Original Math Notes
All Seasons
Season 5
Episode 508: Charlie Don't Surf
Refraction of light rays by a sphere of transparent material, with position- and time-dependent index of refraction
Refraction of light rays by a sphere of transparent material, with position- and time-dependent index of refraction
Scene 17:
Charlie plucks a magnifying glass from a table, catches the
sunlight coming in from the windows.

          This magnifying glass is
          transparent, right?  But when light
          passes through it --

The glass causes a BRIGHT SPOT SURROUNDED BY SHADOW to appear
on a piece of paper on the desk.

                    CHARLIE (cont'd)
          -- the focal point appears to be
          encircled by shadow.
          The paper begins to smoke.

                    CHARLIE (cont'd)
          The curvature of the glass bends
          and focuses the light waves away
          from the dark areas into that one
          blinding point.

Lensmaker's Equation »

Lensmaker's Equation
The lensmaker's equation relates the focal length of a simple lens with the spherical curvature of its two faces, where R1 and R2 represent the radii of curvature of the lens surfaces closest to the light source (on the left) and the object (on the right). The sign of Ri is determined by the location of the center of curvature along the optic axis, with the origin at the center of the lens. Thus for a doubly convex lens, R1 is positive and R2 is negative. The focal length f is positive for a converging lens but negative for a diverging lens, giving a virtual focus, which is indicated by a cone of gray rays.
Scene 17:
          Underwater topography.

          Precisely.  Now here -- yellow --
          medium-size, up to two meters.

          Red is big, up to four meters.


          Purple means the waves were over
          ten meters.

Mathematics of Tsunamis »

Mathematics of Tsunamis
Computer modeling and simulations lead to a better understanding of natural disasters, such as the Indian Ocean Tsunami of 2004, and may prevent loss of life in the future. Using the system of partial differential equations known as the shallow water wave equations, this Demonstration provides a reasonable approximation of the behavior of real ocean waves during a tsunami.
Scene 21:
          So in that sense, ocean waves are
          fundamentally the same as light and
          sound waves -- yes?

          Fundamentally, yes.

          Sound waves travel at seven hundred
          and sixty-one miles per hour --

          At sea level.

          And light waves in the vacuum of
          space travel at --

          -- one hundred and eighty-six
          thousand miles per second.

Measuring the Speed of Light with Marshmallows »

Measuring the Speed of Light with Marshmallows
Place a plate with a single layer of marshmallows in a turntable-free microwave and zap it until some of them just begin to melt. Measure the distance in centimeters between melt spots; it should be roughly 6 cm. Next, find a label with the frequency of the microwave, usually 2450 MHz. With these two pieces of data, the speed of light can be calculated, via velocity = frequency × wavelength. Chocolate chips or another substance that melts nicely may be substituted.
Scene 30:
Without a word and in perfect synch, David and Colby play
rock-scissors-paper.  David's scissors cut Colby's paper.

          I always lose when we do that.  I
          think you cheat.
          How do you cheat at roshambo?


          I wouldn't call it cheating, but
          the chaos school advocates a purely
          random distribution of rock,
          scissors or paper.  However,
          probability analysis has shown that
          a mixed strategy of preconceived
          gambits --
            (catches himself)
          You know what?  I'm going to keep my
          roshambo strategies to myself -- in
          case I have to throw down with one
          of you someday.  

Rock-Scissors-Paper »

Rock breaks scissors, scissors cut paper, and paper wraps rock. Play against the random moves of a robot.
Scene 35:
                    CHARLIE (cont'd)
          I applied a deconvolution operator to
          filter out spectral interference --
          from other plants known to grow on the
          Channel Islands -- and then I created a
          neural network algorithm that could
          learn to recognize known spectral
          patterns based on training sets.
            (beat, proud of himself)
          That was the cool part.

Image Restoration for Degraded Images »

Image Restoration for Degraded Images
This figure shows restoration of original images from blurred images by applying various deconvolution techniques. Photographs of people's faces on television hidden by little squares can be thought of as examples for degraded images. Vary the controls to the optimum to see an interactive restoration of the pristine image from the degraded image.
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