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The proof of Sum of Square Numbers!


Sir Albert Einstein once said ”After a certain high level of technical skill is achieved, science and art tend to coalesce in aesthetics, plasticity, and form. The greatest scientists are artists as well.” And I think that string art is one of the artworks of mathematics. Also, the calculations of string art are really interesting.<p>Here is the menthod to find the intersection envelope for string art ideas: Find the intersiction envelope (on plus.maths.org). Also, we can extend the menthod in the …


Try proving the rules for 7 and 11. It’s fun!


A Fourier series is a way to expand a periodic function in terms of sines and cosines. The Fourier series is named after Joseph Fourier, who introduced the series as he solved for a mathematical way to describe how heat transfers in a metal plate.<p>The GIFs above show the 8-term Fourier series approximations of the square wave and the sawtooth wave.


<b>The Relationship of Numbers</b><p>Today I wanted to share something with you, it’s not complex, but I was reminded of it today in maths class and it struck me as beautiful and nice and I figured I’d remind you of it, too.<p>In the diagram above you can see there are six circles, each which a family of numbers assigned to it. What I find fascinating is the easy steps one can take to move from the most inner circle to the most outer.<p><b>Natural Numbers:</b><br>These were the first numbers we had and used as a species: …

Visual Proofs

¼ + 1/16 + 1/64 + 1/256 + … = 1/3<p>1/3 + 1/9 + 1/27 + 1/81 + … = ½<p>½ + ¼ + 1/8 + 1/16 + … = 1<p>1 + 2 + 3 + … + n = n * (n+1) / 2<p>1 + 3 + 5 + … + (2n − 1) = n2<p>a2 + b2 = c2<p><b>CITATION ( source) :</b><p>Nelsen, R. B. <i>Proofs Without Words: Exercises in Visual Thinking.</i> Washington, DC: Math. Assoc. Amer., 1997.


Mathematics and Architecture<p>The link between math and architecture goes back to ancient times, when the two disciplines were virtually indistinguishable. Pyramids and temples were some of the earliest examples of mathematical principles at work. Today, math continues to feature prominently in building design. We’re not just talking about mere measurements — though elements like that are integral to architecture. Thanks to modern technology, architects can explore a variety of exciting design …


<b>Integral</b> and the difference between: <b>The Riemann-Darboux approach & The Lebesgue approach</b>.<br>Riemann-Darboux’s integration (in blue) and Lebesgue integration (in red). To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain’s volume (above sea level).<i><br>The Riemann-Darboux approach:</i> Divide the base of the mountain into a grid of 1 meter squares. Measure the altitude of the mountain at the center of each square. The volume on a single …



Value of Pi, written in Sanskrit by Arya Bhatta, long before any European even thought of it.


A bit of #math today #circlearea<p>▼ Reshared Post From Richard Green ▼<p><b>Circumference and area of a circle</b><p>The <b>circumference</b> of a circle of radius r is given by the formula 2πr, where π is the famous constant whose value is approximately 3.14159265. This graphic by <b>Margaret Nelson</b> illustrates a simple way in which the formula for the circumference can be used to obtain the formula for the <b>area</b> of a circle, πr^2.<p>The idea is to cut the circle into a large, even number of equal sectors, as if you were …

Seeing Status in the Classroom

In my last post, I discussed the idea of social status and its consequences for classroom teaching and learning. I was introducing you to my way of …

Teaching Direct Variation Conceptually (as a foundation for slope)

OK, the following may rate as one of my best test questions ever. It might take you a second to see the point:<p>If you don’t get it yet, compare it to …


PYTHAGOREAN TRIPLE <br>►TOP◄ A Pythagorean triple consists of three positive integers <br>a, b, and c, such that <i>a</i>2 + <i>b</i>2 = <i>c</i>2.<p>Such a triple is commonly written (a, b, c).<br>• A well-known example is (3, 4, 5).<p>A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. (Pythagorean triple - Wikipedia)<p><i>►</i><b>BOTTOM</b><i>◄</i> A depiction (by Adam Cunningham and John Ringland) of all the primitive Pythagorean triples (a,b,c) with a and b < 1170 and a odd, where a is plotted on the horizontal axis, b …

56 great math websites for students of any age

Below you will find 56 of the best math resource websites available. Parents and teachers of children 3 to 23 who are looking for videos, games, …


Visual power rules.This and more from Prof. Wilson at his blog. Reminds me of Byrne’s beautiful Euclid. Via Don Steward’s Median blog.


Math emotions.<p>Spotted in the Stanford Math Department (photo cred: Jessica Su)<p>No attractive as in fixed points? =(


What is 1 - 2 + 3 - 4…? Before you read on, guess!<p>In mathematics, 1 - 2 + 3 - 4 +… is the infinite series whose terms are the successive positive integers, given alternating signs. The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, …), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler figured out the sum of this infinite series, though a rigorous explanation wouldn’t arrive until much, much later. Leonhard Euler …


What is a Billion?<p>We live in a time of billions; billions of a galaxies, stars, people, light-years…but how do we put something like this into perspective?<p>Can we really understand what a “billion” means, or its significance?


So,triangles seem to be the strongest shape - but why?<br>Simply because their mathematical characteristics; the lengths are fixed and if the joints stay connected,there is no way the angles can change without changing the length of one side. Therefor, the weight has to be distributed between all sites of the triangle.<br>This knowledge is used in industrial design,for example. I a square is the shape of use,the diagonals have to be enforced by bracing, so the square becomes a combination of two …



A rope is supported at its ends. What shape do you think it assumes? Galileo though parabola, in red. Wrong! Let’s cut Galileo some slack though. The right answer, so called catenary, is very close.

Here's an interesting infographic about Pi and Pie! http://t.co/0h1OGi30GD

which is the most perfect geometrical figure?

I’m not sure, Anonymous! I’m just going to say it’s the circle.<p>Anywho, while trying to form an opinion on the answer to your question, I looked up “circle” on Wikipedia and found this picture:<p>“Area enclosed by a circle = π × area of the shaded square.” I know it’s not very impressive, but I’ve just never thought of it like that! Thanks, Anon.