Over the last century-and-a-half, mathematicians found every possible multiplication table.

The largest irreducible multiplication-table, dubbed the Monster Group, contains

808017424794512875886459904961710757005754368000000000
=
2⁴⁶×3²°×5⁹×7⁶×11²×13³×17×19×23×29×31×41×47×59×71

interlocking pieces.

That’s like the number of atoms in Jupiter.

Richard Borcherds

(modified by me)

(Source: ams.org)




Three observations get you there:

  1. min {a,b,c} = − max {−a, −b, −c}
  2. second-from-top {a,b,c,d,e} = max ( {a,b,c,d,e} without max{a,b,c,d,e} )
  3. max {a,b,c} ~ log_t (t^a + t^b + t^c ),   t→∞

Putting these three together you can make a continuous formula approximating the median. Just subtract off the ends until you get to the middle.

It’s ugly. But, now you have a way to view the sort operation—which is discontinuous—in a “smooth” way, even if the smudging/blurring is totally fabricated. You can take derivatives, if that’s something you want to do. I see it as being like q-series: wriggling out from the strictures so the fixed becomes fluid.




√(x²−1)(x²−k²).      x,k∈ℂ

(actually just going over the unit circle, not all of ℂ)

edit: hey, are these showing up as moving gif’s for you?

Read More

(Source: math.berkeley.edu)










 SSSSSSSSSSSSSSSSSSSSSSSSS
 SSSSSSSSSSSSSSSSSSSSSSSSS
 SS        SSSSS        SS
           SSSSS
           SSSSS
           SSSSS
           SSSSS
           SSSSS
           SSSSS
          SSSSSSS
       SSSSSSSSSSSSS

(Source: Wikipedia)










A broader tax base, it is thought, will insure that wealthy suburbanites pay for essential services needed by the poor. No evidence is available to indicate that this actually happens in large cities.

Poor neighborhoods receiving ”services” which are not tailored to their needs may not be better off when increased resources are allocated to their neighborhood. In large collective consumption units, residents of poor neighborhoods may have even less voice about levels and types of services desired than they do in smaller-sized collective consumption units. Increasing the size of the smallest collective consumption unit to which citizens belong may not help solve problems of redistribution.
Vincent Ostrom and Elinor Ostrom, Public Goods and Public Choices

(Source: socsci.colorado.edu)




There’s a collective experience going on at a rock concert that—I’ve always assumed—would probably be what church should be like, if church was what it should be.
Jeff Tweedy

(Source: youtube.com)