3 ms·

Reciprocal Lattice

peter_d_sherman 6y ago
>"In physics, the reciprocal lattice represents the Fourier Transform of another lattice..." [...] "In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space" --- PDS: Which brings up the following idea(s): Speculation (on my part): There must exist a lattice whose Fourier Transformed version -- is exactly identical to itself. In other words, an Identity Lattice... (for lack of a better term...) And of course, the next question after that... Is there a real-world material, whose crystalline structure, matches that Identity Lattice? ? And then the next-to-final question: If so -- then what special properties does that material possess? ? And the final question: What are all of the applications of those properties? ?
- posterboy 6y ago
  The bell curve's and similar functions are neutral under fourier transformation. I have no idea how that translates to lattices.
  - peter_d_sherman 6y ago
    Interesting! Did not know that before! That would point to bell curves (at least in statistics/probability theory) being the Identity Element of the algebraic topology (or perhaps "set/numeric mapping" is a better word) created by applying Fourier Transformations to Statistical Curves... Which just got me to think about the following idea... (Thank you for your post, BTW! -- Without it, wouldn't have thought of the following idea: ) Let's say that we consider the whole of 3D objects in Algebraic Topology (perhaps I am getting my terms wrong here, I am not a professional mathematician, so please correct me if I am wrong), but we consider the whole of objects that are formed by packets of vectors in 3D spaces, or perhaps curves in 3D spaces... OK, so now what happens when those are Fourier Transformed? And, are there any 3D objects (curves, vector packets, etc.) -- whose Fourier Transformation -- is itself? (That is, it's an Identity Element?). Now... same thing, but for objects in higher dimensions (4D, 5D, etc.)... and if you can do all of that (I can't! <g>), then what's the takeaway from that? Note To Future Self: Reinvestigate this...

>"In physics, the reciprocal lattice represents the Fourier Transform of another lattice..." [...] "In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space" --- PDS: Which brings up the following idea(s): Speculation (on my part): There must exist a lattice whose Fourier Transformed version -- is exactly identical to itself. In other words, an Identity Lattice... (for lack of a better term...) And of course, the next question after that... Is there a real-world material, whose crystalline structure, matches that Identity Lattice? ? And then the next-to-final question: If so -- then what special properties does that material possess? ? And the final question: What are all of the applications of those properties? ?

posterboy 6y ago
The bell curve's and similar functions are neutral under fourier transformation. I have no idea how that translates to lattices.
- peter_d_sherman 6y ago
  Interesting! Did not know that before! That would point to bell curves (at least in statistics/probability theory) being the Identity Element of the algebraic topology (or perhaps "set/numeric mapping" is a better word) created by applying Fourier Transformations to Statistical Curves... Which just got me to think about the following idea... (Thank you for your post, BTW! -- Without it, wouldn't have thought of the following idea: ) Let's say that we consider the whole of 3D objects in Algebraic Topology (perhaps I am getting my terms wrong here, I am not a professional mathematician, so please correct me if I am wrong), but we consider the whole of objects that are formed by packets of vectors in 3D spaces, or perhaps curves in 3D spaces... OK, so now what happens when those are Fourier Transformed? And, are there any 3D objects (curves, vector packets, etc.) -- whose Fourier Transformation -- is itself? (That is, it's an Identity Element?). Now... same thing, but for objects in higher dimensions (4D, 5D, etc.)... and if you can do all of that (I can't! <g>), then what's the takeaway from that? Note To Future Self: Reinvestigate this...