*Time* is the first of the *Manifold* trilogy (which has a fourth book, *Phase Space*); the second and third books are *Space* (2000) and *Origin* (2001). Each of the books tells a separate story in a separate universe.

I enjoyed the first book, but I can’t say I was hugely whelmed.

I did check out (in the library sense) *Space* and will check it out (in the giving it a look sense), but it’s possible I’ll put it down and move on. At this point I have a sense that Baxter might be a workhorse author, like Stephen King or Tom Clancy (or John Grisham or …).

That he cranked out the four volumes of this trilogy in four years while also putting out six other novels (and two non-fiction books) in that time says something.

He’s been writing since 1991, so 60 books (not to mention five non-fiction books) divided by 30 years is two books a year. Every year for three decades.

[Given that it’s really 65 books and 29 years, the actual number is 2.24 books per year, plus whatever else he does. That’s some impressive output.]

This might account for the *‘serviceable but not thrilling’* aspect of *Time*. If the rest of his work is this ‘serviceable’ I probably won’t be digging deeply into his catalog. (I may not even finish this trilogy given all the other stuff on my reading list.)

**§**

So my first reaction is that I didn’t find *Time* very compelling or engaging. The first part was better, but I ended up skimming pages in the last third of the book.

Which is a little surprising since I’m usually very forgiving when it comes to hard SF. I revel in the interesting — and often unique — ideas, so I’ll overlook plot and character issues.

I think one problem for me with this book is I didn’t find many new ideas. There were a lot of retreads, though. The novel invoked a number of other novels: Arthur Clarke’s *Rama* series as well as *2001*, or Greg Bear’s *Eon* series. Also of Jack Chalker’s *Well World* series. And David Brin’s *Uplift* series. Nothing really grabbed me as new.

A greater problem for me is scope. I just don’t find SF novels that span galaxies or eons very engaging. Very few, if any, authors can pull of the grandiose, and (for my money) Baxter doesn’t seem to be one of them.

[I always thought Asimov pulled it off in his *Foundation* series. A rare exception. Ironically, his second wife, Janet, wrote (as J.O. Jeppson) an SF novel that’s my canonical example of “scope blowup” silliness. It ends with intelligent galaxies.]

**§**

The *Manifold* series is grandiose in the extreme. I’ll get to that, but it will involve some spoilers.

The main character is Reid Malenfant, who struck me as an Elon Musk analogue. Baxter wrote this in 1999; Musk formed SpaceX in 2002, although he, along with his brother, started Zip2, an online city guide, in 1995. He and brother Kimbal started X.com, which went on to become PayPal, in 1999.

So I *think* the striking similarity between Reid Malenfant and Elon Musk is coincidence. It may be the prescient prediction that some very rich private person (or corporation) would try to do what NASA does.

And, for many reasons, do it better.

Certainly Malenfant (what a name) and Musk share the vision, the brashness, and the willingness to forge ahead no matter what. One of the conflicts in the story is between Malenfant and the government, especially NASA, which Baxter paints as bumbling and territorial.

Malenfant’s vision is to mine the asteroids (using whatever existing hardware he can cobble together to bootstrap the plan). The wealth contained there is so great, access to it would alter the global economy. Asteroids contain everything from water (more than in all Earth’s oceans) to organics to metals.

But Cornelius Taine, an eschatologist mathematician, also a major shareholder in Malenfant’s corporation, captures Malenfant’s interest by explaining the Doomsday argument. This ends up diverting Malenfant’s resources and efforts in an attempt to detect putative signals from the distant future.

The *really* distant future — a time beyond the evaporation of black holes — a time of the heat death of the universe. By then humanity has embedded itself in a “lossless computing substrate” that allows mind to continue, but (since all computational states will eventually be computed) which offers nothing new, ever.

Some portion of this mind has decided that things shouldn’t have turned out this way, so they seek to change the past so it didn’t.

**§**

One result is that Malenfant and Taine detect a message from the future. This message points them to 3753 Cruithne, an asteroid co-orbital with Earth.

Baxter has a section at the end where he explains that many of the ideas in the book are “real” — by which he mean real scientific *theories* (or *ideas*). Some cases, such as 3753 Cruithne, are completely factual. Others, such as messages from the future, are much less so.

Malenfant has had Dan Ystebo, a marine scientist, training genetically enhanced squid to fly the ships that will mine the asteroids. They’ve added intelligence to the squid. Part of the idea is that the squid can be sent on what amount to suicide missions to get the automated asteroid factories started.

Instead, they send the squid to Cruithne. They assume it’s a suicide mission to set up automated robot explorers, but they don’t realize the squid is pregnant. The story of the squid would have made an interesting novel, but they weirdly become side players.

Significant when the plots needs them to be but otherwise not really of any account in the story. Nor do we ever learn much about them beyond the story of Sheena 5, the first squid.

**§**

Explaining much more gets me into spoiler territory, although there really isn’t that much to spoil. Given the set up, the story goes where it has to.

One aspect is the Earth’s population becomes persuaded by the Doomsday argument and existential despair sets in. Which results in some people killing other people, mostly off-screen. (One guy blows up MIT.)

Then the confirmation that we have distant ancestors calms society down, at least until the rumors and conspiracy theories begin.

On top of all this, another consequence of manipulation by our distant ancestors is hyper-intelligent but nearly autistic children (the “blues”) start being born. They are alternately heralded and feared, and their story line is a distinct thread.

Suffice to say the children are instrumental in the finale.

Which involves the destruction of the entire universe by introducing a bubble of true vacuum. Thus ends the book.

In the meantime, the original squid, Sheena 5, and a robot, have gone through the portal found on Cruithne to make exponential jumps into the future to see what becomes of humanity.

Reid, his ex-wife Emma (a sub-plot of its own), and Cornelius, also pass through the portal, but they end up visiting many hundreds of different universes (most of which can’t support life).

In turns out that mind only evolved once, on one planet, in one universe. The rest of the multiverse is empty. The rest of this universe is empty. We’re all we ever find.

**§**

Apparently the next two books take place in two different universes and involve different stories. But Reid Malenfant is the main character in all three. (Based on the list of characters in the Wiki articles for the next two books, Reid is the only character that repeats.)

Supposedly each of the three has the central theme of answering the Fermi Paradox three different ways. In the case of *Time*, the answer is that we’re it. Intelligent life simply never evolved anywhere else.

I’m gonna give *Space* a chance, but with other books clamoring for my attention (including a five-book series Baxter did with Terry Pratchett), we’ll see how it goes.

In any event, now it’s Brin, Bear, Benford, and Baxter, on the list of SF authors who write hard SF and have last names that start with “B”. (Bracketed by Asimov and Clarke.) A cute curiosity. [Tying it together, Benford, Bear, and Brin, all wrote *Foundation* novels.]

*Stay inside reading, my friends!*

∇

]]>When I walk, I try to take a different path every day, only repeating when I’ve exhausted all possibilities. But yesterday I decided that today I’d retrace my steps and take pictures of those signs.

Without further ado, colored chalk wishes to help us smile:

These are in the order encountered from the park entrance:

The messages occur every 50 feet or so, shorter distances at first and longer (and less ornate) as the author got tired or ran out of messages (as you see, there are no repeats).

They made me smile as much today as they did yesterday.

A friend of mine mentioned she’s seen it in her neighborhood, too, so I think maybe it’s going around.

It’s a nice idea. I just wish I’d found some colored chalk of my own.

**§**

So thanks to the local artist who contributed these smiles. Just what we need!

It’s supposed to rain tonight and all day tomorrow, which is why I made a point of retracing my steps today with a camera. Now her work is immortalized. (It is a her. She did sign her work at one point, but I’m assuming she’s young, so I’m protecting her privacy.)

Near the end of my walk, far from the park, on a freeway overpass I found some more chalk messages from (I think) the same artist. I’ll leave you with those:

Hard to see in the bright sunlight, but a good message, I think. (That’s my shadow on the lower right.)

