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Category Archives: Math

**Zeno’s famous Paradoxes** involve the impossibility of arriving somewhere as well as the impossibility of even *starting* to go somewhere. And that flying arrows have to be an illusion. [*Time flies like an arrow, but fruit flies like a banana.*]

If Zeno were alive today, he’d be over 2500 years old and would have seen his paradoxes explained in a variety of ways by a lot of very smart people. Yet at heart they still have some metaphysical *oomph*. And the thing is, at least in some contexts, Zeno was (sort of) right. There *is* something of a paradox here involving space and time.

Or at least something interesting to think about.

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7 Comments | tags: rational numbers, real numbers, Zeno of Ela, Zeno's paradoxes | posted in Brain Bubble, Math, Philosophy

It’s actually obvious and might fall under the *“Duh!”* heading for some, but it only recently sunk in on me that the *Born Rule* is really just another case of **the ***Pythagorean* theorem. The connection is in the way the coefficients of a quantum superposition, when squared, must sum to unity (one).

For that matter, Special Relativity, which is entirely geometric, is yet another example of the Pythagorean theorem, but that’s another story. (One I’ve already told. See: SR #X6: Moving at Light Speed)

The obvious connection is the geometry behind how a quantum state projects onto the basis eigenvectors axes.

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9 Comments | tags: geometry, Pythagoras, QM101, quantum mechanics, trigonometry | posted in Brain Bubble, Math, Physics

Last time I began exploring the similarity between the **Schrödinger equation** and a classical **heat diffusion equation**. In both cases, valid solutions push the *high curvature* parts of their respective functions towards flatness. The effect is generally an averaging out in whatever space the function occupies.

Both equations involve *partial* derivatives, and I ignored that in our simple one-dimensional case. Regular derivatives were sufficient. But math in two dimensions, let alone in three, requires partial derivatives.

Which were yet another hill I faced trying to understand physics math. If they are as opaque to you as they were to me, read on…

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10 Comments | tags: derivatives, exponential function, partial derivatives, QM101, Schrödinger Equation | posted in Math, Physics

This is the first of a series of posts exploring the mysterious Schrödinger Equation — a central player of quantum mechanics. Previous QM-101 posts have covered important foundational topics. Now it’s time to begin exploring that infamous, and perhaps intimidating, equation.

We’ll start with something similar, a classical equation that, among other things, governs how heat diffuses through a material. For simplicity, we’ll first consider a one-dimensional example — a thin metal rod. (Not truly one-dimensional, but reasonably close.)

Traveller’s Advisory: Math and graphs ahead!

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8 Comments | tags: derivatives, exponential function, QM101, quantum, Schrödinger Equation | posted in Math, Physics

I posted a while back about the wonders of Fourier Curves, and I’ve posted many times about Euler’s Formula and other graphical wonders of the complex plane. Recently, a **Numberphile** video introduced me to another graphical wonder: **Euler Spirals**. They’re one of those very simple ideas that results in almost infinite variety (because of chaos).

As it turned out, the video (videos, actually) led to a number of fun diversions that have kept me occupied recently. (Numberphile has inspired more than a few projects over the years. Cool ideas I just had to try for myself.)

This all has to do with virtual turtles.

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8 Comments | tags: geometry, Leonhard Euler, Numberphile, pi | posted in Math, Wednesday Wow

Last February I posted about how my friend Tina, who writes the **Diotima’s Ladder** blog, asked for some help with a set of diagrams for her novel. The intent was to illustrate an aspect of Plato’s Divided Line — an analogy about knowledge from his worldwide hit, the *Republic*. Specifically, to demonstrate that the middle two (of four) segments always have equal lengths.

The diagrams I ended up with outlined a process that works, but I was never entirely happy with the last steps. They depended on using a compass to repeat a length as well as on two points lining up — concrete requirements that depend on drawing accuracy.

Last week I had a lightbulb moment and realized I didn’t need them. Lurking right in front of my eyes is a solid proof that’s simple, clear, and fully abstract.

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2 Comments | tags: algebra, geometry, Plato | posted in Math, Philosophy

Recently my friend Tina, who writes the blog **Diotima’s Ladder**, asked me if I could help her with a diagram for her novel. (Apparently all the math posts I’ve written gave her ideas about my math and geometry skills!)

What she was looking for involved **Plato’s Divided Line**, an analogy from his runaway bestseller, the *Republic* (see her post *Plato’s Divided Line and Cave Allegory* for an explanation; I’m not going to go into it much here). The goal is a geometric diagram proving that the middle two segments (of four) must be equal in length.

This post explores and explains what I came up with.

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18 Comments | tags: algebra, geometry, Plato | posted in Math, Philosophy

I’ve always had a strong curiosity about how things work. My dad used to despair how I’d take things apart but rarely put them back together. My interest was inside — in understanding the mechanism. (The irony is that I began my corporate career arc as a hardware repair technician.)

My curiosity includes a love of discovery, especially unexpected ones, and *extra* especially ones I stumble on myself. It’s one thing to be taught a neat new thing, but a rare delight to figure it out for oneself. It’s like hitting a home run (or at least a base-clearing double).

Recently, I was delighted to discover something amazing about spheres.

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9 Comments | tags: derivatives, geometry, holographic principle | posted in Math, Sideband

*Flat Earth!*

To describe how space could be flat, finite, *and yet unbounded*, science writers sometimes use an analogy involving the surface of a **torus** (the mathematical abstraction of the doughnut shape). Such a surface has no boundary — no edge. And despite being embedded in three-dimensional space, the torus surface, if seen in terms of compensating surface metric, *is* indeed flat.

Yet a natural issue people have is that the three-dimensional embedding is clearly curved, not flat. It’s easy to see how wrapping a flat 2D sheet into a cylinder doesn’t distort it, but hard to see why wrapping a cylinder around a torus doesn’t stretch the outside and compress the inside.

In fact it does, but there are ways to eat our cake (doughnut).

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30 Comments | tags: 2D space, 3D space, cosmology, flat space, Flatland, space, torus | posted in Math

I’ve written a number of posts about **four-dimensional Euclidean space**, usually in the context of one of my favorite geometrical objects, the **tesseract**. I’ve also mentioned 4D Euclidean spaces as just one of many possible multi-dimensional parameter spaces. In both cases, the familiar 2D and 3D spaces generalize to additional dimensions.

This post explores a specialized 4D space that uses complex numbers along each axis of a 2D nominally Euclidean space. Each **X** & **Y** coordinate has two degrees of freedom, a *magnitude* and a *phase*. This doesn’t make 4D spaces easier to visualize, but it can offer a useful way to think about them.

It also connects back to something I wrote about in my QM-101 series.

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11 Comments | tags: 4D, complex numbers, complex plane | posted in Math, Sideband