Math's posts

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**The Conic Sections Inside Quadric Surfaces**

A certain class of surfaces in space, known as quadric surfaces , can be defined with the general polynomial of two variables: \(Ax^2+By^2+Cxy+Dx+Ey+F=0.\) Such an equation can look somewhat intimidating, Our interest isn't in understanding the equation, ...

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**The Conic Sections**

It is widely known that the "conic sections" are the curves of intersection of a plane with a double-napped cone (i.e., two cones placed point-to-point), as illustrated above. The three most important conic sections are the ellipse, the parabola and the hyp...

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**The Traveling Salesman**

One of the most famous problems of math and computer science is the Traveling Salesman Problem . Given a list of \(n\) cities, what is the shortest tour that visits each city exactly once and returns back to the starting city? One of the reasons why this pr...

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**The Mandelbrot Set (Chaos Part II)**

In a previous post , we discussed chaos from a mathematical perspective. In everyday life we use the term "chaos" when events are unfolding in an unpredictable manner. In mathematics, functions and algorithms can exhibit chaotic behavior when it is hard to ...

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**The Mandelbrot Set (Chaos Part II)**

In a previous post , we discussed chaos from a mathematical perspective. In everyday life we use the term "chaos" when events are unfolding in an unpredictable manner. In mathematics, functions and algorithms can exhibit chaotic behavior when it is hard to ...

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**Chaos**

Most things that we encounter in our lives follow some sort of predictable pattern. Microwaving something for 26 seconds instead of 25 makes it just a little bit hotter, though we might not be able to really tell the difference. Depress the accelerator of y...

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**Sonic Booms**

Why does a sonic boom ... boom ? That is, why is it so loud? Above a plane flies at twice the speed of sound (it's Wonder Woman's plane, which is why you can't see it, just its sound waves), and we can see it sound waves dissipate over time/distance. (The s...

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**Mathematical Envelopes**

Start with a circle and let two points go around it, one twice as fast as the other. At each step of the way, draw a line to connect the points. You'll get a picture similar to the one above. This set of lines somehow clearly draws a cardioid. This is an ex...

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**Translations Through Rotations**

In a previous post , we showed a few images in which a shape seems to rotate, though the shape is made up of individual points which actually only translate , that is, move along a straight line. Above, the image gives the impression that each point is movi...

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**Buffon's Needle (and his Noodles)**

It seems strange, but it is true: you can approximate \(\pi\) by throwing needles on the floor. Back in 1733 a French guy named Georges-Louis Leclerc, Comte de Buffon (i.e., "Count Buffon") posed a simple question: If you throw a needle down on a wood plan...

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