*Stay positive, my friends!*

∇

]]>Spirits seem high around here. On my morning walk, in the park I saw that someone had used colored chalk to write good thoughts on the asphalt path: *“Stay Positive!”* *“Nature!”* *“Yay! Vit. D.”* *“Family Time”* *“Exercise!”* (Maybe others will join in. I think I have some colored chalk…)

It’s hard to top the real life wows, but I do have a few interesting items that might at least offer something of a distraction.

You may have heard of the Platonic solids — five three-dimensional shapes called out by one of the more famous of those ancient Greeks.

They are the only 3D shapes comprised of:

- Identical faces (2D polygons)
- Identical angles and sides in each face
- Identical angles and edges in each vertex

Another word for all those identical things is *regular*. These five solids are the only regular 3D shapes.

Since they go back to Plato, one would think they’ve been studied in great detail. One would be correct to think so.

Given the field is so ancient and well-plowed, one might also think there is nothing new to find. On *that*, one would be ** in**correct…

I love that something so simple and — one, *ahem*, might think — so thoroughly understood still held a mystery that took so long to solve.

I also love how computer animation lets us visualize math like this. When it’s done well (as it is here) it also makes the math a lot of fun.

[If you find the path animations fascinating, there’s a bonus video that consists of just animations.]

Don’t let “math” put you off. This is *geometry*, which is way more fun (it’s hardly even “math”).

**§**

Speaking of April Fool’s, there was some excitement last December about a possible new particle or new force.

Just last February, Sixty Symbols put out this video about it:

As the video mentions, these scientists first announced this discovery back in 2016. The December announcement involves a second, different, experiment by the same group.

This video gives a very good overview and explanation of what the experiment was and what (they claim) it implies.

On the one hand, two experiments seem to find the same result. (But two experiments by the same group at the same lab, so there’s that.)

On the other hand, this new particle is only ∼17 MeV (million electron volts), so it’s extremely light and should have been noticed long ago. And there is no evidence of it anywhere except from those two experiments.

So scientists are skeptical for now, and I suspect everyone involved has more important things on their mind right now. As the video suggests, what’s really needed is a third experiment, preferably by a different group. Science is all about repeatability.

[Near the end of the video he mentions how the mass of the proton and neutron isn’t from the mass of the quarks. I’ve been thinking about how actual mass (due to Higgs interaction) is just ∼1% of the mass of atoms, and atoms are just ∼15% of all matter, and matter is just ∼30% of the total mass of the universe. So *actual* mass really isn’t much of a player in the universe.]

**§**

If you’re interested in computer modeling of reality, the Two Minute Papers channel has a lot of good videos. Here’s one about the dynamics of simulating water:

Simulated water may not be wet, but it’s starting to look awfully realistic!

That realism is a bullet point in the Virtual Reality hypothesis. If we were *in* that water simulation, we would find it (simulated) wet. The idea in the VR hypothesis is that, given how good simulations are *now*, just imagine what they’ll be like down the line. A Matrix scenario isn’t unthinkable.

These videos are pretty cool (and nicely short), so here’s another one:

It’s a good illustration of a big point I was making in all those posts about creating virtual realities: Every aspect of the putative reality is up to the code. Every aspect of a virtual reality is arbitrary.

**§**

As a bonus for techno-geeks like me, here’s a tour of the inside of a nuclear reactor (that was built but never turned on):

All I can say to that is: “Wow!”

**§**

I’ll leave you with a very good piece of advice I saw mentioned recently: During these stay-at-home times, be sure to drive your car.

Depending on the age of your battery, you should either invest in a device to keep it healthy or just be sure to go for a drive once a week or so. (I knew a guy who began working from home regularly and was quite surprised when his battery turned up flat.)

Short drives (say to pick up food locally) not only won’t help, they’ll hurt by draining your battery each start. If you’re making a lot of little trips, the need for a long drive once in a while is more important.

Going for a drive keeps you isolated and gets you outside for a change of scenery and maybe some fresh air. (But you’ve all been going for walks, right?)

*Stay distanced, my friends!*

∇

]]>But there are definitely exceptions. Some horror stories — usually comedies or parodies — manage to find a new spin on old tropes. When it comes to storytelling, I am a big fan of new spins, almost regardless of genre.

Which is why I really enjoyed ** The Cabin in the Woods**.

This is a 2012 movie, so I am rather late to the game. (It predates my retirement!) It generated *a lot* of critical regard when it came out (Roger Ebert gave it three-of-four stars; Rotten Tomatoes gives it a 91%/75% rating).

It’s always been on my list of movies to someday see, so I was glad to see it show up on Hulu recently.

I gotta give this a **Wow!** rating. I *definitely* want to watch this again.

If you haven’t seen it and plan to, don’t read the Wiki page, because it’ll spoil the plot. What seems like a straightforward twist has more to it than at first appears.

The setup is traditional: Five young friends go for a vacation in the woods to a cabin that one of them just inherited (or bought? I forget). There is even the traditional weird threatening hillbilly running the gas station they stop at for gas and directions to the cabin.

But as they pass through a small tunnel (that will later be blocked!), we see a force field switch on locking them into the area (a bird crashes into it and vaporizes)…

So right away we learn this isn’t the usual slasher in the woods story. This involves a group of people using technology (ala NASA launch control center) to run some sort of … horror experiment?

The thing is, that’s not a spoiler. It’s what we learn right off. The secret is what’s behind it all.

**§**

The film was written by Joss Whedon (of *Buffy* and *Firefly* fame) and Drew Goddard (who has also done stuff).

Whedon produced, and Goddard directed, so the whole thing is mainly a collaboration between two people. I think that adds clarity and vision to a production. One thing that makes the *Resident Evil* movies so engaging (to me, anyway) is that they are singular visions by Paul W.S. Anderson (they’re also a definite exception to my not caring for horror movies — I love those).

This likewise applies to the *Pitch Black* films (another exception) by David Twohy. The later films aren’t quite as good, but *Pitch Black* is a classic. (Another rather horrific auteur I really like is Quentin Tarantino.)

There is something extra that comes from both writing and directing a film.

**§**

The *Resident Evil* and *Pitch Black* film series both fall into the science fiction horror category, for which (as an SF fan) I’m far more disposed.

I really like the *Alien* series, for example. I’m even okay with the recent entries and crossovers, although some of it does tweak my “this is silly” bone. And I think the original *Predator* film is one of Arnold’s best.

I thought *Event Horizon* (1997) was silly the first time I saw it, but watching it again I kind of enjoyed it. Even bought the DVD when I saw it in the $5 bin. (Now it’s on Hulu or Netflix, I think.) It was directed (but not written) by Paul W.S. Anderson.

Another not bad SF horror movie is *Supernova* (2000), which features a very young James Spader and Robin Tunney.

I’m a bit more inclined towards ghost stories. Monsters and slashers don’t strike me as that interesting, but ghosts do. Maybe because *13 Ghosts* (1960) really scared the crap out of me as a kid.

Even as an adult, *Ju-on* (2000) and *The Ring* (2002) gave me goosebumps.

**§**

On the other hand, I love a good comedy, so I tend to like good comedy horror (not being bothered by the gore).

I really enjoyed the *Scary Movies* film series, for instance. (Although, I didn’t realize there were five. I bought the first four, though.)

The *Scream* series was kinda cute, at least at first. I enjoy deconstruction, too. That’s a big part of *The Cabin in the Woods*.

There was a TV series, *Scream Queens* (2015), that was kind of cute. (But second season had sequel stank.)

Speaking of comedy and Chucky the horror doll, I saw a perfectly on the nose skit from *Astronomy Club* in which a man is being treated as a hero for having defeated a horror doll (ala Chucky). The man didn’t understand why he was being treated as a hero. Man versus doll. He just kicked it across the room and that was it. No biggie.

