### »  Proof confirmed of 400-year-old fruit-stacking problem - physics-math - 12 August 2014 - New Scientist

Tom Hales is the probably the nicest person I’ve met at Pitt, but look at him being all sassy:

"This technology cuts the mathematical referees out of the verification process," says Hales. "Their opinion about the correctness of the proof no longer matters."

An Isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity.

- A ball set on an Isocrone (or Tautochrone) curve will reach the bottom at the same length of time no matter where you place the ball, so long as there is no impeding friction.

[Gif] - Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points’ acceleration along the curve. On the top is the time-position diagram.

[source]

Mathematics and Art - A Penrose Tiling

"The whole is greater than the sum of its parts"
- Aristotle (384 BC – 322 BC)

A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose who investigated these sets in the 1970s.

A Penrose tiling has many remarkable properties, most notably:

• It is non-periodic, which means that it lacks any translational symmetry.
• It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through “inflation” (or “deflation”) and any finite patch from the tiling occurs infinitely many times.
• It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.

The three types of Penrose tiling:The original pentagonal Penrose tiling (P1), Kite and dart tiling (P2) and Rhombus tiling (P3).They have many common features: in each case, the tiles are constructed from shapes related to the pentagon (and hence to the golden ratio), but the basic tile shapes need to be supplemented by matching rules in order to tile aperiodically; these rules may be described using labeled vertices or edges, or patterns on the tile faces – alternatively the edge profile can be modified (e.g. by indentations and protrusions) to obtain an aperiodic set of prototiles.

Image: Penrose Tiling at http://en.wikipedia.org/wiki/Penrose_tiling.

Some substitution rules includes: Overlapping Robinson Triangle I & II - Discovered by P. Gummelt and Danzer’s 7-fold variant.

Rule Image: (I) A substitution rule where the tiles are allowed to overlap. The image left indicates, that the yellow and the green tiles do overlap. It is unknown whether these tilings are mld to the Penrose Rhomb tilings. Rule Image: (II) As Overlapping Robinson Triangles I, this is a variant of the Penrose Rhomb tiling, using only one prototile, and the tiles are allowed to overlap. Here, the overlap happens after applying the substitution rule twice on one tile.

Danzer’s 7-fold variant -  Rule Image:  Substitution tiling with isosceles triangles as prototiles allow several variations: For each tile in the first order supertiles, one can choose whether it is a left-handed or a right-handed version. By playing around with these possibilities, one obtains this variant from Danzer’s 7-fold.

### "There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann."

George Pólya (via ordnungsokonomik)

Howdy, everyone! Tonight we’re rolling out Mathemedia, a compendium of math-related media — and we’re looking for your submissions!

Send us your favorite books, movies, songs, art, essays, and articles (etc.) that prominently feature mathematics or mathematicians — and we’ll add them to Mathemedia. But first! a few ground rules:

1. We prefer that material be accessible to anyone with a love of mathematics and access to Wikipedia — assuming, at most, a typical high school education in math.
2. In general, we’re looking for stories about mathematics — told either from the inside or outside. Fiction is great; non-fiction is good too, as long as it’s designed for more than a mathematically-trained audience. There are several great lists out there on the interwebs of seminal papers and great textbooks and phenomenal websites — this is not one of them.
3. If you have the time, please include a short description of the book, movie, etc. in question. This will help anyone browsing the list figure out what’s most interesting and appealing to them. If the material is legally available on the Internet (e.g., an article or music video), feel free to include a link!

Thanks for helping spread our love of math! We’ll start putting up what you submit later tonight.

This list was inspired by math-is-beautiful's fabulous list of math-related blogs :)

### "In some sense, everything is trivial."

Topology professor (via mathprofessorquotes)

There are 177,147 ways to tie a tie, according to mathematicians.

### "These days, mathematics books tend to be awash with symbols, but mathematical notation no more /is/ mathematics than musical notation /is/ music. A page of sheet music /represents/ a piece of music; the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience; the music exists not on the printed page, but in our minds. The same is true for mathematics; the symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive- the mathematics lives and breathes in the mind of the reader like some abstract symphony."

Keith Devlin- The Language of Mathematics: Making the Invisible Visible

Enjoying my lovely birthday gift from exponentiate <3

(via a-heart-of-calcifer)