bobbycaputo:

**Incredibly Intricate Modernist Sand Castles**

If you grew up near the beach — or just vacationed there as a kid — chances are you’ve made a sand castle. You may have even entered a sand castle-building contest. But we doubt you’ve seen anything like the work of Calvin Seibert, who shapes sand into impossibly sharp edges and smooth curves, creating geometric structures that both suggest modernist architecture and, in their surprising stairways, evoke the work of M.C. Escher. Click through for some of our favorite Seibert sand castles, which we discovered via Colossal, and visit the artist’s Flickr page to see more of his creations.

(Via. Flavorwire)

themathkid:

In the 1950s, Solomon W. Golomb investigated the question: how few cells can you remove from an 8×8 square to exclude the shape of a given polyomino? His book Polyominoes shows minimal exclusions for all polyominoes up through order 5. Presented here are the minimal known exclusions of hexominoes from an 8×8 square. The solutions shown are not necessarily unique.

themathkid:

symmetricalstars:

themathkid:

The Catalan numbers give the number of ways to tile an n-sized stair case with exactly n rectangles.

That’s so interesting!

Continuing on my previous post:

The 42 tilings here also represent the Catalan number C7. Surely there is a bijection from these tilings to the heptagon triangle dissections [see previous post]. Help me find it?

themathkid:

Found this on Tumblr a while ago. Didn’t reblog and subsequently lost it. Glad to have finally found it again!

quantumaniac:

**The Poincaré Conjecture**

Imaginestretching a rubber band around the surface of an apple, then shrinking it down slowly. This shrinking could occur without tearing the rubber band or breaking the apple - and the band would never have to leave the surface. However, if this rubber band were to be stretched across, say, a tire - there is no way to shrink to a point without breaking one or the other. The surface of such an apple is “simply connected,” but the tire is not. Henri Poincaré (shown below), during the early twentieth century - knew that two dimensional spheres had this ‘connected’ property - and he asked if the same applied for three dimensional spheres.

The conjecture turned out to be immensely difficult to prove. After more than a century, Grigori Perelman finally devised a solution. In 2006, Perelman was awarded the Fields Medal for this contribution, but he decided to turn it down, stating that:

“I’m not interested in money or fame, I don’t want to be on display like an animal in a zoo.”

quantumaniac:

**Spiral of Theodorus**

First constructed by Theodorus of Cyrene, the spiral (also called the square root spiral, Einstein spiral or Pythagorean spiral) is composed of contiguous right triangles. It is begun with an isosceles right triangle with each leg having a length of 1 - then another right triangle is formed next to it with one leg being the hypotenuse of the prior triangle and the other leg having a length of 1.Thus, each successive nth triangle has side lengths √*n* and 1, with a hypotenuse of √(*n* + 1).