One of the best comedy horror movies I ever saw was *Tucker & Dale vs. Evil* (2010). Talk about deconstruction and role reversal. I highly recommend it.

Best comedy zombie movie has to be *Shaun of the Dead* (2004), but *Cockneys vs Zombies* (2012) gives it a bit of a run for the money. *Fido* (2006) is also pretty good. All three are must-see for zombie fans.

The current Walking Dead and general zombie craze owes a lot to those films.

**§**

As I say, horror isn’t really my genre, but I really got a kick out of *The Cabin in the Woods* and thought I’d recommend it for all those of us house-bound.

Thought I’d also mention some other movies you might find diverting depending on your tastes.

I’ll try to think of some other out-of-the-way movies and books to explore (check those links for previous “reviews” of might-be-interesting stuff).

*Stay horrified, my friends!*

∇

]]>This is one of those places where something that seems complicated turns out to have a fairly simple (and kinda cool) way to see it when approached the right way. In this case, it’s the way multiplication *rotates* points on the complex plane. This allow us to actually visualize certain equations.

With that, we’re ready to move on to the “heart” of the matter…

To begin with, here’s what inspired these posts:

Don’t worry, it’s not as bad as it looks (although I admit I tip-toed around it myself). It turns out there’s a way of thinking about it visually that makes it a lot less mysterious.

What the formula describes is a **cardioid curve** drawn on the complex plane. When graphed, it looks like this:

That red curve happens to also be the boundary of the main cardioid of the Mandelbrot. All points inside the curve are guaranteed to be in the Mandelbrot set. (Which is why I was interested in all this. Normally these points have to be calculated to the max iteration limit to show they’re in the set. That takes time.)

In the equation above, the two variables, ** c** and

[The *unit circle* is just a circle with a radius of one centered on the origin. It is related to the *unit square* mentioned last time in that points inside operate differently from points outside.]

We can’t compute with infinite sets, so how this actually works is that, as explained in the last post, we generate a bunch of points for ** u** — the more we generate, the finer the resolution (the smoother the curve) of the cardioid we’ll create.

We might, for example, generate a point for every degree around the circle. That gives us 360 points (complex numbers). Then we execute the formula above with each point, which gives us 360 results — 360 points that are in the cardioid.

We can do this with more or fewer points, depending on how smooth we want the cardioid curve to be. (The graph above was done with 360 points along each curve.)

**§**

None of which involves the *simple* way to see what’s happening.

It describes the algorithmic process of generating a cardioid curve, but it doesn’t provide much sense of how or why the curve emerges.

To see that we need to think visually about what the formula *means*.

**§ §**

We start with the set of infinite points represented by ** u** — the unit circle.

When we see ** u** in the equation, we visualize a circle with a radius of one centered on the origin. Then we ask what the equation does to that circle.

The first thing that happens is that ** u**, in both cases, is divided by

One logical result is to make the circle bigger or smaller. Dividing is inverse multiplying, so it would likewise affect the circle’s size.

Therefore, ** u**/

The graph above shows the half-size circle in gray.

**§**

Now we can consider the part inside the parentheses. What does *one-minus-a-half-circle* mean?

We can again think about this in a different form: what about *adding* one to the circle? What would that do?

There’s a fine point in the section above I skipped past. Previously I said that multiplying points on the complex plane *rotates* them, but here I said multiplying the point of the unit circle changes the *size*.

Recall that, in addition to rotating, multiplying also changes the distance from the center (“magnitude”) unless one (or both) of the points has a magnitude of **1.0**. By definition, all points in the unit circle do have a magnitude of **1.0** — that’s what the unit circle *is*.

When we divide by **2**, what we’re really dividing by is the complex number (**+2**, **0 i**), which lies on the positive

Likewise, when we add one, we’re really adding the complex number (**+1**, **0 i**). Addition is member-wise, so this just adds

*Subtracting* the circle from (**+1.0**, **0 i**) has a slightly different effect. Think about what we get with four key points (the four cardinal points):

- “12 o’clock”: (
**+1.0**,**0.0**) – (*i***0.0**,**+0.5**) = (*i***+1.0**,**-0.5**)*i* - “3 o’clock”: (
**+1.0**,**0.0**) – (*i***+0.5**,**0.0**) = (*i***+0.5**,**0.0**)*i* - “6 o’clock”: (
**+1.0**,**0.0**) – (*i***0.0**,**-0.5**) = (*i***+1.0**,**+0.5**)*i* - “9 o’clock”: (
**+1.0**,**0.0**) – (*i***-0.5**,**0,0**) = (*i***+1.5**,**0.0**)*i*

The right edge of the circle stays where it is, the center moves right to **1.0**, and the left edge moves from **-0.5** all the way right to **+1.5**. Also, the top is flipped to the bottom and vice versa.

The circle has again moved one unit to the right, and it has been reversed both horizontally and vertically.

On the gray half-size circle (also on the light blue unit circle), the point at “3 o’clock” is the *start* of the circle, the point with the angle zero. The angle of the circle increases moving counter-clockwise, so “12 o’clock” is 90°, “9 o’clock” is 180°, and “6 o’clock” is 270°.

The red circle is reversed, so its zero angle is at its “9 o’clock”. The angle still increases in the counter-clockwise direction, so “6 o’clock” is 90° and so on.

Remember that the zero-angle point of both circles is the same — it’s where the two circles touch.

So our equation amounts to a half-size circle multiplied by another half-size circle that’s been reflected to the right.

All we have to do now is figure out what it means to multiply one circle by another.

**§**

Previously we modified the circle with a fixed quantity — dividing by two or adding one. Now we’re multiplying two circles. What does that even mean?

It means we multiply points in one by the *matching* points in the other. How do we match up points to multiply? By their angle. We multiply together points with the same angle.

What makes it a little interesting is that, while the points on the centered half-size circle (gray) have the angles just described, since the reflected circle (red) is no longer centered, the angle *to* its points is different:

The diagram above shows the angle to eight points around the reflected circle. Angle is always from the origin. (So is magnitude.)

Remember that the red circle “starts” at the “9 o’clock” position and goes counter-clockwise from there. So the *point angle* slowly increases in the *negative* direction and then returns to zero at the “3 o’clock” position.

The angle to the points on the upper half of the circle is positive, again slowly increasing and then returning to zero.

Note that when determining which points match, we use the original angles as first described above. But when multiplying points, we consider their actual angles as described here.

Note also the lengths (magnitudes) of the red lines. Roughly half of them have a magnitude less than **1.0** and half of them are greater. The ones that are less will shrink the magnitudes of points we multiply, and the ones that are greater will increase the magnitude. This is part of where the shape of the cardioid comes from.

The idea is the same as described for generating points to draw the cardioid. Conceptually infinite points, but we pick as many actual points as makes sense for our need.

It turns out that when we multiply these points, the resulting set of points describe a cardioid curve.

**§ §**

Remember that multiplying one point by another creates a new point that sums the angles and multiplies the magnitudes.

On the centered half-size circle (gray) the magnitude is always **0.5**, so one part of that multiplication is one-half times something. The other part, the reflected circle (red), as we saw above, offers magnitudes that change from **0.5** to **1.5** and back.

This immediately gives us some sense that the resulting curve will be lopsided, starting with a smaller magnitude that grows to maximum at the 180° point and then gets smaller. Exactly like a cardioid.

We can also see that the angle of the cardioid points is retarded in the first half (by the negative angle to the red circle) and advanced (by the positive angle) in the second half. These effects on magnitude and angle give us the cardioid curve.

**§**

Some specific cases to illustrate how we do this:

The first points are where the two half-sized circles touch, the (original) zero angle. The point is: [**+0.5**, **0 i**].

