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The magic circle, or Mobius strip, named after a German mathematician, is a loop with only one surface and one edge. Impossible? Well, not quite.

Definition on Wikipedia:

The Möbius strip or Möbius band (UK /ˈmɜrbiəs/ or US /ˈmoʊbiəs/; German: [ˈmøːbi̯ʊs]), also Mobius or Moebius, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as 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 two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. That is to say, it is a chiral object with "handedness" (right-handed or left-handed).

The Möbius band (equally known as the Möbius strip) is not a surface of only one geometry (i.e., of only one exact size and shape), such as the half-twisted paper strip depicted in the illustration to the right. Rather, mathematicians refer to the (closed) Möbius band as any surface that is homeomorphic to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any closed rectangle with length L and width W can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in 3-dimensional space, and others cannot (see section Fattest rectangular Möbius strip in 3-space below). Yet another example is the complete open Möbius band (see section Open Möbius band below). Topologically, this is slightly different from the more usual — closed — Möbius band, in that any open Möbius band has no boundary.

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface (it has zero Gaussian curvature). A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution.

http://en.wikipedia.org/wiki/M%C3%B6bius_strip

The Euler characteristic of the Möbius strip is zero.

And there is some really interesting maths behind, check it out here:

http://mathworld.wolfram.com/MoebiusStrip.html
http://www.cut-the-knot.org/do_you_know/moebius.shtml

#maths   #mathematics   #moebius   #geometry   #amazing  
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Cool.!!!
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Brandon Konrad

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You guys should check out this channel, he has a lot of useful videos covering various math topics.
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Brandon Konrad's profile photoDA BLACKMAN'S's profile photoNur Johari Bin Salleh's profile photoJason Mitchell's profile photo
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I'm using his videos to try and understand calculus topics before I go into Calculus class.

My curiosity about Integral and Derivative can't wait a year. 
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Assignment Expert

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Laveshsingh Ramjunum's profile photoAishvarya Sampath Raghavan's profile photoSavidya Ranmini's profile photoIsabella Södergren's profile photo
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Did you know the half of it
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Sasha Prieto

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some useful multiplication strategies
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Avi Taksha's profile photoNorman Simon Rodriguez's profile photoLyle Hardin's profile photoAli Habeeb's profile photo
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Some of the ways my kids have learned math seems so confusing. I'm with +George Ioannou . Rote memorization does the trick. Though, I admit, the above examples are good (except maybe the number line). That number line reminds me of some of the common core stuff. 
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Math

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Pascal's mechanical calculator
In 1642, French physicist, mathematician, and all-around smart guy Blaise Pascal devised a gear-driven mechanical calculator that proved a novel concept -machines could do math.

How the Pascaline Works

The mechanism is enclosed in a rectangular box which top panel can be divided into two main parts: the input dials and the accumulators.  The input dials are used to inscribe the number to be added or subtracted.  They are composed of spiked wheels, mounted on independent axels, each one corresponding to a given order of a scientific or monetary number.  Each wheel works like a rotary phone dial, with an outside numbered wheel and a metal stop which are fixed, mounted on the box itself, and a spiked wheel which is mobile.  To inscribe a digit, the operator must insert a stylus in between the two spikes that surround the digit written on the outer wheel and he must turn the wheel with the stylus all the way to the stop lever.  Each spike corresponds to one unit in the base of the wheel (decimal or monetary).  The wheel for “denier” has 12 spikes, the one for « sol » has 20 spikes and a decimal wheel has 10 spikes!  The wheel rotation powers the mechanism which in itself is extremely simple.  One can find in it some lantern gears, which transmit the data to the accumulators, pawls and ratchets which allow for the proper positioning of the various wheels, and a set of “sautoirs” which are dedicated to transmitting carries from one wheel to the next wheel of higher order.
It is believed that Pascal built twenty machines. Nine have survived. Most are in public collections, except for two of them.

Pascal's calculators 
Distinguishing originals from replicas
http://www.ami19.org/Pascaline/IndexPascaline-English.html
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Anderson Scott's profile photoJoan de Gracia's profile photoIsabella Södergren's profile photoPaulo Santos's profile photo
 
And Christian philosopher, dont forget. Science doesnt always represent the polar opposite of religion ;) Pascal is one of my tech heroes.
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Katherine Xavier

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Subtraction poem
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Nate Bitton's profile photoStudying From Home's profile photoNavya Gowda's profile photoRakesh Jain's profile photo
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Catchy poem.. cool! 
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Vijay Krishnan

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And this is why, the wonders of Mathematics never cease to amaze me :))
 
To see what underlies this… Also suggest people see our gold nugget and Riemann Hypothesis videos… 

Why -1/12 is a gold nugget
Million Dollar Math Problem - Numberphile
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Zach Cox

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This video should end with the following assignment:  "Now your homework is to show why multiplication of two NxN matrices is NOT commutative!"
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Million Young

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math is a language
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maybe, Million Young don't have to wirte an enciclopedia or an extensive and precise exposition of terms to make a simple post based on a basic philosophical discern on Google +..... "Math is a language" or maybe, if you believe on conmutativity (=)...you could speculate that "Language is a Math"... Or persist on believe on a Monadic existence of abstract entities...
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Math

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The arc-length parametrization of an Archimedean spiral visualized with colors.
It is really interesting how the colors are bent around. It seems that the distribution is quite non-uniform, even though the spiral is rather uniform in growth.

