Comments on: Rolling Circles and Balls (Part 1)

By: Alexander Bogomolny

Alexander Bogomolny — Wed, 02 Jan 2013 16:07:16 +0000

I can add to the list of books Vasilyev and Gutenmacher “Lines and Curves: A Practical Geometry Handbook”, Birkhauser; 1 edition (July 23, 2004) and C. Zwikker’s “The Advanced Geometry of Plane Curves and Their Applications”, Dover, 2005. At my site http://www.cut-the-knot.org there are several interactive Java gadgets that explore exactly those properties of cardioids that you mentioned.

By: amarashiki

amarashiki — Wed, 02 Jan 2013 01:42:07 +0000

Order and chaos are two sides of the same coin. Indeed, the words “cosmos” and “chaos” share the same greek origin and mean curiously such a different “notions”, likely complementary “a la Böhr”, since order is regularity and chaos is mainly associated to nonlinear and “wild” phenomena. In fact, you should know I have always preferred my own variation of those words “Chaosmos” (Chaotic Cosmos, as portmanteau) to call the main features of this Universe (or Multiverse/Polyverse if you believe there are other bubble Universes beyond the Lemaitre limit, aka the observable radius of our known Universe).

By: Rolling Circles and Balls (Part 4) « Azimuth

Rolling Circles and Balls (Part 4) « Azimuth — Wed, 02 Jan 2013 00:38:08 +0000

In Part 1 we rolled a circle on a circle that’s the same size [...]

By: Bruys

Bruys — Wed, 10 Oct 2012 07:17:13 +0000

One way that helps me to picture it is to first imagine the moving circle moving around the stationary one but with the same point of the moving circle touching the stationary one. In this case, it is apparent that the moving circle will have rotated only once.

Second, picture the moving circle rotating while its centre remains fixed.

Lastly, the actual example is a combination of these two motions, as the moving circle rolls without slipping around the stationary circle. The combination of the two types of rotation produces two rotations in all.

By: Colin

Colin — Thu, 13 Sep 2012 08:22:08 +0000

It’s a strange sort of optical illusion (perhaps well known by magicians?). I too found it impossible to mentally see from the animation. Even with mathematica demonstrations “frame by frame” I could not see it. I was about to use 2 table coasters to help me out when I remembered that in V. Hart’s video she used paper with lines. Only by looking closely at her video I got it.

Really readily understandable, fun and fascinating this entire series. From geometry to the center of the universe astronomy to engine design and also with some nice user contributed proofs and facts, its a “little fractal” of science – I think this would be quite a good talk to high-school students.

By: John Baez

John Baez — Wed, 12 Sep 2012 00:44:35 +0000

The point on the rolling circle that's initially at (1,0) on the rolling circle faces right twice as that circle rolls once around big one. If the rolling circle were the Earth and the center of the big circle were the Sun, you'd see the stars go around the sky twice each year. So, we can say the rolling circle rotates twice as it revolves once. (Of course, the Sun would be actually touching the Earth, but never mind!) You can probably get what I'm asking in Puzzle 1 and Puzzle 2 now. It's also good to figure out how many times you see the stars go around the sky during a single year.

By: Ricky

Ricky — Tue, 11 Sep 2012 19:24:14 +0000

“Can you see how the rolling circle rotates twice as it rolls around the fixed circle once? Do you understand why?”

OK, sorry for the basic question, but for the life of me I can’t see how the rolling circle rotates twice – the point on the rolling circle that’s at (1,0) or 0 deg ends up at the same spot after the cardioid has been completed, and it only reaches that spot once. ? This must be a dumb question but I would really appreciate some insight.

By: Rolling Circles and Balls (Part 3) « Azimuth

Rolling Circles and Balls (Part 3) « Azimuth — Tue, 11 Sep 2012 09:18:07 +0000

In Part 1 and Part 2 we looked at the delightful curves you get by rolling one circle on another. Now let’s see what happens when you roll one circle inside another!

By: Rolling Balls and Circles (Part 2) « Azimuth

Rolling Balls and Circles (Part 2) « Azimuth — Mon, 03 Sep 2012 03:40:53 +0000

Last time we rolled a circle on another circle the same size, and looked at the curve traced out by a point on the rolling circle. It’s called a cardioid. But suppose we roll a circle on another circle that’s twice as big. Then we get a nephroid! [...]

By: John Baez

John Baez — Sun, 02 Sep 2012 12:47:49 +0000

Thanks! I’m reading your blog. It’s a lot of fun!