Both points have zero angle, both have magnitudes of **0.5**. Multiplying them rotates by zero degrees. Multiplying the magnitudes gives us **0.25**, so the new point is [**+0.25**, **0 i**]. And, sure enough, that’s the “widow’s peak” of the cardioid.

To keep things easy, if we jump to the opposite side of both circles, the first point is [**-0.5**, **0 i**] (with an angle of 180°) and the second point is [

We’re multiplying a magnitude of **-0.5** times **+1.5**, which gives us **-0.75**, so the resulting point, halfway around the cardioid, is at [**-0.75**, **0 i**] (which turns out to be a special point in the Mandelbrot; more on that another time).

Now a slightly harder one, the “12 o’clock” point on the first circle matches the “6 o’clock” point on the second. The first point has an angle of 90° and the other (from the origin to the bottom of the red circle) has an angle of just over **-26°** and a magnitude just under **1.12**.

So the new point is rotated backwards (clockwise) by **-26°** and has a magnitude of about **0.56**. Here’s what that looks like:

The purple point is the point on the cardioid curve. Its angle is the the 90° of the blue point plus the negative angle of the red point.

The magnitude of the red point is a bit greater than **1.0**, so the purple point has a larger magnitude than the blue point. You can see it’s just beyond the half-size circle boundary.

**§**

This turned out to be harder to explain than I’d anticipated, so I made an animation to show the process:

Note how the purple point is always between the blue and red points (because it combines their angles).

All-in-all, I thought it was an interesting illustration of the geometric nature of the complex plane. It’s also a good example of how, at least sometimes, we can think about math visually.

BTW: If you look at the Wiki page for cardioid, you’ll find there are many ways to generate this curve. It is, in some regards, a fundamental emergent structure.

*Stay heartful, my friends!*

∇

]]>Which, in turn, is a big stepping stone to a fun fact about the Mandelbrot I want to write about. (But we all have to get there, first.) I think it’s a worthwhile journey — understanding the complex plane opens the door to more than just the Mandelbrot. (For instance, Euler’s beautiful “sonnet” also lives on the complex plane.)

As it turns out, the complex numbers cause this plane to “fly” a little bit differently than the regular X-Y plane does.

The two do have a lot in common, but the magic of ** i** changes the math we do on points. (Mainly the multiplication of complex numbers.)

Recall that a complex number has the form:

+abi

Where ** a** and

Since the two terms aren’t compatible (a plain number and one with an imaginary unit), we can’t carry out the addition. The two parts are forever separate. This makes complex numbers *compound* numbers.

Note that, if ** b**=

**§**

That the real and imaginary parts ** a** and

That’s all the complex plane is, an X-Y graph in which we use a complex number’s real component, ** a**, as

In the graph above, the blue points and numbers on the horizontal *x*-axis call out the integer points on the real number line. The red points and numbers on the vertical *y*-axis call out the integer multiples of ** i**. The purple points and numbers show some combined real and imaginary values.

The “unit square” at the center separates an important behavior. The value **1.0** tends to be hugely significant in math — it’s the foundation of the integers and, more significantly here, the multiplicative identity. Points inside the square, when multiplied, tend to get closer to the origin (**0**,**0 i**), and points outside, when multiplied, tend to get further from it.

The same basic behavior occurs with real numbers. Multiplying two numbers, both above **1.0**, results in a number *larger* than either. Multiplying two numbers, both less than **1.0**, results in a number *smaller* than either.

On a graph, *smaller* usually means closer to the center, and *larger* usually means further away. A multi-dimension graph provides the notation of *magnitude* — the distance from the center.

Note that when *drawing* the graph we ignore ** i** and use only the

**§ §**

Let’s take a look at that math magic.

I’ll start by reversing the order of the real and imaginary components *and* swapping the letters:

+aib

This doesn’t change anything. It’s still the same (complex) value. I want the imaginary part first, but I still want the letters to read from left to right just because that’s what we’re used to.

(Remember that what we use for ** ‘a’** and

The reason for all the swapping is to highlight how the equation looks a lot like a low-order polynomial:

+ax(=bax^{1}+bx^{0})

Which suggests that we treat complex numbers as polynomials when it comes to doing math with them. This is handy, because we already know how to do math that way — it’s basic algebra.

**§**

Adding the two complex numbers…

+aib

+cid

…(where ** a**,

(

+ai)+(b+ci)d=(+ai)+(ci+b)d=(+a)c+(i+b)d

The real numbers, ** a**+

Multiplying those two complex numbers is where it gets a little more interesting. We’re multiplying polynomials, which uses the *inside-outside-left-right* protocol from high school algebra class.

It ultimately boils down to this:

(

+ai) × (b+ci)d=(+bc)ad+ (i)ac+i^{2}bd

And since *i*^{2} = **-1**, we can reduce it to:

(

+bc)ad+(i(-1)+ac)bd=(+bc)ad+(i–bd)ac

Again, once we do the math on the real numbers (** a**,

If this section made your eyes glaze, that’s okay. The only important part is this:

(

+ai) × (b+ci)d=(+bc)ad+(i–bd)ac

Multiplying two complex numbers together results in a certain specific result (which is a new complex number). As it turns out, this multiplication has certain interesting properties I’ll come back to.

**§**

Compare this with how we might think to multiply two ordinary X-Y coordinates.

In this case, both ** x** and

[

,x] × [y,u]v=[,xu]yv

Which is entirely legit. But it obviously delivers different results than we get with complex number multiplication.

To make the comparison apples-to-apples, here is the complex number — *treated as a graph coordinate* — multiplication result:

[

,x] × [yi,u]vi=[(–xu), (yv+xv)yu]i

Note that we’re back to the original (** a**,

Remember that the ** i** in the equation just helps distinguish the real and imaginary parts and reminds us we’re working with complex numbers.

This different math matters because of what happens when we multiply points on the complex plane. A simple illustration is shown to the left.

The red and blue points are the two points we’re going to multiply. The red point is at [+3.2, +1.2** i**] and the blue point is at [+2.5, +1.9

The black point shows what we get if we multiply them simply as ordinary [** x**,

[3.2,1.2] × [2.5, 1.9] = [3.2×2.5, 1.2×1.9]

=[8,2.28]

The purple point shows the complex multiplication result:

[(3.2×2.5)-(1.2×1.9), ((3.2×1.9)+(1.2×2.5))

]i=[5.72,9.08]i

So the points end up in different places, but something a bit more important also happens.

**§**

If we look at the lines (vectors) drawn from origin to the points, those lines all have an angle with respect to the x-axis.

What might not be obvious (but is true) is that the angle of the purple line is the sum of the angles of the blue and red lines.

We can say either that the purple point is the result of rotating the blue point by the red point’s angle, or the result of rotating the red point by the blue point’s angle. It amounts to the same thing.

*The crucial point is that multiplying complex points results in rotation.*

Both the purple and black points are further away from the center because both the red and blue points are outside the unit square (the upper-right quadrant of which is the single square in the lower left).

**§ §**

This rotation aspect gives us a new way to find points on the unit circle.

We start with an easy point, the one at [**+1**, **0 i**]. It sits right on the

Then we decide how many points around the circle we want. The easiest approach is to pick a small angle and find a point for each increment of that angle around the circle. For instance, if we pick an angle of 1° there will be 360 points around the circle.

Now we find the point for that increment angle. That’s simply:

[

cos(θ) ,sin(θ)]=[cos(1°),sin(1°)]=[+0.9998,+0.0175]i

We could have gotten the first point this way, too:

[

cos(0°),sin(0°)]=[+1.00,0.00]i

But since it’s a “well-known” point, we didn’t bother.

Both of these are points on the unit circle, so by definition their magnitude is **1.0**. Therefore, multiplying them will keep that magnitude constant.

