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What's this?  +Xah Lee made this picture, and I explain it on my blog, Visual Insight:

After you read it you can tackle this:

Puzzle: what's the catacaustic of an astroid?

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I'm sorry. Can you please translate that? I know those are English words, but they do not compute. LOL

This reminds me of the time that I picked up a textbook of a housemate who was in grad school for math. Talk about a foreign language!
They're explained in the blog article.  Briefly:

A deltoid is the curve traced out by a circle rolling inside a circle 3 times as big. 

An astroid is the curve traced out by a circle rolling inside a circle 4 times as big.

If you shine line at a curve and let it reflect, the rays bunch up along a curve called a catacaustic of the original curve.

The picture here shows that the catacaustic of a deltoid is an astroid!  So you should wonder: what's the catacaustic of an astroid?
Hi. There was so much material that came up when you covered your visual insight topics here on g+. Why didn't the links make it to the ams blog? Btw, your paper really is a fun one, what a quality starndard for a visual insight...
Xah Lee
+John Baez thank you! i have a question. Did you ever cover the proof or can you indicate a proof? Thanks.

it was difficult back then for me (almost 20 years ago) when i was learning calculus. I recall there are general theories about envelopes using differential equation. I think i'd be able to understand now.
I think I see it whenever I dig out and watch a VHS
Your link for deltoid took me to the generic listing (includes deltoid muscle) which took me to deltoid curve. Other than that, got the general idea (though my internal brain RAM isn't big enough to hold the concepts simultaneously). Demanding know-nothing critic says "could you make an animated gif (or whatever? html5?) that demos each term separately (text for term, image for term), sequentially then the combination?
+Refurio Anachro - I'm trying to keep the AMS blog articles pretty short, so I can keep cranking them out without getting bored.   But if you have a link you want me to add to an article, just let me know.

In this blog article, the link to 'Rolling circles and balls' goes to lots of material on catacaustics, deltoids, astroids, etc.   The Wikipedia links give you a lot more material, too.
+James Lamb - if you want something more leisurely, try this:

My Visual Insight blog is hosted by the American Mathematical Society, so the main audience is professional mathematicians; I'm using that excuse to talk about a lot of intense math at a fairly rapid clip.  My posts here are (usually) aimed at something more like ordinary folks.
I posed this puzzle: what's the catacaustic of an astroid?

Greg Egan put a picture of the answer here:

and he wrote:

For a source at (-infinity,0) and an astroid with cusps on the coordinate axes, up to a choice of scale, a curve with parametric form:

(cos(3t) + 3 (4 cos(t)+cos(5t)), 2 sin(2t) (cos(t)-9 cos(3t)) - 4 cos(2t) (sin(t)-3 sin(3t))))

This is a six-cusped curve, but it not a hypocycloid.

In other words: unlike the deltoid and astroid, it's not formed by rolling a circle in a circle!
Xah Lee
+John Baez excellent, thanks for the link to Greg Egan's article.
+John Baez I googled the answer since I was to lazy to find my ruler. Wolfram had a different perspective on it but it seems they construct the catacaustic differently.

(Edited: Reason, didn't see at first glance Wolfram uses a generalized definition for the catacaustic so my original post didn't make much sense.)
Thanks for your answer +John Baez! No, i meant linking to your g+ posts. Nevermind. Your links do lead to interesting material and that's cool, too. (no irony, that paper really is a nice supplement) I also do like the crisp format. Just wondering...

Not sure of you asked this but i'll tell anyways. I'd very much like to contribute! I know, i'm invited to send an email as everyone else. But it'd be more exciting for me if i could go for something you might consider special fun. You've been a great inspiration for me and i'd be thrilled to give back.

So if you see something i did, please feel invited to use any stuff you find interesting! I won't start mailing you half my posts, that'd be wasting all sides. But in case i get the message, i might even invest and see if i can do a custom picture myself. Now you know.
+Refurio Anachro - I'll check out your posts, thanks!  I'm looking for images that are very beautiful and illustrate serious mathematical ideas.
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