http://en.wikipedia.org/wiki/Differential_geometry_of_curves#Length_and_natural_parametrization

http://en.wikipedia.org/wiki/Archimedean_spiral

http://1ucasvb.tumblr.com/post/44972646489/the-arc-length-parametrization-of-an-archimedean
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Jacob Wiles's profile photoLeo Faria's profile photoMOOn Jello's profile photoMARIA DE LOURDES LEANDRO's profile photo
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[√(2 + √3)]^x + [√(2 - √3)]^x = 4
[√(2 - √3)]^x = t
[√(2 + √3)]^x = 1/t
1/t  + t = 4
1 + t² = 4t
t² - 4t + 1 = 0 <=>
∆ = b² - 4ac
∆ = (-4)² -  4 · 1 ·1
∆ = 16 -12
∆= 12
√∆ = 2√3
x₁ = (-b - √∆)/2a = (4 - 2√3)/2 = 2 - √3
x₂ = (-b + √∆)/2a = (4 + 2√3)/2 = 2 + √3
 
[√(2 - √3)]^x = t₁
[√(2 - √3)]^x = 2 - √3
(2 - √3)^(x/2) = (2 - √3)¹
x/2 = 1
x = 2
 
[√(2+ √3)]^x = 1/t₂
[√(2+ √3)]^x = 1/(2 + √3)
(2 + √3)^(x/2) = (2 + √3)⁻¹
x/2 = -1
x = - 2
 
[√(2 + √3)]^x + [√(2 - √3)]^x = 4  <=> x = -2 v x = 2


x = -2
L = [√(2 + √3)]^x + [√(2 - √3)]^x
L = [√(2 + √3)]^(-2) + [√(2 - √3)]^(-2)
L = {1/[√(2 + √3)]}^2 +{1/ [√(2 - √3)]}^2
L = 1/(2 + √3) + 1/(2 - √3)
L = (2 - √3)/[(2 + √3) (2 - √3)] +  (2 + √3)/[(2 + √3) (2 - √3)]
L = (2 - √3 +2 + √3 )/[(2 + √3) (2 - √3)]
L = 4/[(2 + √3) (2 - √3)]
L = 4/(2² - √3²)
L = 4/(4-3)
L = 4/1
L = 4,   R = 4, L = R

v

x = 2
L = [√(2 + √3)]^x + [√(2 - √3)]^x
L = [√(2 + √3)]^2 + [√(2 - √3)]^2
L = 2 + √3 + 2 - √3
L = 4,   R = 4, L = R
 
[√(2 + √3)]^x + [√(2 - √3)]^x = 4  <=> x = -2 v x = 2

Solutions: x = -2, v x=2
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Ameya Salankar

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You Mathematicians, you just don't know your limits......!
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Math

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A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.

http://xahlee.info/SpecialPlaneCurves_dir/Hypotrochoid_dir/hypotrochoid.html
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tarun purohit's profile photoLuis Restrepo's profile photoAsta Muratti's profile photoThomas Wilson's profile photo
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I'm enjoying these lovely animations - what tool made this one?
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espetacular   gracias
 ·  Translate
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Corina Marinescu

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Get your best paper, cut a circle and fold it, fold it so that the circumference falls on a fixed point inside. Repeat, using random folds. Now see the creases. This is how you paper-fold an ellipse. =)

Mathani creation
 http://mathani.tumblr.com/
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Kuvan G.C's profile photoIvan Teliatnikov's profile photoandy merrill's profile photoNirvana Zoraya's profile photo
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Thanks +Corina Marinescu! I will be sure to show my origami friends. 
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Ulises M. Alvarez

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"Discover what it takes to move from a loose theory or idea to a universally convincing proof."
Video: +TED-Ed 
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Michael Harrison

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This is such a fun problem.  Give someone a plain circle and ask them to find the center.  It's a nice reminder that classic geometry can still beat analytic geometry from time to time.
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Arthur Panos's profile photoBill Kohler's profile photoИрина Киреева's profile photoAnders Vorum's profile photo
 
i have one theory that one day every thing will be calculable without using analysis.
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Asif Raza

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A sip of comfort..
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I would suggest that a link back to the source would be a sufficient gesture to address any concerns regarding etiquette. Considering the social context, this appears to be fair use. No need for name calling.
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Socratica

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Matrices are a great example of infinite, non-abelian groups.  In our latest Abstract Algebra video we introduce both the General Linear and Special Linear Groups.
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Math

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Tessellations have been popular decorations for hundreds of years, as this tiled ceiling of the Sheikh Lotfollah Mosque in Iran (1602-1619) shows. Any shape or shapes that can be repeated to fill a 2D plane can be considered tessellations; so, equilateral triangles, squares, rectangles and hexagons are all simple shapes that can be tessellated.

http://tessellations.org, lots examples of tessellations and describes different methods for creating your own.

http://en.wikipedia.org/wiki/Sheikh_Lotfollah_Mosque

http://www.360cities.net/image/iran-isfahan-meidan-naqshe-jahan-square-shah-imam-east-masjed-sheikh-lotfallah-mosque#82.79,-4.41,75.6
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Valdis Kletnieks's profile photoДмитрио Трийдми's profile photoCosmin Dumitrescu's profile photoDmitry Shintyakov's profile photo
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+Valdis Kletnieks ...&? Dali painted melted clock... they are not alive, nor sentient... sure there are many clocks in same painting but they are not same shape - hence not really repetitive - yet very appreciated...
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