Penrose Tiling

[Dark_Side]

Hello - first post and whatnot. I checked and I don't think anyone's mentioned this yet. If they have and I just missed it, then let me go ahead and apologize.

I just saw the second episode for the first time tonight (Curse real life!), so this wikipedia article really stood out to me:

http://en.wikipedia.org/wiki/Penrose_tiling

"The Penrose tiling, the Fibonacci sequence and the golden ratio are intricately related and perhaps they should be considered as different aspects of the same phenomenon."

So on top of all the references to the Fibonacci sequence and Phi, we now have a missing baddie whose practically named for them - and one with ties to Walter, no less.

[D-Roc]

Oh wow, great connection Dark_Side.

A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tilings are considered aperiodic tilings.[1] Among the infinitely many possible tilings there are two that possess both mirror symmetry and fivefold rotational symmetry, as in the diagram at the right, and the term Penrose tiling usually refers to them.
A Penrose tiling has many remarkable properties, most notably:

• It is nonperiodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original exactly.
• Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be trivially true of a tiling with translational symmetry but is non-trivial when applied to the non-periodic Penrose tilings.
• It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction; the diffractogram reveals both the underlying fivefold symmetry and the long range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation."

The Penrose tiling, the Fibonacci sequence and the golden ratio are intricately related and perhaps they should be considered as different aspects of the same phenomenon.

• the ratio of thick T to thin t rhombuses in the infinite tile is the golden ratio T/t = φ = 1.618..
• the Conway worms, sequences of neighbouring rhombuses with parallel sides, are Fibonacci ordered appearances of T and t and thus the Ammann bars also form Fibonacci ordered grids
• around each 5T − star a segmented Fibonacci spiral is formed by the sides of rhombuses [1]
• the distances between repeated finite motifs in the tiling grow as Fibonacci numbers when the size of the motif increases
• the distribution of oscillation frequencies in a Penrose tiling shows bands and gaps whose widths are in proportions expressed by φ.[10]
• the substitution scheme introduces φ as a scaling factor;its matrix is the square of the Fibonacci substitution matrix; implemented as a symbol sequence ( e.g. 1→101, 0→10) this substitution produces a series of words with lengths which are the Fibonacci numbers with odd index, F(2n+1) for n=1,2,3.., the limit being the infinite Fibonacci binary sequence
• the eigenvalues of the substitution matrix are φ+1 (=φ²) and 2-φ (=1/φ²)

The references to 'symmetry' and 'infinity' leap out at me, considering how many duplicate Christopher Penroses were in the MD spawn room. Also, the Fib connection, as you point out, is also a neat one.

[Deknor]

AH! I wonder if JJ wanted us to see that from the Pen Rose image on the newspaper machine thing, or if he thought we'd get the connection from the name. OR perhaps he gave both of them to us to reaffirm that it is something to look at.

[mygoodeye]

can someone explain Penrose Tiling in laymans? I'm not that dumb but anything maths related goes right over my head! i didnt understand a word of that. nice catch tho

[Rocky]

can someone explain Penrose Tiling in laymans? I'm not that dumb but anything maths related goes right over my head! i didnt understand a word of that. nice catch tho

If you understand what a regular tessellation is, then you can understand the Penrose tiling a lot more easily if you consider the shape of the tiles. Basic tessellations use the same shape repeated over and over again to make a pattern. See this Escher example:



The Penrose tiling is then different(aperiodic) tiles being used to make a pattern. Example:
Notice the different shaped tiles.

Hopefully I correctly explained the Penrose tiling, anf hopefully you understand it a little better now.

[Deknor]

New Penrose things to consider!

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special case. Specifically, if (M, g) is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass m, and A is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition.
This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of inverse mean curvature flow, which they developed. In 1999, Hubert Bray gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001.

