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 moving right to left (a translation), while in fact, each point is actually moving in a circle (rotating).

The most basic form of showing "translation through rotation" is just to line up a bunch of circles (where "a bunch of" could mean "an infinite number of") but only show one point on each circle. There are lots of ways to pick this one point; a basic one is where a circle B, one unit to the right of circle A, shows a point 10\(^\circ\) farther around the circle that circle A. (This idea was given to us by NichG in a comment on the previously mentioned post, who also generated this image.)

Here is our take on this concept:

It may be hard to believe that each point is just moving in a circle, so we provide this image as "proof":

Each of the circles shown above has a center that lies on a horizontal line. We can put these circle centers on other curves to get different results. Below, we put the centers on a sine wave.

This is the basic structure of the image at the top of this post; we just varied the color a little according to its height to give the impression of a white-capped ocean wave.

Other Shapes

What if each point does not follow a circle, but some other path? Below, each point follows a square of unit length.

Again, since it can be hard to believe, we offer this image as proof:

We can also vary the way that we traverse this square. The red dot above travels at a constant speed around the square, covering each side in 1 time unit. In the following image, we traverse the square again at a rate of 1 side length per time unit, but slow down at the corners and speed up in the middle.

Notice how the sharp corners are smoothed out. To show the difference of the "traversing of the square," consider the following, where the two dots follow the same path at two different rates.

They arrive at each corner at different times, but the red dot slows down at the corners and speeds up in the middle, while the white dot travels at a constant speed. 

We also considered traveling along a triangle.

We have worked our way down from circles (which have "infinite sides"), to squares (with 4 sides) to triangles (3 sides, but you didn't need us to tell you that). To get even lazier with our motion, lets just move dots up and down.

The red dot is moving up/down at a constant speed; varying this so it goes up/down along a sine wave gives ...

... a sine wave. 

We tried one more, moving a dot along a 5 pointed star.

We're not sure there is much point to this one, but we thought it was fun to try.

We would be interested in seeing any images like these that you think look cool. Post it somewhere and send us a link.

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Our Code

We've gotten requests to show our code in the past. We haven't done it so far mostly out of shyness/embarrassment about the quality of our coding, but we'll show some of it here. We write in Mathematica.

All of the images are created using the same basic workflow. Use a Manipulate environment to test the image; the change the "Manipulate" to "Table" to create a set of frames, then Export these frames to a gif.

Here is the code to do the basic circles at the top. Change f[x_]:=0 to f[x_]:=Sin[x] to get the circles with centers along the sine wave.

f[x_] := 0

Manipulate[

 Show[ListPlot[

   Table[{Cos[i - a], Sin[i - a]} + {i, f[i]}, {i, -5, 5, .5}], 

   PlotRange -> {{-5, 5}, {-1.5, 1.5}}, 

   PlotStyle -> {White, PointSize[.02]}, Axes -> None, 

   AspectRatio -> Automatic, Background -> Black], 

  ParametricPlot[{{Cos[t], Sin[t]}, {Cos[t - 3] - 3, 

     Sin[t - 3]}, {Cos[t + 3] + 3, Sin[t + 3]}}, {t, 0, 2 Pi}, 

   PlotStyle -> {{White}}], 

  ParametricPlot[{{Cos[t], Sin[t]}, {Cos[t - 3] - 3, 

     Sin[t - 3]}, {Cos[t + 3] + 3, Sin[t + 3]}}, {t, 0, 2 Pi}, 

   PlotStyle -> {{Thickness[.005], White, Opacity[.6]}}], 

  ListPlot[{{Cos[-a], Sin[-a]}, {Cos[-3 - a] - 3, 

     Sin[-a - 3]}, {Cos[-a + 3] + 3, Sin[-a + 3]}}, 

   PlotStyle -> {{PointSize[.025], Red}}], Axes -> None], {a, 0, 2 Pi,2 Pi/30}]

basicRotationframes = 

  Table[Show[

    ListPlot[

     Table[{Cos[i - a], Sin[i - a]} + {i, f[i]}, {i, -5, 5, .5}], 

     PlotRange -> {{-5, 5}, {-1.5, 1.5}}, 

     PlotStyle -> {White, PointSize[.02]}, Axes -> None, 

     AspectRatio -> Automatic, Background -> Black], 

    ParametricPlot[{{Cos[t], Sin[t]}, {Cos[t - 3] - 3, 

       Sin[t - 3]}, {Cos[t + 3] + 3, Sin[t + 3]}}, {t, 0, 2 Pi}, 

     PlotStyle -> {{White}}], 

    ParametricPlot[{{Cos[t], Sin[t]}, {Cos[t - 3] - 3, 

       Sin[t - 3]}, {Cos[t + 3] + 3, Sin[t + 3]}}, {t, 0, 2 Pi}, 

     PlotStyle -> {{Thickness[.005], White, Opacity[.6]}}], 

    ListPlot[{{Cos[-a], Sin[-a]}, {Cos[-3 - a] - 3, 

       Sin[-a - 3]}, {Cos[-a + 3] + 3, Sin[-a + 3]}}, 

     PlotStyle -> {{PointSize[.025], Red}}], Axes -> None], {a, 0, 2 Pi - 2 Pi/30, 2 Pi/30}];

Export["08_basic_rotation_with_circle.gif", basicRotationframes]

We used the following to create waves from squares. In the Table command, change square to square2 to follow the square at the other speed.

