#
Category Archives: Math

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.

Continue reading

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).

Continue reading

28 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.

Continue reading

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

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.

Continue reading

5 Comments | tags: bra-ket notation, inner product, matrix multiplication, outer product, QM101, quantum mechanics | posted in Math, Physics

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…

Continue reading

15 Comments | tags: derivatives, equinox, sine wave, spring equinox, vernal equinox | posted in Life, Math

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.

Continue reading

4 Comments | tags: Bloch sphere, QM101, quantum computing, quantum mechanics, quantum spin | posted in Math, Physics

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.

Continue reading

35 Comments | tags: QM101, quantum mechanics, quantum spin | posted in Math, Physics

Unless one has a strong mathematical background, one new and perhaps puzzling concept in quantum mechanics is all the talk of *eigenvalues* and *eigenvectors*.

Making it even more confusing is that physicists tend to call eigenvectors *eigenstates* or *eigenfunctions*, and sometimes even refer to an *eigenbasis*.

So the obvious first question is, “What (or who) is an *eigen*?” (It turns out to be a what. In this case there was no famous physicist named Eigen.)

Continue reading

13 Comments | tags: eigenstate, eigenvalue, eigenvector, matrix transform, QM101, quantum mechanics | posted in Math, Physics

In quantum mechanics, one hears much talk about *operators*. The Wikipedia page for operators (a good page to know for those interested in QM) first has a section about operators in classical mechanics. The larger quantum section begins by saying: *“The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator.”*

Operators represent the *observables* of a quantum system. All measurable properties are represented mathematically by an operator.

But they’re a bit difficult to explain with plain words.

Continue reading

4 Comments | tags: QM101, quantum mechanics, quantum operator | posted in Math, Physics

**Trigonometry** is infamously something most normal people fear and loath. Or at least don’t understand and don’t particularly want to deal with. (In fairness, it doesn’t pop up much in regular life.) As with matrix math, trig often remains opaque even for those who do have a basic grasp of other parts of math.

Excellent and thorough tutorials exist for those interested in digging into either topic, but (as with matrix math) I thought a high-altitude flyover might be helpful in pointing out important concepts.

The irony, as it turns out, is that trig is actually pretty easy!

Continue reading

30 Comments | tags: cosine, sine, sine wave, trigonometry | posted in Math, Sideband