Impossible Cookware and Other Triumphs of the Penrose Tile - Issue 13: Symmetry - Nautilus

Impossible Cookware and Other Triumphs of the Penrose Tile - Issue 13: Symmetry - Nautilus: One of the curious aspects of aperiodic division of the plane is that information about positioning is somehow communicated across great distances—a Penrose tile placed in one position prevents the placement of other pieces hundreds (and thousands and millions) of tiles away. “Somehow a local constraint imposes a global constraint,” says Harriss. “You impose that at no scale will these tiles give you something that is periodic..."

It turns out crystals don’t always form atom-by-atom. “In very complex intermetallic compounds, the units are huge. It’s not local,” says Shechtman. When large chunks of crystal form at once, rather than through gradual atom accretion, atoms that are far apart can affect one another’s position, exactly as do Penrose tiles.

Mathematics of Eternity Prove The Universe Must Have Had A Beginning -- Part II

Mathematics of Eternity Prove The Universe Must Have Had A Beginning -- Part II:  "To make the point simply, imagine Hilbertville, a one-dimensional semi-infinite city, whose border is at x = 0: The population is infinite and uniformly fills the positive axis x > 0: Each citizen has an identical telescope with a finite power. Each wants to know if there is a boundary to the city. It is obvious that only a finite number of citizens can see the boundary at x = 0. For the infinite majority the city might just as well extend to the infinite negative axis...

Thus, assuming he is typical, a citizen who has not yet studied the situation should bet with great confidence that he cannot detect a boundary. This conclusion is independent of the power of the telescopes as long as it is finite..."

He finishes with this: "We may conclude that there is a beginning, but in any kind of inflating cosmology the odds strongly (infinitely) favor the beginning to be so far in the past that it is eff ectively at minus infinity."

Mathematics of Eternity Prove The Universe Must Have Had A Beginning - Technology Review

Mathematics of Eternity Prove The Universe Must Have Had A Beginning - Technology Review: Today, Audrey Mithani and Alexander Vilenkin at Tufts University in Massachusetts say that these models are mathematically incompatible with an eternal past. Indeed, their analysis suggests that these three models of the universe must have had a beginning too.

Their argument focuses on the mathematical properties of eternity--a universe with no beginning and no end. Such a universe must contain trajectories that stretch infinitely into the past.

However, Mithani and Vilenkin point to a proof dating from 2003 that these kind of past trajectories cannot be infinite if they are part of a universe that expands in a specific way.

'Infinity Computer' Calculates Area Of Sierpinski Carpet Exactly - Technology Review

'Infinity Computer' Calculates Area Of Sierpinski Carpet Exactly - Technology Review  Sergeyev begins by adding a new axiom to the axiom of real numbers, which he calls the infinite unit axiom. This introduces grossone--the infinite unit.

Because it is governed by the other axioms of real numbers, grossone behaves much like one too. So it's possible to multiply grossone, divide it, add to it and subtract from it, just as is possible with other real numbers.

That suddenly makes working at infinity much easier by using a computing process that Sergeyev calls the infinity computer, which has the additional axiom built in. "The introduction of grossone gives a possibility to work with finite, infinite and infinitesimal quantities numerically," he says.

To show off its power, he works through the Sierpinski carpet examples given above, revealing how it's possible to keep track of the number of iterations at infinity simply by adding or subtracting real numbers from grossone. If a square can created in grossone steps, a square doughnut can be created in -grossone minus 1- steps. In this way, it's a simple matter to differentiate between any of the shapes in carpet sequence.

Ultimate logic: To infinity and beyond - New Scientist - New Scientist

Ultimate logic: To infinity and beyond: At the same time, though, Gödel had a crazy-sounding hunch about how you might fill in most of these cracks in mathematics' underlying logical structure: you simply build more levels of infinity on top of it. That goes against anything we might think of as a sound building code, yet Gödel's guess turned out to be inspired. He proved his point in 1938. By starting from a simple conception of sets compatible with Zermelo and Fraenkel's rules and then carefully tailoring its infinite superstructure, he created a mathematical environment in which both the axiom of choice and the continuum hypothesis are simultaneously true. He dubbed his new world the "constructible universe" - or simply "L"...

As Cohen was able to show, in some logically possible worlds the hypothesis is true and there is no intermediate level of infinity between the countable and the continuum; in others, there is one; in still others, there are infinitely many. With mathematical logic as we know it, there is simply no way of finding out which sort of world we occupy...

Among other things, ultimate L provides for the first time a definitive account of the spectrum of subsets of the real numbers: for every forking point between worlds that Cohen's methods open up, only one possible route is compatible with Woodin's map. In particular it implies Cantor's hypothesis to be true, ruling out anything between countable infinity and the continuum.