square[x_] := 

 With[{xx = Mod[x, 4]}, 

  Piecewise[{{{-.5 + xx, .5}, 0 <= xx < 1}, {{.5, .5 - (xx - 1)}, 

     1 <= xx < 2}, {{.5 - (xx - 2), -.5}, 

     2 <= xx < 3}, {{-.5, -.5 + (xx - 3)}, 3 <= xx <= 4}}]]

pacing[x_] := (-Cos[x*Pi] + 1)/2

square2[x_] := 

 With[{xx = Mod[x, 4]}, 

  Piecewise[{{{-.5 + pacing[xx], .5}, 

     0 <= xx < 1}, {{.5, .5 - pacing[(xx - 1)]}, 

     1 <= xx < 2}, {{.5 - pacing[(xx - 2)], -.5}, 

     2 <= xx < 3}, {{-.5, -.5 + pacing[(xx - 3)]}, 3 <= xx <= 4}}]]

squarewave1frames = 

  Table[Show[

    ListPlot[Table[square[-i + a] + {1 i, ff[i]}, {i, -8, 8, .2}], 

     PlotRange -> {{-2, 2}, {-1.1, 1.1}}, 

     PlotStyle -> {White, PointSize[.02]}, Axes -> None, 

     Background -> Black, AspectRatio -> Automatic],ParametricPlot[

    square[x],{x,0,5},PlotStyle\[Rule]White],ParametricPlot[square[

    x],{x,0,5},PlotStyle\[Rule]{White,Thickness[.005],Opacity[.6]}],

    ListPlot[{square[a]},PlotStyle\[Rule]{PointSize[.025],

    Red}]], {a, 0, 4 - .1, .1}];

Finally, this draws the star:

outerstar = Table[{Sin[t], Cos[t]}, {t, 0, 2 Pi, 2 Pi/5}];

innerstar = Table[.5 {Sin[t], Cos[t]}, {t, 2 Pi/10, 2 Pi, 2 Pi/5}];

starpoints = Riffle[outerstar, innerstar];

starlinedata = Partition[starpoints, 2, 1];

starfunction[xx_] := 

 With[{x = Mod[xx, 10], t = Floor[Mod[xx, 10]]}, 

  starlinedata[[t + 1, 

    1]] + (x - t)*(starlinedata[[t + 1, 2]] - 

      starlinedata[[t + 1, 1]])]

Replace square in the above table with starfunction to see its results. (You'll have to change the plot range, etc.)

We hope this helps. Have fun. Again, show us what you come up with.

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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 plank floor, what is the probability that the needle will cross a line between planks?

It turns out that the probability, if the needle is the same length as the width of the planks, is \(P=2/\pi\approx 0.63662\). That means that if you throw said needle 100 times on the floor, you'd expect about 64 of them to be across a line; throwing 50 needles, like in the animation above, should give about 32 crossings. 

This gives a way of approximating \(\pi\): since \(\pi=2/P\), throw a needle \(n\) times on the floor and count the number of crossings \(c\). This means that \(P\approx c/n\), and so \[\pi = 2/P \approx 2/(c/n)= 2n/c.\]

In the above gif, we find \(\pi \approx 2\cdot 50/32 =3.125.\) That is the best approximation we can arrive at with only 50 tosses. Makes you wonder if we rigged the animation. (We did, sort of. The needle tosses were done at random with Mathematica; we lucked out that we got 32 crossings when we were ready to make a .gif and chose to not change it.) Speaking of "rigging it," there is an interesting story in this, starting with the following generalization:

If the needle has length \(\ell\) and the width of the planks is \(d\), then \(P = 2\ell/(\pi d)\) (where \(\ell\leq d\)). Turning this into an approximation for \(\pi\) with \(n\) tosses and \(c\) crosses, we have \[\pi \approx 2\ell n/(c d).\]

Back in 1901, a mathematician named Mario Lazzarini published the results of his tossing a needle of length 2.5cm onto lines spaced 3cm apart. His claimed results are amazing: in his tosses of 2000, 3000 and 4000 needles, his reported number of crossings are within 1 of the optimal number. Also, on toss 3408, he reportedly had 1808 crossings, which gives the approximation \(\pi \approx 3.1415929\), which is accurate up to the last "9." Most people find it hard to believe that 1) he actually tossed a needle 4000 times, 2) he did it in any way that was truly random (after all, people can hardly do anything randomly), and 3) his results were so close to "perfect." But his paper did get published, and maybe that earned him tenure at some institution of higher education.

Noodles

In 1969, J. F. Ramalay published in the American Mathematical Monthly an article entitled "Buffon's Noodle Problem," in which he generalizes the needle problem further. Instead of tossing needles of length \(\ell\), he solved the problem of tossing "noodles" of length \(\ell\) - that is, arbitrary curves with arc length \(\ell\). Because arbitrary curves are ... curvy, one noodle can cross a line more than once. He found that the expected number of total crossings \(E\) is \[E = 2\ell/(\pi d),\] effectively the same formula found before (and here \(\ell\) can be greater than \(d\)). The difference between "probability" \(P\) and "expected number" \(E\) is subtle yet significant, but here we treat them as the same. If \(P=.3\) or \(E=.3\), then in each case you'd expect about 30 crossings after 100 throws. In the traditional case of needles, that means 30 needles cross a line; with noodles, that means there are 30 crossings, likely from less than 30 noodles.

We demonstrate this with the following 2 gifs. In the first, we toss circles (donuts?) onto a floor where the circumference (arc length) of each circle is 1, the distance between lines. 

Note how each circle, when it crosses a line, gives two crossings. (And we didn't rig this one to give ourselves the optimal 32 crossings.)

Below, we throw arbitrary curves onto our board. Each curve is color-coded by the number of crossings it gives: there is one curve that gives 3 and a couple that give 2.

Thanks to Stan Wagon of Macalester College; he coded a Wolfram Demonstration on the topic and we cribbed some of his code to make our curves and count the crossings. 

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