If we multiply the first point (call it * U* for Unit circle) by the second point (

[Recall how this works: We are either rotating * U* by

Now we multiply ** U** by

We continue to multiply * U* by

If we want more points, we pick a smaller angle. A larger angle gives us fewer points. Below is an example that uses **3°** as the angle, which results in 120 points around the circle (360/3 = 120).

Included in the diagram is a line (in red) to the **R**(otation) point illustrating the 3° angle. The original * U* point is the one directly below it.

As you can see, each point is spaced by that angle. Each time we multiply a given point, the new point is rotated 3° counterclockwise.

**§ §**

That’s enough (possibly, for some, *more than*) for this time.

But now we finally have the groundwork to explore the heart of the Mandelbrot.

*Stay on the plane, my friends!*

∇

]]>This is a little detour before the main event. The first post of this series, which explained why the imaginary unit, ** i**, is important to math, was long enough; I didn’t want to make it longer. However there is a simple visual way of illustrating exactly why it seems, at least initially, that the original premise isn’t right.

There is also a visual way to illustrate the solution, but it requires four dimensions to display. Three dimensions can get us there if we use some creative color shading, but we’re still stuck displaying it on a two-dimensional screen, so it’ll take a little imagination on our part.

And while the solution might not be super obvious, the *problem* sure is.

Let’s review the original premise. It involves equations like this:

The premise is that — so long as ** a** isn’t zero — there is always some value for

**§**

Let’s start by simplifying things down to the basics. If we set ** a**=

And, of course, if ** x**=

We see this visually by looking at a graph for that equation. It traces out a parabola (in fact, *“x-squared”* is one definition of a parabola):

We can see that the curve touches the *x*-axis at ** x**=

What the premise is essentially saying is that all polynomial curves touch the *x*-axis at some point. There is always some point along ** x** where the equation is equal to zero. (In some cases, as you’ll see next, multiple points.)

**§**

Let’s look at the curve for another example. Let’s leave ** a**=

Then we get:

If we graph that equation we get:

As you see, setting ** b**=

Sure enough, the values ** x**=

Setting ** b** to a negative value pulls the parabola down, which guarantees the curve crosses the

Changing the value of ** a** just makes the parabola narrower (if

The point is, changing the value of ** a** doesn’t shift the parabola up or down. Only the value of

By the way: In the first post, I set ** b**=

In that case, the solution is ** x**=

That’s where that curve crosses the *x*-axis when we pull it down that far.

I used **4** that time because it’s the square of **2**. (I wanted a whole number answer.) To have the curve cross the x-axis at **±3**, we would set ** b**=

The relationship is pretty obvious. The distance from the vertical center of the parabola to where it crosses the *x*-axis is the square root of the distance the parabola reaches below the *x*-axis.

I used ** b**=

**§**

Finally, let’s look at what happens if we (still leaving ** a**=

Now we have:

If you recall, we ran into trouble when both ** a** and

Based on what we’ve seen about the value of ** b**, we might imagine this pulls the curve upwards. Now the graph looks like this:

And it doesn’t appear to touch the *x*-axis anywhere!

This, then, is the visual representation of the trouble we ran into before. It initially seemed there is no value of ** x** that satisfies the equation. Here it seems the line never touches the

It’s the same situation seen two different ways.

(I again used **+1** here rather than **+4** to keep the curve close to the center. Had I used **+4**, the lowest part of the curve would have been way up at **+4**.)

**§ §**

This is the point where ** i** steps in to save the day. By moving our parabola into the domain of complex numbers, we can satisfy the equation.

If you recall, for the equation with ** b**=

The problem illustrating this visually is that it requires four dimensions.

The graphs above show the behavior of the real numbers, which are embedded in the complex numbers (as the real number line — the *x*-axis itself).

The graphs can show both the input (on the *x*-axis) and output (on the *y*-axis). That’s the advantage of those graphs — visualizing how the output changes with different inputs.

If we expand our scope to the complex numbers, then in an equation like:

=yfunction()x

Both the input, ** x**, and the output,

We can use three-dimensional rendering to display one of those as the *z*-axis, but that still leaves one we can’t display.

**§**

We can sneak up on the problem by looking at this table of results:

x |
-2i |
-1i |
0i |
1i |
2i |
---|---|---|---|---|---|

-5.0 | +21, +20i | +24, +10i | +25, +0i | +24, -10i | +21, -20i |

-4.0 | +12, +16i | +15, +8i | +16, +0i | +15, -8i | +12, -16i |

-3.0 | +5, +12i | +8, +6i | +9, +0i | +8, -6i | +5, -12i |

-2.0 | +0, +8i | +3, +4i | +4, +0i | +3, -4i | +0, -8i |

-1.0 | -3, +4i | +0, +2i | +1, +0i | +0, -2i | -3, -4i |

0.0 | -4, +0i | -1, +0i | +0, +0i | -1, +0i | -4, +0i |

+1.0 | -3, -4i | +0, -2i | +1, +0i | +0, +2i | -3, +4i |

+2.0 | +0, -8i | +3, -4i | +4, +0i | +3, +4i | +0, +8i |

+3.0 | +5, -12i | +8, -6i | +9, +0i | +8, +6i | +5, +12i |

+4.0 | +12, -16i | +15, -8i | +16, +0i | +15, +8i | +12, +16i |

+5.0 | +21, -20i | +24, -10i | +25, +0i | +24, +10i | +21, +20i |

The table shows the result of ** x**-squared for integer values of

The situation we’ve been considering is essentially the blue column down the middle — where ** yi**=

In that column, ** yi** is always zero in the result, because we’re operating strictly on the real number line. The real value, as expected, is

In the other columns, the squaring behavior of the real part is the same, but notice how the ** x** values move downwards by the square of

What’s different is that ** +yi** and

The more crucial change is what happens to the imaginary part of the result. It’s only zero when ** x**=

**§**

That got a bit complicated and what do all those numbers mean, anyway?

Visually, as ** yi** gets larger, both positively and negatively, two things happen:

Firstly, the curve is pulled downwards on the *y*-axis. The amount it moves is the square of ** yi**.

Secondly, the parabolic curve, which used to lie completely in the 2D plane of the real numbers, starts to twist. The part of the curve with positive ** x** goes one way (up or down into the third dimension) while part of the curve with negative

The curve itself remains flat, but it twists like a propeller. The direction of twist depends on whether (result) ** yi** is positive or negative.

Here’s one way we can try to render this:

The plot above shows three parabolic curves, each with different imaginary values for input. The black curve is (** x**,

The vertical *z*-axis is the imaginary value (** yi**) of the result (the second value in the table data — the first value, the real part, is the

Note how the blue curve starts down at **-20** for ** x**=

The red curve does the opposite and matches the left-most column of the table. The middle level, where the black curve floats, is where (output) ** yi**=

These curves, by the way, are for the original equation from the first post:

The blue and red curves above use **+2 i** and

What’s going on is a little more apparent when viewed from above:

Here we can see how the black curve never reaches below **+4**. Without complex numbers there is no solution.

But the blue and red curves are pulled downwards. In fact they touch the *x*-axis illustrating that, at least in complex space, there is a solution.

We can also view the plot from the side:

This view shows how the blue and red curves twist. The plane containing the black curve is the real number ** xy** plane.

**§**

I’ll leave you with one more 3D graph. This one plots curves for multiple values of ** yi** (more than the five shown on the table above):

It shows how ** i** twists the curve into complex space and pulls it down the

The curves are black when ** i** is zero or very low. They turn blue as

*Stay parabolic, my friends!*

∇

]]>This post is kind of an origin story. It seeks to explain why something rather mind-bending — the so-called *“imaginary numbers”* — are actually vital members of the mathematical family despite being based on what seems an impossibility.

The truth is, math would be a bit stuck without them.

Suppose I told you that equations such as the one below *always* have a value for ** x** that satisfies the equation (that makes it true)?

