6d(d-1) or 3d(d-1)?

### John Baez

Shared publicly -Check out Greg Egan's new page on caustics! The blue curve here is an ellipse. When we shine light on it, the reflected rays bunch up along a curve called a

http://www.gregegan.net/SCIENCE/Catacaustics/Catacaustics.html

#geometry

**catacaustic**. The ellipse is described by a quadratic equation, but its catacaustic is described by a polynomial equation of degree 6. This is an example of a general pattern discussed by Egan: the catacaustic of a curve of degree d has degree at most 3d(d-1). The math is serious, but the pictures will delight you!http://www.gregegan.net/SCIENCE/Catacaustics/Catacaustics.html

#geometry

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12 comments

Craig Froehle

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6d6? That's the damage roll for my magic broadsword in D&D.

About the tough calculations in this article, Egan wrote: "In fact, I swear that Mathematica actually burnt out the hard drive on my computer by requiring so much virtual memory and grinding it so hard for a week when I tried to compute an explicit formula for the catacaustic of a generic quadratic with symbolic coefficients!"

+Pablo Bleyer - whoops, I meant 3d(d-1). Fixed - thanks!

Excellent! On sunny mornings, I am always endlessly entertained by the caustics produced by my glass coffee cup.

I was wondering what the caustic on the left side was doing there, but from reading Egan's page, I think it's a "virtual catacaustic", analogous to a virtual image: the rays are reflected outside of the curve and you wouldn't actually see that catacaustic, but you can obtain it by extrapolating the reflected rays backward through the surface.

Is the burnt out hard drive hyperbole or not? If it literally happened then I am curious if the code can be optimized to save the hard drive (or maybe it would be better to run it on some server somewhere)

Mathematica can be pretty memory-intensive. I remember doing my doctoral thesis work with it on a poor old Power Mac 6100 and making the hard drive thrash and thrash.

+Sheila Miguez - Egan doesn't tend toward hyperbole. I doubt the hard drive literally caught on fire, but I think it really did stop working while he was doing a week-long calculation.

Apparently there's a virus that can make your printer catch on fire:

http://www.infobarrel.com/New_Virus_Sets_Printers_on_Fire_Is_Your_Computer_Secure

Apparently there's a virus that can make your printer catch on fire:

http://www.infobarrel.com/New_Virus_Sets_Printers_on_Fire_Is_Your_Computer_Secure

+Matt McIrvin - I woke up at night and remembered I wanted to add that explanation, but you beat me to it. The mathematical construction of a catacaustic as the solutions of a polynomial equation automatically throws in those 'virtual' points.

Sheila Miguez - Maybe implementing breaks allowing the drive to cool down would do the job.

Alternatively Egan should invest in a special AV (or server) harddisk if he wants to try this again. They're specially designed for working 24/7. I learnt about them in the VDR Portal. I'm running such a Linux-based DIY video recorder - with normal harddisks. My VDR is running 24/7, but obviously not recording all the time.

Alternatively Egan should invest in a special AV (or server) harddisk if he wants to try this again. They're specially designed for working 24/7. I learnt about them in the VDR Portal. I'm running such a Linux-based DIY video recorder - with normal harddisks. My VDR is running 24/7, but obviously not recording all the time.

Steve Esterly

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+John Baez, it seems the printer fire risk was hype, and although the printer security risk is real, there is hardware protection against the fire:

http://nakedsecurity.sophos.com/2011/11/30/flaming-retort-putting-out-the-hp-printer-fires/

http://nakedsecurity.sophos.com/2011/11/30/flaming-retort-putting-out-the-hp-printer-fires/

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