A Catastrophe

Here we show the p,q-plane (red) together with the zeros of the polynomial
x^3 + px + q (yellow) and its discriminant 4x^3+27y^2=0 (black on red). The white line shows the zeros corresponding to a given choice of p and q.

Notice how
the polynomials corresponding to points left of the disciminant have only one real zero and the polynomials to the right have three distinct ones. On the discriminat we have polynomials with two real zeros (one of them double).

This surface also plays a role in catastrophe theory, where it is called the cusp catastrophe. The catastrophe occurs when
the top black circle merges with the one in the middle, leaving only the bottom one. One images, that the economy or the climate slowly follows the top black circle up the yellow surface just to drop back down catastrophically when the discriminant is passed the second time.

This animation was made by Hans-Christian and Oliver using surfex.

This animation was #4 on our geometric animations advent calendar:

http://www.calendar.algebraicsurface.net

Category:

Howto & Style

Tags:

• geometry
• mathematics
• cusp
• fold
• discriminant
• catastrophe
• advent
• calendar
• bothmer
• hannover
• oliverlabs
• surfex
• cg

[phil10300]

Why is it a catastrophe?

[bothmer]

Because the black point on the top drops down to the lower level when passing over the edge. Imagine that the top black ring represents the state of the economy. Even though the whole surface of possible states is smooth, there can be a catastrophic change from one instant to the next.

[mrSpa88]

Nop.

catastrophe in greek means "full turn".

cata (KATA= down or ultimate)

strophe (ΣΤΡΟΦΗ (it is spealing "strofi")= turning)

[WillowFox]

Ah my bad, I thought it was supposed to be some kind of deep emo thing cuse some emo kid sent it to me. Sorry bout the slam there lol.

[awhoergek]

nice illustration of cusp catastrophe

[Koy124]

@bothmer WTF ??

[lovehorrordotde]

I hope this will not happen to anybody ;o(

[deoxy999]

I don't care about the math put into these, I just like the shape.

[VolkColopatrion]

oh it drops VERY SHARPLY

[Raminber]

I know hin. He is a good prof.

[XBONESXx]

dude i didn't get a word in your duiscription but i hope to one day

[artem1002]

lol you got pwned bothmer

[DJJP510]

yea good for you

[pokemonkiller20]

random