is:       6,174.
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reblogged

Magnetic wormhole created inside Laboratory

physicists have crafted a wormhole that tunnels a magnetic field through space.
“This device can transmit the magnetic field from one point in space to another point, through a path that is magnetically invisible,” said study co-author Jordi Prat-Camps, a doctoral candidate in physics at the Autonomous University of Barcelona in Spain. “From a magnetic point of view, this device acts like a wormhole, as if the magnetic field was transferred through an extra special dimension.”
The idea of a wormhole comes from Albert Einstein’s theories. In 1935, Einstein and colleague Nathan Rosen realized that the general theory of relativity allowed for the existence of bridges that could link two different points in space-time. Theoretically these Einstein-Rosen bridges, or wormholes, could allow something to tunnel instantly between great distances (though the tunnels in this theory are extremely tiny, so ordinarily wouldn’t fit a space traveler). So far, no one has found evidence that space-time wormholes actually exist.
The new wormhole isn’t a space-time wormhole per se, but is instead a realization of a futuristic “invisibility cloak” first proposed in 2007 in the journal Physical Review Letters. This type of wormhole would hide electromagnetic waves from view from the outside. The trouble was, to make the method work for light required materials that are extremely impractical and difficult to work with, Prat said.
Magnetic wormhole But it turned out the materials to make a magnetic wormhole already exist and are much simpler to come by. In particular, superconductors, which can carry high levels of current, or charged particles, expel magnetic field lines from their interiors, essentially bending or distorting these lines. This essentially allows the magnetic field to do something different from its surrounding 3D environment, which is the first step in concealing the disturbance in a magnetic field.
So the team designed a three-layer object, consisting of two concentric spheres with an interior spiral-cylinder. The interior layer essentially transmitted a magnetic field from one end to the other, while the other two layers acted to conceal the field’s existence.
The inner cylinder was made of a ferromagnetic mu-metal. Ferromagnetic materials exhibit the strongest form of magnetism, while mu-metals are highly permeable and are often used for shielding electronic devices.
A thin shell made up of a high-temperature superconducting material called yttrium barium copper oxide lined the inner cylinder, bending the magnetic field that traveled through the interior
The final shell was made of another mu-metal, but composed of 150 pieces cut and placed to perfectly cancel out the bending of the magnetic field by the superconducting shell. The whole device was placed in a liquid-nitrogen bath (high-temperature superconductors require the low temperatures of liquid nitrogen to work).
Normally, magnetic field lines radiate out from a certain location and decay over time, but the presence of the magnetic field should be detectable from points all around it. However, the new magnetic wormhole funnels the magnetic field from one side of the cylinder to another so that it is “invisible” while in transit, seeming to pop out of nowhere on the exit side of the tube, the researchers report today (Aug. 20) in the journal Scientific Reports.
“From a magnetic point of view, you have the magnetic field from the magnet disappearing at one end of the wormhole and appearing again at the other end of the wormhole,” Prat told Live Science.
Broader applications There’s no way to know if similar magnetic wormholes lurk in space, but the technology could have applications on Earth, Prat said. For instance, magnetic resonance imaging (MRI) machines use a giant magnet and require people to be in a tightly enclosed central tube for diagnostic imaging.
But if a device could funnel a magnetic field from one spot to the other, it would be possible to take pictures of the body with the strong magnet placed far away, freeing people from the claustrophobic environment of an MRI machine, Prat said.
To do that, the researchers would need to modify the shape of their magnetic wormhole device. A sphere is the simplest shape to model, but a cylindrical outer shell would be the most useful, Prat said.
“If you want to apply this to medical techniques or medical equipment, for sure you will be interested in directing toward any given direction,” Prat said. “A spherical shape is not the most practical geometry.”
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The sines as numbers

sin(pi/6): 1/2
sin(pi/4): 1/sqrt(2)
sin(pi/3): sqrt(3)/2
sin(pi/2): 1
sin(pi): 0
sin(3pi/2): -1
sin(2pi): 0
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“One who cannot love her smallest creations, cannot claim to stand in front of Nature.”

The world lost a true master today.  Takashi Amano was an imaginative innovator and world-class artist.  He inspired enough people to start an artistic global movement and he will be sorely missed.

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once i had a dream that my cat was working at mcdonalds w/ me and she had a lil uniform and she kept getting fur in the fries and everyone was yelling at me and saying “ur cat sucks on fries” and i was like “shes just a cat give her a break!” and i woke up crying

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