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.

The GIFs above show the 8-term Fourier series approximations of the square wave and the sawtooth wave.

Mathematica code:

f[t_] := SawtoothWave[t]
T = 1; 
nmax = 18; 
a0 = (2/T)*Integrate[f[t], {t, -(T/2), T/2}]
anlist = Table[(2/T)*Integrate[f[t]*Cos[(2*Pi*n*t)/T], 
     {t, -(T/2), T/2}], {n, 1, nmax}]
bnlist = Table[(2/T)*Integrate[f[t]*Sin[(2*Pi*n*t)/T], 
     {t, -(T/2), T/2}], {n, 1, nmax}]
fs[t_, nmax_] := a0/2 + Sum[anlist[[n]]*Cos[(2*Pi*n*t)/T] + 
     bnlist[[n]]*Sin[(2*Pi*n*t)/T], {n, 1, nmax}]
Manipulate[Column[{Plot[{f[t], fs[t, nmax0]}, {t, -1, 1}, 
     PlotRange -> All, AxesLabel -> {"t", "f(t)"}, 
     PlotStyle -> {{Thick, Black}, {Thick, Red}}, 
     ImageSize -> 700, AspectRatio -> 1/2.8], 
     Row[{"f(t)=", fs[t, nmax0]}]}], {nmax0, 1, nmax, 1}]

(via visualizingmath)


The Chance To Dance Again

by Michael Keller

We highlighted the TED talk of Hugh Herr a couple of weeks ago. But his work is too important and beautiful to leave to just one post.

The MIT associate professor of media arts and sciences is making prosthetic limbs and exoskeletons that restore function in those who have lost legs from injury or disease. This set of gifs focuses on his team’s BiOM powered ankle and foot prosthesis

"Bionics is not only about making people stronger and faster," he said during the talk. "Our expression, our humanity can be embedded into electromechanics."

To prove his point, Herr and fellow researchers studied dance movement to replace the lower leg that professional dancer Adrianne Haslet-Davis lost after last year’s Boston marathon bombing. He concluded his talk by bringing Haslet-Davis on the stage to perform a bionic rumba. 

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World’s Fair 50:

The Automatic Language Translator

Another World’s Fair crowd pleaser was the IBM Automatic Language Translator. In a live demonstration, the computer translated Russian text into English in a matter of seconds.

The most amazing part was that the translation wasn’t created from a computerized ‘dictionary search’ but from the analysis of both languages’ complex nuances and shades of meaning, syntax and grammar. To think that 50 years later, we have smart phones with translation apps for just about every language spoken.  Очень здорово. Translation: Very cool.  


(via fastcompany)