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.
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.
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.
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.
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).
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.
When I was in high school, bras were of great interest to me — mostly in regards to trying to remove them from my girlfriends. That was my errant youth and it slightly tickles my sense of the absurd that they’ve once again become a topic of interest, although in this case it’s a whole other kind of bra.
These days it’s all about Paul Dirac’s useful Bra-Ket notation, which is used throughout quantum mechanics. I’ve used it a bit in this series, and I thought it was high time to dig into the details.
Understanding them is one of the many important steps to climb.
Today is the first Earth-Solar event of 2021 — the Vernal Equinox. It happened early in the USA: 5:37 AM on the east coast, 2:37 AM on the west coast. Here in Minnesota, it happened at 4:37 AM. It marks the first official day of Spring — time to switch from winter coats to lighter jackets!
Have you ever thought the Solstices seem more static than the Equinoxes? The Winter Solstice particularly, awaiting the sun’s return, does it seem like the change in sunrise and sunset time seems stalled?
If you have, you’re not wrong. Here’s why…
One small hill I had to climb involved the object I’ve been using as the header image in these posts. It’s called the Bloch sphere, and it depicts a two-level quantum system. It’s heavily used in quantum computing because qubits typically are two-level systems.
So is quantum spin, which I wrote about last time. The sphere idea dates back to 1892 when Henri Poincaré defined the Poincaré sphere to describe light polarization (which is the quantum spin of photons).
All in all, it’s a handy device for visualizing these quantum states.
Popular treatments of quantum mechanics often treat quantum spin lightly. It reminds me of the weak force, which science writers often mention only in passing as ‘related to radioactive decay’ (true enough). There’s an implication it’s too complicated to explain.
With quantum spin, the handwave is that it is ‘similar to classical angular momentum’ (similar to actual physical spinning objects), but different in mysterious quantum ways too complicated to explain.
Ironically, it’s one of the simpler quantum systems, mathematically.