curiosamathematica
curiosamathematica:

Sperner’s lemma
Color the vertices of a triangulated triangle with three colors such that:
each vertex of the main triangle has a different color;
each vertex on an edge of the main triangle is colored with one of the two colors at the end of its edge;
then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.
This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer’s fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.

curiosamathematica:

Sperner’s lemma

Color the vertices of a triangulated triangle with three colors such that:

  • each vertex of the main triangle has a different color;
  • each vertex on an edge of the main triangle is colored with one of the two colors at the end of its edge;

then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.

This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer’s fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.

insidious-gestalt-aggregation

insidious-gestalt-aggregation:

These dreary text posts standard action of repressed brain and ideology. Stifled. Stymied. Strangulated. Siphoning those resplendent, candid, altruistic tendencies. Dichotomy, schism. Hailing battle cries across this gorge. Dissociated personality. Not in clinical sense. Ha ha ha, laugh at your own depravity. Reconciliation, nigh impossible. Grasping, tearing, pulling down, into the depths between. Death will surely come. Regardless of implied action.

insidious-gestalt-aggregation

insidious-gestalt-aggregation:

We’re all trying to make others smile. Those genuine shared smiles. The moment of release and gratification. The truths exemplified in those stark moments. Stripped of intellectual weight. The back story, the etymology of the interaction. Fatalist view on end result of pure enlivened chaos,…

insidious-gestalt-aggregation

insidious-gestalt-aggregation:

That unchoreographed dance between minds. Nestled in the everyday connection. The emotion brought from both personal experience and human biology. All the preceding incidents of emotion. memory, self-identification, come into play after the first sentence uttered. Every word having purpose. The…

danieltoumine

sleeping in a jar

danieltoumine:

Murder out my Yaris® and send me off to Antarctica.
I wanna be the lowest satellite the world will never see.

Forgotten ///\/\///
                  \/\ and \ \  \/\/    \//                               Severed. 

///// \             /
\//       //\/ 

Sloughed off long before the flakes ever put a tick on screen—
fingers still twirlin’ right in front of their eyes all ‘phhhzzsss…’

in a spiralling gesture leaving them all bronze, bitter, and cold.
Cuz statues always stand in a self absorbed kinda way anyway.


- @DanielToumine