Graphing Congruences in Modular Arithmetic And Turning a Cartesian Graph Into a Polar One

Even though this isn't anywhere close to Fields Medal level stuff, and it's not very deep, I think it's still pretty cool.  I haven't seen it before, but I'm pretty sure someone somewhere already did something like this.

What if we want to transform a graph in cartesian coordinates into one that looks like polar coordinates, where the x-values circle around some center point, and the y-values measure a distance from that center?  X would be comparable to Theta and Y would be comparable to R.  So we'd do a transformation as follows:

Theta = X times (( 2 times pi ) over a modulus)

R = Y

This modulus would cut the distance we have to travel around the center point into equal pieces.  We'll use the modulus 7 for this demonstration.

So if I wanted to take the arbitrary function Y=X^7 (this is an unrelated 7 - sorry I'm using the same number twice) and graph it in this system, I could do so, but first I'd have to do some easy algebra and express X in terms of Theta instead of Theta in terms of X.  Just divide both sides by (( 2 times pi ) over the modulus 7).

We end up getting R = ((7 times Theta) over (2 times pi)) raised to the power of 7.  That is something we can graph in Desmos.

Now, if I cut the distance around the origin into 7 parts, it would just be the angle Theta = (( 2 times pi ) over 7).  For some reason Demos really doesn't like me graphing that and won't let me do it, so I use the graph Y = tan(( 2 times pi ) over 7) times x, since that's the same thing.

So this line shows one-seventh of the distance around the origin.  But only the upper right part of the line stemming from the origin.  The lower left part is just a continuation of it, and does not represent another portion of 7 parts around the line as far as I know.

Modular arithmetic would tell us that once our Y=X^7 equation crosses this line, that equation is congruent to 1 (mod 7) at the R value where it crosses.  When the equation crosses the right side of the x-axis, that's when it's congruent to 0 (mod 7).  We could do this with other values as well if we were willing to draw more lines (and I'm not).

Anyway, we can test it out.  Here in this picture, we see that our X to the 7 function, the red curve, crosses the 1 (mod 7) line, the green line, right where that blue circle is.  The blue circle is just R = 8^7.  A pretty big number.  Over 2 million.  Is it congruent to 1 (mod 7)?  Indeed it is.  It's equal to 299,593 times 7 plus 1.  

Let's try another one.  This time we see the graph crosses the line at another purple circle, much larger than the last blue circle.  In fact, the blue circle at R = 8^7 looks almost like a small dot here.  This purple circle is R = 15^7.  Is this also congruent to 1 (mod 7)?  Sure is.  It's equal to 24,408,482 times 7 plus 1.  Nice to see computers confirming what the math shows.  And it's fun to graph this thing and zoom in and out on it.  Reminds me of the old Spirograph toy from a long time ago.

And that's all I've got to say about modular arithmetic here.

Now I want to do a conversion of a cartesian graph into a polar one using this same transformation, just because I think it looks interesting.

Let's do a nice and easy graph of a system of linear equations and note their intersection point.  Here the diagonal green and blue lines intersect at the red x=1 line and the orange y=7 line.

Applying our transformation to this system, we see the red x=1 line is now diagonal at the origin, the orange y=7 line is now a circle, and the two diagonal lines that cross each other there have become some kind of crazy spirals.  Very complicated looking.

So I just took something nice and easy and made it way more complicated for no reason.  I just think it's something kind of worth looking at, at least for me.

Proverbs 21:30

Proverbs 21:30 states "There is no wisdom, no insight, no plan that can succeed against the Lord." 

This verse is really about the futility of going against divine authority, but for some reason it reminds me of a common implication that if there is no final authority, then anything is permitted.  This is a common argument in apologetics I believe.

And vacuous truths, which I mentioned not too long ago, look somewhat similar.  If the premise is false, then any conclusion is permitted.

So if there isn't a God, apologists would say anything goes using their usual arguments.

But also if there is one, and we declare there to not be, again we'd reach the conclusion that anything goes, using a vacuous truth argument.  (Or if Proverbs 21:30 turned out to be true but we believed we had insight that was against the Lord.  Then any conclusion would go there too.)  

So in declaring God as a falsehood, whether accurate in actuality or not, apologists are going to reach the idea that anything goes.  Either using a standard argument or a vacuous truth one.

Certainly not a brilliant observation or anything, but a questionable and minor observation easy to put in a personal journal for a day. 

The Black Sun

I was fortunate enough to witness the April 8, 2024 solar eclipse in person.  I had to travel a bit to reach the path of totality, but it wasn't terribly far.

The experience was a bit different than I pictured it would be.  Instead of it feeling like nighttime in the middle of the day, it was more like a sunset in the middle of the day, where it didn't get near as dark as night, but still much darker than it usually is during the day.  It was enough to hear the sound of roosters crowing in the distance.  

But it was odd to look up at the sun and see it completely black in color.  I took a picture with my phone, but I guess since I didn't have any special photography equipment, it didn't capture the moment correctly at all, and the sun looks pretty much as bright as normal in the picture.  

Still, as short as the experience was, and as disappointing as my documentation of it was, the memory of the sun completely black in the sky with a glowing ring around it sticks with me.  Definitely a once in a lifetime experience.  I'll never see it again.

Why There's No Quintic Formula - The Easiest Explanation I Have So Far...

Here's currently the best explanation I have of why there is no general formula in radicals for the roots of a quintic polynomial. It's more for my quick reference than anything else. It may not be totally accurate but maybe it's close.

Vieta's formulas put the roots of a general polynomial equation with unspecified variable coefficients in a position to be swappable, but potentially roots nested inside other roots can pop up, and they can't be easily swapped if they're not conjugates. If a general quintic formula in radicals existed, it would violate this conjugate swapping structure.

I think the biggest pitfall I've gotten caught in is not realizing that the only requirement for SPECIFIC examples where we already know the coefficients is that they respect the nested conjugate rule and don't violate it. Their roots don't have to be all swappable with each other. But when we don't KNOW the coefficients and are dealing with a general variable-coefficient case, we have to allow all roots to be swappable because a specific polynomial may potentially have that property.

These ideas build on a video I uploaded a while back here: https://www.youtube.com/watch?v=qOHkF26EKfg