Physics and Integer Computation with Eisenstein Integers

In summary, the conversation discusses the use of Gaussian and Eisenstein integer domains for geometry research. While Gaussian integers can be described using rational integers, Eisenstein integers require non-integers for computational geometry. The use of irrational numbers, such as √3, is necessary due to the limitations of computer hardware and the way floating point numbers are stored in memory. The possibility of designing a new type of computer that can work with symbolic representations of numbers is also mentioned.
  • #1
Ventrella
29
4
I realize this question may not have an obvious answer, but I am curious: I am using Gaussian and Eisenstein integer domains for geometry research. The Gaussian integers can be described using pairs of rational integers (referring to the real and imaginary dimensions of the complex plane). And so I can do purely Diophantine math (only integers: no real numbers required).

But Eisenstein integers (occupying a triangular lattice) require non-integers for doing any computational geometry (specifically, ½ and √3). My scheme uses only rational integers for a compact and efficient set of parameters. In the case of the Eisenstein domain, I must apply a transformation requiring the irrational number √3 to map points in the plane.

This is not impeding my work, but I am curious: is it the physical nature of computer hardware that creates a constraint that requires the irrational number √3 to be used in the case of triangular lattices? Physics and nature prefer triangular (hexagonal) arrangements over orthogonal (square) ones, and yet our computers are not able to precisely represent these arrangements without the use of an irrational number.

If the answer to my question requires the design of a new kind of computer, then I would be curious how (or if) that can be done! (I suspect it is not possible).

Meanwhile, I will have to make do with the fact that all geometry defined with Eisenstein integers can never be as precise (or computationally compact) as with the Gaussian integers. This is obvious in the pragmatic sense, but the fundamental reason is unclear - and it may fall into the domains of meta-math, physics, and ontology.
 
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  • #2
Ventrella said:
is it the physical nature of computer hardware that creates a constraint that requires the irrational number √3 to be used in the case of triangular lattices? Physics and nature prefer triangular (hexagonal) arrangements over orthogonal (square) ones, and yet our computers are not able to precisely represent these arrangements without the use of an irrational number.
Because of the way floating point numbers are stored in memory, computers work exclusively with rational numbers, and some rational numbers can't be represented exactly in hardware. For example, numbers such as 0.1 and 0.2 are stored as approximations. There are software libraries that can store floating point numbers with much greater precision, and there probably are libraries that can work with symbolic representations of numbers, such as ##\sqrt{3}##, but I don't know about them.

Ventrella said:
If the answer to my question requires the design of a new kind of computer, then I would be curious how (or if) that can be done! (I suspect it is not possible).
 

FAQ: Physics and Integer Computation with Eisenstein Integers

What are Eisenstein integers and how are they used in physics?

Eisenstein integers are complex numbers of the form a + bω, where a and b are integers and ω is the cube root of unity. They are used in physics to represent points in a triangular lattice, which is useful in understanding crystal structures and other physical phenomena.

How are Eisenstein integers different from regular integers?

Eisenstein integers are different from regular integers in that they are complex numbers, meaning they have both a real and an imaginary component. Regular integers, on the other hand, only have a real component.

What is the significance of using Eisenstein integers in integer computation?

Using Eisenstein integers in integer computation allows for a more efficient and accurate representation of complex numbers, as they have a smaller magnitude compared to regular complex numbers. This is useful in applications such as signal processing and error correction.

Can Eisenstein integers be used in all areas of physics?

Eisenstein integers can be used in various areas of physics, particularly in fields where the concept of a triangular lattice is relevant. This includes crystallography, electromagnetism, and quantum mechanics.

Are there any limitations to using Eisenstein integers in physics and integer computation?

One limitation of using Eisenstein integers is that they are only applicable to situations where a triangular lattice is present. Additionally, their use may be limited by the computational resources available, as they may be more complex to work with compared to regular integers.

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