Comments on: Let’s Do Some Math: Scrambler

By: D MacKinnon

D MacKinnon — Fri, 08 Feb 2013 18:34:40 +0000

Nice post and a neat topic.

Speaking of the counterintuitive shapes that can be obtained, I was inspired to look at scrambler paths a while back:

http://www.mathrecreation.com/2009/08/hypocycloid-scrambler.html

http://www.mathrecreation.com/2009/09/scrambler-fractal.html

Sorry for the unrendered latex on those old pages – I’ll have to fix that at some point.

By: Martin VÃ©rot

Martin VÃ©rot — Thu, 07 Feb 2013 12:51:49 +0000

About the problem of the overall shape obtained : This problem is quite interesting and reminds me of epicycles and the history of astronomy.

Mathematicians used a combination of two circles with their own rotation angle to obtain various shapes.
You can even obtain rectangles with this kind of device.
http://www.clas.ufl.edu/users/ufhatch/HIS-SCI-STUDY-GUIDE/0034_summaryPtolemiacAstron.html

It allowed them for example to explain the retrograde movement of Mars.
http://www.lasalle.edu/~smithsc/Astronomy/retrograd.html

The big idea of Kepler was to get rid of the epicycle and introduce ellipses to simplify the description of the movement with a unique and minimal description.

To conclude : composed movements can be really counterintuitive especially with quite a lot of parameters to play with (relative angle velocity and relative ratio of the two circles).

By: Alec Wilson

Alec Wilson — Mon, 04 Feb 2013 08:08:37 +0000

Similar to Dan Pearcy, I’ve made an applet in Geogebra but to show the locus of the Scrambler seat:

http://www.geogebratube.org/student/m29179

Using 4 sliders and the following in the input line.

Curve[cos(2*pi*n/V) + r*cos(2*pi*n/v),sin(2*pi*n/V)+r*sin(2*pi*t/v),n,0,t]

Vectors & ensuing parametric equations made this quite…

By: Matt Vaudrey

Matt Vaudrey — Sun, 03 Feb 2013 03:56:56 +0000

Has anybody mentioned the path as a polar function yet? It brings to mind the Spirographs from when we were young.

By: Geogebra Applet for Dan Meyer’s Scrambler Problem | Teaching Mathematics

Fri, 01 Feb 2013 17:26:55 +0000

[…] applet aids investigation of Dan Meyer’s Scrambler Problem. Click on the picture to go to the […]

By: Dan Pearcy

Dan Pearcy — Fri, 01 Feb 2013 17:11:47 +0000

I’ve also just whipped up an applet using Geogebra.

http://www.geogebratube.org/student/m28985

By: Dan Meyer

Dan Meyer — Fri, 01 Feb 2013 08:59:45 +0000

Lots of interesting solutions here. I showed it to a few teachers here in Singapore and I was very surprised at how many people (like Rachel) predicted the locus was an ellipse. My intuition fails me there. So I added the answer video to the main page, along with a nice Desmos calculator that lets you play around with different parameters.

By: l hodge

l hodge — Thu, 31 Jan 2013 20:51:19 +0000

I really like this problem – lots of possible approaches and extensions.

What happens if the 4 rides on the arm rotate much faster than the main body rotates? What if the main body rotates much faster than the arm rotates? The ratio of the rotation rates provided seems to be quite unique as far as the path (locus) of the red seat goes.

If you play around with the ratio of the rates you get some very interesting paths:

https://www.desmos.com/calculator/ifdmbriuqj

I am not 100% sure that these equations model the path of the red seat.

By: Eileen

Eileen — Thu, 31 Jan 2013 18:02:23 +0000

Do we know how long the ride is?

By: James McKee

James McKee — Thu, 31 Jan 2013 16:17:53 +0000

Now I see it. I can definitely see in myself the tendency to pull things out of a moving frame of reference and see them against the larger (stationary) frame of reference. Definitely and interesting aspect to the problem! When I created the Geogebra simulator, both animations were relative to the larger, stationary background, so it seemed to me that both rotations had to have the same period (6 seconds), so that’s how I viewed the problem. Thanks for the clarification!