In the equation, ** a** and

If ** a** is zero, the first term is always zero, and it doesn’t matter what

So when ** a** is zero, the premise doesn’t apply because

But when ** a** is

**§ §**

Let’s start with a simple example: ** a**=

When ** x** is

In fact, when ** b** is zero,

(In a way, this is the exception to the exception about ** a** being non-zero — the equation is always satisfied when

**§**

When ** b** (but not

For example, suppose: ** a**=

Which means that ** x** has to be

In fact, in this case there are two answers, **+2** and **-2**, because squaring either gives us **+4**.

If ** a** is something other than

The premise, so far, seems to hold.

**§**

We can also satisfy the equation when ** a** (but not

For example, suppose: ** a**=

And ** x** has to be

If ** a** has some other negative value, then

For example, if ** a**=

As demonstrated, ** x** has to be

**§**

But what about a version where both ** a** and

For example, suppose: ** a**=

Which seems problematic. Any value for ** x** is squared and ends up positive which, added to

There seems no way to satisfy the equation.

It appears, then, the premise must be false?

**§ §**

It would seem so, unless we can come up with a way to square a number and end up with a negative value.

Which is something that our arithmetic, *as we understand it so far*, says never happens. A squared number is always positive.

But we need this:

As a general principle, what we really need is:

Because if we can figure *that* out, we can get any other negative value just by multiplying with any value we want.

If we solve for ** x** (by taking the square root of both sides) we end up with:

We call this value ** i** (tragically labeled as the “imaginary unit”), and we can use it to solve that unsolvable equation above. (And many others.)

That’s the big deal about ** i** — that’s what makes it useful. It provides a mechanism that allows important equations to work. (As it turns out, since many of those are basic physics equations, it appears

We can’t actually calculate the square root of a negative number, but if we accept that there is such a thing — take it on faith — then we can use it to make these equations work.

As you’ll see, sometimes ** i** is a bit of magic we stick into an equation (to give it special properties), and sometimes

For example, getting back to our problematic equation, if we set ** x**=

As you see, ** i** is squared away, and the

The bottom line is that our imaginary friend allows us to keep our mathematical truth! (Make of that what you will.)

**§ §**

Our imaginary friend enables a new type of number: the **complex numbers**. (Sometimes called the “imaginary” numbers.)

The natural numbers (ℕ) are natural. They enable a new kind of number, integers (ℤ), which include natural numbers but add negative numbers. (And definitely includes zero. It’s optional in the natural numbers, but I prefer the definition that begins at zero rather than one.)

The integers enable the rational numbers (ℚ) (who have the form: ** p**/

The real numbers (ℝ) are a new kind of number, distinguished for being uncountably infinite. They include the countable number kinds, but add the concept of a smooth (uncountable) continuum.

**§**

The complex numbers (ℂ), then, which are enabled by ** i**, include the real numbers (which include,… etc). They are a new kind of number with some different properties from other numbers.

One of those properties is that each complex number contains two parts — they have an inherent two-dimensional nature that turns out to be quite useful (and crucial to where I’m headed with all this).

Admittedly, if one is seeking a place to declare that *“math is made up”* the divide between the countable and uncountable is one intriguing line in the sand. The idea of ** i** is even more challenging. I know people who find

But, as with many abstract mathematical concepts, it demands recognition for having such valuable application to the world we live in. Math would be incomplete without complex numbers.

[As an aside, the hierarchy of number types doesn’t end with complex numbers. There are quaternions (ℍ), which have *three* (*different!*) ‘imaginary” components, and octonions (too obscure for a Unicode character), which have *seven*.]

**§
**

The two-dimensional nature of complex numbers comes simply from having two parts, a *real* part and an *imaginary* part.

We usually write a complex number like this:

+abi

The first part is the real part, the second is the imaginary part. The presence of ** i** in the imaginary part means the addition can’t really be done, so the two-part combination

Various math operations on such numbers typically resolve to the same form. Math with complex numbers results in new complex numbers.

The two real numbers, ** a** (the

When ** b** is zero, the imaginary part is zero, and the number is essentially the real number

This two-part nature leads to seeing complex numbers as 2D coordinates, in which case, we often write them like this:

[

,x]yi

(Using ** x** &

If ** x** and

(When plotting we ignore ** i** — it’s just there to visually remind us the vertical axis (

When we do this, we refer to the ** xy** graph as the

**§ §**

That’s enough for this time.

Next I’ll explore what makes the complex plane — as opposed to just the ** xy** plane — so useful. (For one thing, it’s really handy for drawing images. It’s especially handy when it comes to

Before that I’ll take a detour to show you *visually* why ** i** is necessary. It’s the same territory we just covered, but with pictures.

Finally we’ll get to our ultimate goal: the heart of the Mandelbrot.

*Stay complex, my friends!*

∇

]]>I dither about three because one of them was wasn’t new, it was season two I started of ** Siempre Bruja**. But I hadn’t yet seen any of

But first you should know about **(Your) CloudLibrary**!

I first ran into it as an app that caught my eye in the Apple App Store, but you can access it as a website, and you can get apps for iOS, Mac OS, Android, Windows, Fire, and Nook.

What it *does* is give you online access to ebooks from your local library.

It does take a library card to log in, but if you don’t already have one, you can get one from your local library for free. Just ask — they’ll be glad to have you.

I’ll mention that the system is interesting in that books are checked out, and checkouts are apparently limited. Once you check a book out, you have a time limit of 21 days in which to read it.

I think I noticed a feature that lets you extend a check out if needed, but I don’t really know what happens at the end of your time. (The book might just go away from your Reading list.)

I don’t know if there’s a limit to how many books you can check out. (I would guess yes.)

It’s possible, just as with physical books, for a book to not be “on the shelves” so to speak. You can, also as with physical books, put a book on hold for when it becomes available. (I haven’t tried this yet, so can’t report on how it works.)

There is also a Save list you can use as a queue for books you’d like to read later. And, of course, you can browse through available books.

I downloaded the iOS app ages ago, but never used it. With the whole staying home thing, I thought I’d check it out again, and I’m glad I did. There are a number of books I’d love to read, but I’m not interested enough to actually buy them.

For instance, I’ve found some Ellery Queen — only two, unfortunately (and *no* Earle Stanley Gardner, damn it) — but when I finish those, I’ve got a bunch of Anne Hillerman queued up (I’m a huge fan of her dad’s Navajo Tribal Police series, and she’s continuing it) along with the Kim Stanley Robinson Mars trilogy and Douglas Adams’ *The Salmon of Doubt*.

So I’m pretty excited about the app and thought it well worth mentioning.

**§**

I watched season one of ** Siempre Bruja** (

The stars are all attractive, the setting of Cartagena, Colombia, is both relaxing and energizing (like a tropical vacation), and there is a joy and innocence flowing through the story. And the Spanish language is so pleasing to the ear.

I can’t applaud loudly enough for a delightful show with a strong black female lead.

With foreign stories (the show is made in Colombia), it’s hard to know what cultural references one is missing or what is missed in translation of subtitles — I suspect a significant number of things just don’t cross over well. That can make the story seem to skip parts one would expect or to provide behaviors that don’t quite make sense.

I enjoyed the first season, mostly on an emotional level, but I didn’t always follow the story details. Re-watching season one for season two did slot more things into place, but I still think the rules of witchcraft are a little *ad hoc* in places. I can’t commend the show for its world-building.

But it’s definitely tropical drink tasty.

The story stars Carmen Eguiluz (Angely Gaviria, who is a delight), a witch being burned at the stake in 1646 Cartagena. But she vanishes from that time and appears in 2019 Cartagena, and we discover she’s been sent on a mission by the sorcerer, Aldemar. But she’ll have to watch out for another sorcerer, Lucien.

There are some twists and turns, and the season, of course, ends with a showdown between witches and wizards. (Credit good sense, and perhaps budget, for not going with a lot of flashy CGI but keeping it more human.)