As well as:

The Penrose process (also called Penrose mechanism) is a process theorised by Roger Penrose wherein energy can be extracted from a rotating black hole. That extraction is made possible by the existence of a region of the Kerr spacetime called the ergoregion, a region in which a particle is necessarily propelled in locomotive concurrence with the rotating spacetime. In the process, a lump of matter enters into the ergoregion of the black hole, and once it enters the ergoregion, is split into two. The momentum of the two pieces of matter can be arranged so that one piece escapes to infinity, whilst the other falls past the outer event horizon into the hole. The escaping piece of matter can possibly have greater mass-energy than the original infalling piece of matter. In summary, the process results in a decrease in the angular momentum of the black hole, and that reduction corresponds to a transference of energy whereby the momentum lost is converted to energy extracted.
The process obeys the laws of black hole mechanics. A consequence of these laws is that if the process is performed repeatedly, the black hole can eventually lose all of its angular momentum, becoming rotationally stationary

.


The use of black holes has often been referred to when thinking of some sort of warp through space and time situation. And Penrose did a lot of work with black holes.

[mygoodeye]

Ok, ive just about got a grasp on the tiling and how it relates to fibonacci etc...then you went and lost me on mathematical gerneral relativity!

ive actually just been delvinging into a bit of general relativity...trying to come up with a plausable idea for an invisibility suit. dont ask

[Dark_Side]

Oh wow, great connection Dark_Side.

The references to 'symmetry' and 'infinity' leap out at me, considering how many duplicate Christopher Penroses were in the MD spawn room. Also, the Fib connection, as you point out, is also a neat one.

Thanks. Yeah, I was intrigued by the mention of "infinity" as well, but I didn't think to make the connection to Christopher. Good catch. This just gets better all the time.

Oh, and thanks for the extra info, Deknor. I really gotta read up on the real Penrose now.

[Deknor]

What if it refers instead to one of Penrose's intelligent relatives also? There are quite a few Penroses with notoriety.

Lionel Sharples Penrose (11 June 1898 - 12 May 1972) was a British psychiatrist, medical geneticist, mathematician and chess theorist, who carried out pioneering work on the genetics of mental retardation. He was educated at the Quaker Leighton Park School and Cambridge University.
Penrose's "Colchester Survey" of 1938 was the earliest serious attempt to study the genetics of mental retardation. He found that the relatives of patients with severe mental retardation were usually unaffected but some of them were affected with similar severity to the original patient, whereas the relatives of patients with mild mental retardation tended mostly to have mild or borderline disability. Penrose went on to identify and study many of the genetic and chromosomal causes of mental retardation (then called mental deficency). This remarkable body of work culminated in the classic book, The Biology of Mental Defect (Sidgwick and Jackson, Ltd., London, U.K., 1949).
Penrose was a central figure in British medical genetics following World War II. From 1945 to 1965 he occupied the Galton Chair at the Galton Laboratory at University College, London. He received a number of awards and honors including the 1962 Albert Lasker Award for Basic Medical Research.[1], sometimes called "America's Nobel Prize." The Lasker citation read: "Professor Penrose and his associates have been responsible over the years for studies which touch all aspects of human genetics, include genetic analyses of most of the known hereditary diseases, contributions to mathematical genetics, biochemical genetics, the study of gene linkage in man, and theoretical work on the mutagenic effect of ionizing radiations. Most recently their attention has been turned to abnormalities of human chromosomes associated with congenital defects, particularly mongolism (Down syndrome)."
In British psychiatry, 'Penrose's Law' states that the population size of prisons and psychiatric hospitals are inversely related, although this is generally viewed as something of an oversimplification.

As well as the Moore-Penrose psuedoinverse.
http://en.wikipedia.org/wiki/Moore-Penrose_inverse

And Newman-Penrose formalism
http://en.wikipedia.org/wiki/Newman-Penrose_formalism

[Wondermind]

What if it refers instead to one of Penrose's intelligent relatives also? There are quite a few Penroses with notoriety.



As well as the Moore-Penrose psuedoinverse.
http://en.wikipedia.org/wiki/Moore-Penrose_inverse

And Newman-Penrose formalism
http://en.wikipedia.org/wiki/Newman-Penrose_formalism

Here's another one for the Penrose collection which ties in with the leaf glyph which holds the triangle symbol. I also added this to the "Photos from episode 2" post as well.
http://en.wikipedia.org/wiki/Penrose_triangle

Rocky: your Penrose tilling ties into what Walter said: "ramdom meaningless input which forms into a logical pattern!"