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Category Archives: Math

Mandelbrot Antennae

*[click for big]*

I realized that, if I’m going to do the **Mandelbrot** in May, I’d better get a move on it. This ties to the main theme of Mind in May only in being about *computation* — but not about computation**alism** or consciousness. (Other than in the *subjective appreciation* of its sheer beauty.)
I’ve heard it called “the most complex” mathematical object, but that’s a hard title to earn, let alone hold. It’s complexity does have attractive and fascinating aspects, though. For most, its visceral visual beauty puts it miles ahead of the cool intellectual poetry of Euler’s Identity (both beauties live on the same block, though).

For me, the cool thing about the Mandelbrot is that it’s a computation that can never be fully computed.

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9 Comments | tags: computation, computer model, computer program, fractals, Mandelbrot, Mandelbrot fractal, mathematics, Turing Halting Problem | posted in Computers, Math

Previously, I wrote that I’m skeptical of interpretation as an analytic tool. In physical reality, generally speaking, I think there is a single correct interpretation (more of a *true account* than an interpretation). Every other interpretation is a fiction, usually made obvious by complexity and entropy.

I recently encountered an argument *for* interpretation that involved the truth table for the boolean logical **AND** being seen — if one inverts the **interpretation** of all the values — as the truth table for the logical **OR**.

It turns out to be a tautology. A logical **AND** *mirrors* a logical **OR**.

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8 Comments | tags: algorithm, AND gate, boolean logic, computation, computationalism, consciousness, human consciousness, interpretation, logic gate, NAND gate, NOR gate, OR gate, Theory of Consciousness, theory of mind, truth table | posted in Computers, Math, Philosophy

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

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 Sideband, Math

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