The second season seems a bit tighter and more controlled to me. I haven’t finished it yet (three episodes to go), but I’ve found the story more down to earth and engaging somehow.

Overall, it’s a very attractive diversion. I give it an **Ah!** rating.

**§**

I’ve been debating whether or not to watch the new ** Lost in Space**. Until last night, I was giving it a miss.

I was a huge fan of the original TV show when it aired, and Judy Robinson is one of my first big screen crushes. (While we’re on the topic of witches, Samantha Stevens from *Bewitched* was another.)

So I watched two episodes, and I’m not sure I’ll be watching more. (Maybe. They managed to sink some plot hooks that now I’m vaguely curious about, damnit.)

The problem I had is the extremely high level of pure bullshit that constantly took me out of the story. A constant cliche-ridden stream of “Yeah, but…” The show is high on visual spectacle, but low on narrative substance.

Let me put it this way: Until Will met the robot, I was *convinced* I was watching a VR test the Robinsons were undergoing in preparation for their trip. There’s even a flashback line about Will coming out of the tank. The level of bullshit was so high, I wasn’t buying any of it as a real story. I was certain it was some kind of *“it was all a dream”* fake out.

I fully expected major characters to die and then wake up in their VR chambers for the debriefing, because it was all so absurdly ridiculous: Spaceships crashing, problems with the ice, magnesium fire (!!), the forest fire, an alien robot (that can’t get out of a tree), the projectile story, and the lyin’ eyes of Doctor Smith. Holy Cow, talk about piling on.

And the parents are kind of cliche. The absent dad, the mom cheating and reminding me of college scandals. The almost divorced couple forced to be together and who will, no doubt, find themselves again.

It’s one trope after another, cliche after cliche, plot convenience after plot convenience. (Lots of convenient radio comms.) The robot has handy heat coils when necessary. Bets on whether it ever uses them again?

Underlying the visuals is an over-amped music track desperately trying to convince me how serious and stirring all this is. (The music got to be so annoying I’m tempted to watch with sound off.)

One thing that really caught my eye: No one puts life craft on the inside of the main craft. The fleeing of the *Resolute* illustrated exactly why. That design is insane.

And entirely characteristic of the style-over-substance (or logic) approach modern storytelling — especially modern visual SF — gorges on and which I disdain. The robot is another good example of “looks great, but WTF” storytelling.

I could go on. Suffice to say I’ll have to be pretty bored to return for more episodes. Overall I have to give the show a **Meh!** rating (very nearly a **Nah!** rating). I was *seriously* underwhelmed (and frequently hooting in derision).

I think part of the problem is, for being so filled with bullshit, the show takes itself way too seriously. If it acknowledged how silly it’s being (this is Irwin Allen, for crying out loud), it would be a lot easier to swallow. And do something about that damn music track.

**§**

When it comes to ** Chilling Adventures of Sabrina**, I have the same problem I had with

It isn’t the show so much as the fans that I wonder about. And I will say that Kiernan Shipka (Sabrina) doesn’t have that sexual tang that Sarah Michelle Gellar (and Alyson Hannigan) did. I gave up on *Buffy* in season two; it just made my skin crawl too much. That show got too overtly sexual for my taste.

The original Sabrina TV show (with Melissa Joan Hart) was a family sitcom, no sexuality at all. The Netflix reboot manages to delve into the (non-sitcom!) horror side nicely, but has (at least in the first episode) de-emphasized the sexuality.

Horror isn’t really my cup of tea, but I found the first episode of the show engaging enough that I’ll return for at least a few more. I’ll give it a tentative low **Ah!** rating for now.

I will say that co-stars Lucy Davis and Miranda Otto look so much like Caroline Rhea and Beth Broderick that for a moment I wasn’t entirely sure they hadn’t brought them back (through a time machine, obviously).

I also got a kick of out seeing Michelle Gomez. I really enjoyed her as The Master in *Doctor Who*. Excellent villain! (I really can’t stand the current The Master, and I’ve just about concluded the Chris Chibnall era is a fail for me.)

**§**

Too bad Parker Posey (Dr. Smith) isn’t playing a witch. This post could have been about three witches, and covens often do follow the rule of three.

Instead there is one show I recommend, one I don’t, and one I’m not sure about yet, but which seems okay so far.

There is also the ninth season of *Archer* (they’re doing *Firefly* this season!), but that’s a whole other thing.

*Stay bewitched, my friends!*

∇

]]>Wow, for the third time this month (third time in a *week*) I’ve realized the day calls for a post I hadn’t planned. The first time was when the MLB delayed the baseball season. The second time — the very next day — was Pi Day and Albert Einstein’s birthday.

This time it’s the **equinox** (and a friend’s birthday; *shout out!*). For those of us in the northern hemisphere it’s the spring (vernal) equinox, and that’s my favorite of the four annual solar node points (two equinoxes; two solstices). It means we have a whole half a year of light ahead.

So I just had to post something.

The word *vernal* means spring, which in general we have long used as a metaphor for birth and new life. (You know about the whole Maypole thing, yes?)

The very idea of spring is evocative. It’s a season of new life, but all sorts of things spring forth, even ideas. (Another kind of spring brings forth new water.)

Just as Christmas was deliberately aligned with the winter solstice celebration (a *huge* pagan party!), Easter — the Resurrection — was aligned with the spring celebration (another huge pagan party).

Of the four annual solar node points I mentioned, those are the two worth celebrating. They are the two good ones.

The solstice means the days will finally start getting long again. (*“Hooray! The Star Dragon isn’t going to eat the sun completely. It’ll grow back! We’re going to live!”*)

The vernal equinox means plants start growing again and hibernating animals wake up. (*“Hooray! There’s going to be new food again and we can stop eating old moldy stuff. We’re going to live!”*)

**§ §**

Back in 2018 I became absolutely enthralled by the Kīlauea volcano eruption.

[See: *Kilauea, Hawaii, USA: Wow!* and *2018: Hawaii Gets Bigger!*]

As it turned out, it was a historical event. It started that April (in the spring!) and covered over 13 square miles with lava by early August when it abruptly ended. Spilling into the ocean at the coastline, it added almost 900 acres of new land to the big island.

It’s been quiet since, although the underground heat lingers and causes problems as it spreads through the ground evaporating ground water.

**§**

So two years ago, in the spring, lava was about to spring forth from the ground in devastating amounts. Molten rock flowing like water.

Lava isn’t the *hottest* thing, but it’s still really, really hot. When it comes to molten stuff:

- Brass: 600° C (1100° F)
- Lava: 1000°–1250° C (1800°–2300° F)
- Iron: 1200° C (2200° F)
- Steel: 1300°–1500° C (2400°–2700° F)
- Glass: 1400°–1600° C (2550°–3000° F)

It’s not something you can stand next to. (Let alone fight someone while standing on a rock chunk floating in it.)

The operative phrase here would be: *“Instantly burnt to a crisp.”*

**§**

It will be two years this August that the lava stopped flowing.

While hot spots remain, the molten has become rock, and new plant life is already growing in the lava field.

I have two videos to share that I thought to include in a Wednesday Wow! post. (That first post of mine in 2018 was filed as a Wow! post, but wasn’t explicitly called out as one although it is in the title.)

I almost included them in the post I just published, but I don’t like to bog down a page with too many video player frames.

So this is both an extension of last Wednesday’s post and a spring equinox thing.

That said, there is also a hellish aspect to this. This guy hikes through what was, less than two years ago, a field of molten lava. As you’ll see, some parts are still smoking.

It would make a good depiction of Hell: Forever wandering a barren field of raw rock with green life over yonder just out of reach…

So I admit my choice may be a little macabre here, but the sights along the hike — especially if you have any interest in geology at all — are rather jaw-dropping. The mineral colors and crystals alone are worth the watch.

