Re: Practical applications of complex numbers

On 01/08/2026 10:37 PM, Stefan Ram wrote:
> Johann 'Myrkraverk' Oskarsson <[email protected]> wrote or quoted:
>> This is some beautiful ASCII art rendering of waves.  Do you have a tool
>> that does this, or do you keep this pre-rendered in a text file for just
>> such occasions?
> 
>    I've written a Python script that renders pixels to an array
>    and then tries to match rectangles with such pixels to ASCII
>    characters; it is using a specific raster font. The raw results
>    of this approach did not look very good, and I found out that
>    I can improve the result by restricting the set of characters to
>    just a few selected characters like ".". Also, I take the slope
>    of the curve into account. For example, the downward moving
>    accent "`" is only used where the curve does move downward with
>    approximately this angle (as can be seen in the sine plots).
>    But this Python script is not yet ready for publication. I also
>    edited two characters of the plots manually in my previous post.

I see.  And you can see some of my old hand drawn curves at this art
gallery.

   https://asciiart.website/search.php?q=myrkraverk&sort_by=random

I never thought to automate it until your graph inspired me.  It's going
to be a nice side project.  And no worries about your Python script, I
prefer to write my own code.  I appreciate the tip about the slope.

> 
>>> If we combine /two/ plates of thickness d/2 we still get a multipli-
>>> cation by -1. So what does /one/ plate of thickness d/2 multiply
>>> the amplitude with?
>> I have a feeling the answer should be /i/, but I'm not sure.  Feel free
>> to recommend books, websites, or PDF files where I can brush up on light
>> physics.
> 
>    This example was taken from a book about quantum physics that is
>    as easy and readable as a book about this topic can possibly be:
> 
> "Quantum Processes, Systems, and Information" (2010) -
> Benjamin Schumacher and Michael D. Westmoreland.
> 
>    (Schumacher is known for his coinage of the word "qubit".)
> 
>    The authors write in section 2.1:
> 
> |Glass plates can be made in a continuous range of thicknesses,
> |producing a continuous range of phase shifts. For this to be
> |possible, the beam phases a must be complex quantities, with
> |both real and imaginary parts. A plate with thickness d/2 may
> |multiply the amplitude by a factor of i = sqrt −1. This does not
> |change the magnitude of the complex phase a, since |a| = |ia|.
> |Two such plates (or a single plate of thickness d) multiply
> |the phase by i^2 = −1, as required.
> 

Thank you for the recommendation.  I've added it to my list of books
to read in the near future.

-- 
Johann | email: invalid -> com | http://www.myrkraverk.com/blog/
I'm not from the Internet, I just work there. | via Easynews.com