Archive for April, 2007

Better Know a Theorem: Solvability of Polynomial Equations

26Apr07

Let’s say I hand you a bunch of polynomial equations/inequalities and some variables, e.g., {x > 0, y < x, x2-y3+z=0, z4+4z>16} in variables x, y, z. Either there exists a triple of values for which all the equations and inequalities are satisfied, or there doesn’t. How can you tell? Is there an efficient procedure […]

Fressen

20Apr07

Asparagus Originally uploaded by BellaTheCat. I went to a Toronto restaurant called fressen for the second time yesterday. If you want to check out a larger supply of photos from the place (6 of them), click here. It’s a vegan restaurant (this fact is key). The first time was pretty random, my friend Rachel and […]

More Roundahedra

19Apr07

After figuring out why the equation in the last post described a tetrahedron, I was able to apply the same technique to generate the four other Platonic solids. By popular demand, here they are! The rounded cube has equation (1-x2)(1-y2)+(1-y2)(1-z2)+(1-z2)(1-x2)=0 and the rounded octahedron is 1-2(x2+y2+z2)-2(x2y2+z2x2+y2z2)+x4+y4+z4=0. I won’t post the icosahedron and dodecahedron’s formulas because […]

Party Time!

12Apr07

artemis Originally uploaded by BellaTheCat. The term’s finally done! I snuck onto Dave’s computer, now that he’s not using it all the time, and found this picture from one of his students. (The picture is of Artemis, and was sent by her owner, the student). It was starting to warm up here but got cold […]

Rounded Pyramid

10Apr07

Updates: Found an interactive Java applet with this shape (up to an affine transformation). My picture ignores four infinite “pointy dishes” that come out the corners. A nicer equation is x2+y2+z2+2xyz=1: also see a 1933 paper by A.S. Merrill. Every conic section can be obtained as a cross-section! A nice picture happened to come up […]

Hilarity

08Apr07

There seems to be a Simpsons marathon on Global today — at least three episodes so far. Given how many times I’ve seen this it’s great to have just “got” a joke for the first time: Kent Brockman: It’s literally the eleventh hour, 10 PM! Just two days ago I accidentally set my alarm for […]