I was gonna give us all the day off today, honestly, I was! My Minnesota Twins start their second game in about an hour, and I really planned to just kick back, watch the game, have a couple of beers, and enjoy the day. And since tomorrow’s March wrap-up post is done and queued, more of the same tomorrow.

But this is too relevant to the posts just posted, and it’s about **Special Relativity**, which is a March thing to me (because **Einstein**), so it kinda *has* to go here. Now or never, so to speak. And it’ll be brief, I think. Just one more reason I’m so taken with matrix math recently; it’s providing all kinds of answers for me.

Last night I realized how to use matrix transforms on spacetime diagrams!

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Leave a comment | tags: Albert Einstein, Einstein, faster than light, frame of reference, light, light speed, light year, matrix math, matrix transform, simultaneity, spacetime, Special Relativity, speed of light | posted in Math, Physics

In the last installment I introduced the idea of a *transformation matrix* — a square matrix that we view as a set of (vertically written) vectors describing a new *basis* for a transformed space. Points in the original space have the same relationship to the original basis as points in the transformed space have to the transformed basis.

When we left off, I had just introduced the idea of a *rotation matrix*. Two immediate questions were: How do we create a rotation matrix, and how do we use it. (By extension, how do we create and use *any* matrix?)

This is where our story resumes…

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1 Comment | tags: 3Blue1Brown, column vector, linear algebra, matrix math, matrix multiplication, rotation, unit vector, vectors | posted in Math, Sideband

For me, the star attraction of March Mathness is **matrix rotation**. It’s a new toy (um, *tool*) for me that’s exciting on two levels: Firstly, it answers key questions I’ve had about rotation, especially with regard to **4D** (let alone **3D** or easy peasy **2D**). Secondly, I’ve never had a handle on matrix math, and thanks to an extraordinary YouTube channel, now I see it in a whole new light.

Literally (and I do mean “literally” literally), I will never look at a matrix the same way again. Knowing how to look at them changes everything. That they turned out to be exactly what I needed to understand rotation makes the whole thing kinda wondrous.

I’m going to try to provide an overview of what I learned and then point to a great set of YouTube videos if you want to learn, too. Continue reading

16 Comments | tags: 3Blue1Brown, column vector, complex numbers, linear algebra, matrix math, matrix multiplication, rotation, trigonometry, unit vector, vectors | posted in Math, Sideband

I’ve been hinting all month about **rotation**, and the time has finally come to dig into the topic. As mentioned, my interest began with wanting to understand what it means to rotate a **tesseract** — particularly what’s *really* going on in a common animation that I’ve seen. What’s the math there?

This interest in rotation is part of a larger interest: trying to wrap my head around the idea of a fourth *physical* dimension. (Time is sometimes called the fourth dimension, but not here.) To make it as easy as possible, for now I’m focusing only on tesseractae, because “squares” are an easy shape.

After chewing at this for a while (the tesseract post was late 2016), just recently new doors opened up, and I think this journey is almost over!

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2 Comments | tags: 2D, 3D, 3D space, 4D, coordinates, cube, dimensions, geometry, line, point, rectangular coordinates, right angle, rotation, square, tesseract | posted in Math

To start the last week of March Mathness, because it’s a Monday, I’m going to go easy on y’all with some light, easy topics. (Maybe I can lull you into paying attention for the major topic of the month: matrix rotation.)

It has occurred to me that, if I’m talking about math in March, I absolutely must mention one of my all-time favorite mathematical objects, the **Mandelbrot**. I’ll try to get to that today, but the main topic is a simple something that I ran into while working on my 3D model of the big island of Hawaii.

The question was: *How many miles are there per degree of ***latitude**?

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10 Comments | tags: circle of latitude, cosine, Earth, latitude, longitude, Mandelbrot, Maths Town, nautical mile, radius | posted in Math

Last week we celebrated Albert Einstein’s birthday (he turned 140). Now we need another cake so we can celebrate the *other* March major mathematician’s birthday — **Emmy Noether** turns 137 today.

To my regret, despite that I frequently invoke her name (she co-starred with Albert in the Special Relativity series), her work in mathematics is pretty far above my head, and I’m simply not qualified to write about it. I can say that her work connects mathematical symmetry with physical conservation laws. She also made significant contributions to abstract algebra.

Just recently, I’ve begun to nibble at the edges of the latter in the form of group theory as a part of studying rotation.

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20 Comments | tags: algebra, cube, Emmy Noether, group theory, mathematics, rotation, square, tesseract | posted in Math

Back at the start of March Mathness I promised the math would be “fun” (really!), but anyone would be forgiven for thinking the previous two posts about Special Relativity weren’t all that much “fun.” (I really enjoy stuff like that, so it’s fun for *me*, but there’s no question it’s not everyone’s cup of tea.)

Trying to reach for something a bit lighter and potentially more appealing as the promised “fun,” I present, for your dining and dancing pleasure, a trio of number games that anyone can play and which might just tug at the corners of your enjoyment.

We can start with **277777788888899** (and why it’s special).

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2 Comments | tags: 277777788888899, algorithm, calculation, Collatz conjecture, computation, Mandelbrot, mathematics, multiplicative persistence, Numberphile | posted in Math

In the March Mathness post I mentioned that one reason I love March is that it contains the **Vernal Equinox**, the official astronomical start of Spring. More importantly to me, it means six months of more daylight than darkness, and as much as I’m a night person, I prefer long, sunny days.

Well, today is the day! The equinox happened at **21:58 UTC** (two minutes before 5:00 PM locally). What’s better is that, after all the miserable bitter cold and all that snow in February and into March, the weather is indeed finally turning. Deeply embedded in our mythologies is the idea of spring rebirth; New Year’s parties aside, *this*, today, is the true new year.

And the forecast is for muon showers!

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17 Comments | tags: Albert Einstein, equinox, muon, Special Relativity, spring, spring equinox, tau, vernal equinox | posted in Physics

Speaking of **Special Relativity**, back when I wrote the SR series, one topic I left along the wayside was the concept of the *spacetime interval*. It wasn’t necessary for the goals of the series, and there’s only so much one can fit in. (And back then, the diagrams I wanted to make would have been a challenge with the tool I was using.)

But now that we’re basking in the warm, friendly glow of March Mathness and reflecting on Special Relativity *anyway*, it seems like a good time to loop back and catch up on the **spacetime interval**, because it’s an important concept in SR.

It concerns what is *invariant* to all observers when both time and space measurements depend on relative motion.

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1 Comment | tags: Albert Einstein, Einstein, faster than light, frame of reference, light, light speed, light year, simultaneity, spacetime, Special Relativity, speed of light | posted in Physics

Earlier, in the March Mathness post, I mentioned **Albert Einstein** was born on March 14th. That’s also Pi Day, which deserved its own *pi* post (about pizza *pi*), so old Al had to wait for me to address a topic I’ve needed to address for several months.

To wit: Some guy was wrong on the internet.

That guy was me.

Back in 2015 (also celebrating Einstein’s birthday), I wrote a series of posts exploring Special Relativity. Near the end of the series, writing **about FTL radio**, I said (assuming an “ansible” existed) I wasn’t convinced it violated causality *if the frames of reference were matched*.

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9 Comments | tags: Albert Einstein, ansible, causality, causality violation, Einstein, Emmy Noether, faster than light, frame of reference, FTL, FTL radio, light, light speed, light year, simultaneity, spacetime, Special Relativity, speed of light | posted in Physics