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Category Archives: Science

Last time we considered a cube-shaped room where we could indicate our opinion about Neapolitan ice cream with a single marker. That worked well because we were dealing with three flavors and the room has three dimensions: east-west, north-south, up-down.

Later I’ll explore other examples of a 3D “room” but while we’re talking ice cream, I want to give you an idea where this goes, I want to jump ahead for a moment and consider good old **Baskin-Robbins**, who famously featured “31 flavors!”

So now the question is, can we set a marker for all 31 flavors?

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4 Comments | tags: 3D space, configuration space, ice cream, multiple dimensions, Neapolitan ice cream, parameter space, phase space, vector space | posted in Basics, Science

Have you ever had (or at least seen) **Neapolitan ice cream**? It’s the kind with chocolate, vanilla, and strawberry, usually as separate layers in one package. As a kid, I didn’t care for the strawberry. I loved the chocolate, and was fine with the vanilla (wouldn’t usually *choose* it, but don’t disdain it).

That’s just my take on it: one flavor liked, one not liked, and one that’s just okay. Someone else might have the same pattern with different flavors. Or love them all equally, or want just the strawberry. Some might not like ice cream at all — any combination is possible.

What if we wanted to describe our feeling about Neapolitan *as a whole*?

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10 Comments | tags: 3D space, configuration space, ice cream, multiple dimensions, Neapolitan ice cream, parameter space, phase space, vector space | posted in Basics, Science

This is a Sideband to the previous post, **The 4th Dimension**. It’s for those who want to know more about the rotation discussed in that post, specifically with regard to axes *involved with* rotation versus axes *about which* rotation occurs.

The latter, rotation about (or around) an axis, is what we usually mean when we refer to a *rotation axis*. A key characteristic of such an axis is that coordinate values on that axis *don’t change* during rotation. Rotating about (or on or around) the **Y** axis means that the **Y** coordinate values never change.

In contrast, an axis *involved with* rotation changes its associated coordinate values according to the angle of rotation. The difference is starkly apparent when we look at rotation matrices.

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3 Comments | tags: 2D, 3D, 4D, column vector, matrix math, matrix transform, rotation, rotation matrix, unit vector, vectors | posted in Math, Sideband

Here on the 4^{th} day of the 4^{th} month, I feel I really should be writing about the 4^{th} dimension. I did say that I would during March Mathness, and I tried to set the math foundation here and here.

But two problems: Firstly, I’m kinda burned out. Those three posts were a bit of work, diagrams & models & math (oh, my!), and then trying to explain them clearly. Secondly, obviously no one finds this interesting except me, so not much motivation for the effort involved. Which was expected (kinda the story of my life). I also said these posts were as much recording my notes as attempts to share.

But it is 4/4 (and no Twins game today), so I thought I’d try winging it anyway.

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15 Comments | tags: 2D, 3D, 3D space, 4D, coordinates, cube, dimensions, square, tesseract | posted in Science

An old saying has it that *“March comes in like a lion and goes out like a lamb.”* That was certainly the case for us this year. February and early March were full-on old-fashioned winter, yet when baseball season started (in the USA) this past Thursday, the snow was mostly gone, and temps were in the 50s. (That’s the thing about winter: spring is pretty sweet.)

The end of March means the official end of the Mathness, but it’s not exactly the end of the math. The whole point of the rotation study was trying to understand 4D rotation, and I haven’t explored that, yet. I plan to, and soon.

But today, as an exit March, I want to talk about math phobia.

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9 Comments | tags: math anxiety, math phobia, mathematics | posted in Math

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