A linked article about an einstein (one-piece) aperiodic tile contains a beautifully simple animation of the continuum of tiles that morph as you scroll down the page.
> it’s impossible to tell any two tilings apart by examining any local area. That’s because every finite patch of any tiling, no matter how large, will show up somewhere in every other tiling
I was about to pose a question but it turns out that's what the article is about!
> Surely they mean algebraic numbers not real numbers.
that's an extremely subtle distinction! I don't really understand this distinction to my own satisfaction.
in my mind the concept I need to understand better is the generalized notion of 'decimal point'
this presents me 2 avenues of further investigation:
'decimal' which just means ten which is already known as the general concept of the base of a positional numeral system (so this links to exponents and modulo arithmetic and the modern algebraic framework)
and the other avenue is the 'point'. so here's my question: up to which extent should I attempt to realize the notion of "fixed point" (like in Tarski's theorem) and the specific application of a "fractional"-point as an instance (where 'decimal' is the specific applied case n=10=2*5) of a "fixpoint" used as part of a positional numeral value determination system?
At the risk of telling you something you don’t already know: algebraic numbers are an extension of integers¹ which adds in those numbers which are zeros of finite polynomials with integer coefficients.² So, to pick a trivial example, ±√2 are algebraic number because they are solutions to x²-2=0.
But there are many real numbers which are not algebraic. π and e are the best known non-algebraic numbers (we call these numbers transcendental. In common parlance, unless otherwise specified, the term transcendental number is generally used to refer to the set ℝ\ where refers to the set of algebraic numbers in this instance³). In fact, statistically speaking, the probability that any given real is algebraic is 0.⁴
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1. It’s worth noting that while it’s easy to assume that algebraic numbers are a subset of the reals, the fact that it includes complex numbers (i is algebraic since it’s a solution to x²+1=0) means it’s not. n.b., just as not all reals are not algebraic, not all complex numbers are algebraic.
2. Rational coefficients of a polynomial can be trivially eliminated by simply multiplying all terms of the polynomial by the lcm of the denominators of any non-integer polynomial, so the set of zeros of polynomials with integer coefficients and the set of zeros of polynomials with rational coefficients will be identical.
3. I’m aware is also often used to label the adele ring, but I’m happy to borrow it for the purposes of a Hacker News comment.
4. The proof of this statement is left as an exercise to the reader.
One way to generate aperiodic tiling is to take a regular lattice and cut it with a plane with a transcendental slope. Project each lattice point down onto the plane and then choose tiles appropriately [0].
Algebraic numbers are those as roots of (finite) integer/rational polynomials. Though things get convoluted depending on what domain you're talking about, the algebraic structure, in this context, if I have it right, means there's periodicity. Aperiodicity requires there not to be such structure.
I only read the headlines so my apologies if I'm way off base.
There are many ways to construct an aperiodic set of points with algebraic coordinates, or even rational, or integer, coordinates.
Penrose tilings have vertices with algebraic coordinates. The new aperiodic "hat" tiling does too.
In fact, the hat tiling has an underlying periodic sub-tiling of kites, and you can use this fact to distort the hat tiling into an aperiodic tiling with integer coordinates.
The grain of truth in your argument is that algebraic numbers are more structured. Crucially, they can always be represented exactly with a finite amount of data. The Penrose vertex coordinates are particularly nice because they're rational linear combinations of 1 and sqrt(5), iirc, so you can represent them as a pair of pairs of integers.
(Edit: I'm also pretty certain the slopes of the cut-and-project plane are algebraic. You can obtain a nice one-dimensional aperiodic tiling by cutting a 2D grid at a slope of the golden ratio, which is obviously algebraic.)
Yep, you're absolutely right. My confusion came from mixing up irrational and transcendental. I think the cut and paste method works for sqrt(2) which is clearly algebraic.
I have to admit I don't immediately see how that would work, since the proof uses properties of Penrose tilings other than aperiodicity, but if you flesh out the details I think it would be a great paper.
https://www.quantamagazine.org/hobbyist-finds-maths-elusive-...