**§**

Without further ado:

This next one actually gets kind of intense (but, again, the sights are really something to behold — being inside a new lava tube like that… yikes!):

One thing I really appreciate is that the guy doesn’t say a word during either video. There simply are no words that could do it justice, and a narration would just be a distraction.

Just put these videos full screen on your largest screen and sit back and appreciate the stunning beauty and awesome power of Pele.

**§**

I hope you enjoyed that as much as I did. If nothing else, it’s a nice quiet distraction by some very “concrete” natural reality.

Whatever else is going on, at least we’ll have more light than dark now.

And no one in the USA is being threatened by lava, so there’s that.

*Stay vernal, my friends!*

∇

]]>The problem is that I’m jaded and have seen a lot, so I can be hard to impress. Not lots of things raise to my highest rating, **Wow!** Fortunately, I’m not so far gone I can’t still see a world filled with wonder, some of which drops my jaw.

The theme, such as it is, concerns measurements, especially tiny and precise ones. Like, for instance, Planck Length tiny.

I suppose I could try to tie this in to, for example, a tiny virus, but the idea is to think about *other things* once in a while, so I won’t. (And a virus is pretty gigantic compared to the Planck Length.)

In fact, there is something of a wow factor in the comparison. I posted a while back about how the Planck Length compares to the size of a hydrogen atom the same way an amoeba compares to the Milky Way galaxy.

If the Planck Length is that small compared to the smallest atom, imagine how small it is compared to a virus that has (depending on the virus) many tens of thousands of atoms. It takes the comparison up to amoeba and galactic super-cluster.

Which makes the first two entries here all that more awesome.

**§**

This first one involves the five best measurements in science (according to these guys, and I don’t see any reason to disagree):

I got a big kick out of the notion about how many times humanity has tested the belief *“the sun rises in the morning.”* (In epistemology, that’s almost a canonical exercise — justifying that true belief about the sun’s rising.)

(The video also touches on the important notions of accuracy and precision, which caught my eye. I posted about those recently, too.)

It’s also fascinating to me to think about that in context of justifying the Standard Model. In some sense, every moment of every day for everyone justifies it, but the LHC tests are rather more precise.

(In fact, they’re currently chasing down what appears to be an anomaly.)

What gets me is that we know the Standard Model is, at best, incomplete. And yet that incomplete model works amazingly well. (But then, so did Newton until we looked really closely at Mercury. On the other hand, we spotted Mercury’s “misbehavior” pretty early. But so far, except for that possible weak force anomaly, the Standard Model works extremely well.)

Ironically, after introducing the Standard Model, the video pivots to General Relativity, our other extremely well-tested science theory, and so far we haven’t found the slightest hint that GR might be wrong. We’ve now measured stars orbiting very close to black holes — and extreme gravity regime — and they behaved exactly as expected.

In any event, when I watched this video, I knew immediately it could be the centerpiece of a Wednesday Wow post. If you watch only one of the videos today, watch this one.

**§**

This next video takes us to Planck’s constant (from which we get the Planck Length, the Planck Energy, the Planck Mass, etc.) and demonstrates an interesting and easy way that just about anyone can measure it:

It spends the first few minutes on a really good explanation of what Planck’s constant is and how it first came about. That alone is worth watching.

Granted, the technique illustrated probably isn’t something one would do at home just for fun (unless maybe, as some of us do, one happens to have that sort of gear lying around). It’s probably more something for high school science class.

(I even used to have a diffraction grating, although I don’t recall any label specifying its resolution. I definitely have some colored pens.)

I thought it was very cool how we can use physics knowledge to fairly casually measure something so tiny and fundamental.

**§**

In this video, a guy with an electron microscope (now there’s a nice piece of home gear) creates an animation of a record stylus tracking a vinyl record groove:

Keep in mind it is an animation — he stitched a series of still images together.

I also wonder if removing the stylus from the assembly made its movements too free. Do the magnets being in coils restrict the movement more?

But it’s still a very cool video. I’ve written many times, in comparing analog versus digital, about how the sound waves are visible in a record groove. This video shows that amazingly well.

In comparison, he also shows some digital media, a CED, a CD-ROM, and a DVD. You can’t see the sound waves in those. (Comparing the track size of the CED with a record stylus was impressive.)

[*Switched-On Bach*. Ha! I (still!) have that record album.]

**§**

Finally, as a bonus “tiny” here’s Imogen Heap in an NPR Tiny Desk concert that made me go wow! It’s the third song she does:

The first two songs didn’t get me all that lit (although it was interesting how she used completely different side players and instruments on each).

It’s the thing with the VR gloves that blew me away (there are other videos available of Heap using the gloves).

She explains how they work starting at the **9:15** minute mark, and the performance itself begins at **12:45**. I’ve watched this several times now, and it blows me away every time. (Part of the fun is trying to figure out exactly what she’s doing with the gestures.)

The thing that struck me was: How long until we see this technology in common play? How long until there’s a group on stage — with no instruments except the gloves — making gestures to play “drums” and “guitars” and so on?

The *technology* (and theory) of music has long fascinated me. It’s one of our oldest traits; it may even predate language. (Think of all the animals that “sing” but have no real language.)

Music speaks to us on a very primal level — hence its power.

**§**

So there was some stuff about the tiny that hopefully entertained you as you “shelter in place” during these strange times.

The whole thing does provide an excellent opportunity to catch up on whatever you need catching up on. Read those books you’ve been putting off. Binge on shows you always meant to get around to watching.

(Clean the house? Nah. It’ll just get dirty again.)

*Stay tiny, my friends!*

∇

]]>Which is a whole other story. I mention it because many of these music makers are sweet, gentle, loving people who just want everyone else to be sweet, gentle, and loving. It’s a common sentiment. Banish the bad forever!

But balance is required. There is a Yin-Yang aspect to life.

Humans are capable of greatness — that’s kind of the thing about humans. In a very short span of time we’ve gone from being little more than another animal to exploring nearly every corner of Earth (and even some of nearby space). We happily inhabit a variety of ecological niches.

Someone once equated our greatness with the greatness of cyanobacteria on the basis that the bacteria had (very usefully) altered the Earth in creating the oxygen atmosphere current life thrives in. But it took them many hundreds of millions of years to do it, they are restricted to their ecological niche, and they never did anything *else*, so I don’t see *that* much greatness.

Humans, in a mere 10,000 years or so, haven’t just expanded to fill the planet. We have altered it significantly. (Possibly to our own peril. Certainly to our peril in small ways in toxins and plastics.) We make ourselves at home from the equator to both poles.

We sail above and under all the seas, we’ve choked near Earth orbit with techno-junk, and we’ve sent robots flying off into space to explore our Solar System (and beyond). We dream of colonies on Mars. We wonder about living under the light of a distant star.

We’ve also created a vast body of art, literature, music, mathematics, and science.

(So, a bit greater than cyanobacteria, is my point. )

**§**

It’s our great minds that give us this great power we wield, and that power can go in a great many directions.

The gentle wish for a world in which power only works in good directions, but who sets that standard, and more importantly what kind of imbalance does that create?

Is it even possible to constrain our power for greatness?

Just think about all our chances to be great:

- Great Good ⇔ Great Evil
- Great Beauty ⇔ Great Ugliness
- Great Understanding ⇔ Great Ignorance
- Great Pleasure ⇔ Great Pain
- Great Peace ⇔ Great War
- Great Healing ⇔ Great Harm
- Great Love ⇔ Great Hate
- Great Gain ⇔ Great Loss
- Great Joy ⇔ Great Sadness

I just don’t believe we can have the Yang without the Yin. It’s our capacity for greatness that gives us access to the wonderful, but which also opens the door to the terrible.

While it might seem great to only have the highs, I’m not sure that’s possible. When one has coin to spend, one can spend it on anything. Our capacity for greatness is our coin.

If you’ve ever known someone who is bipolar and on medication, a common complaint is how the meds “flatten out” the world and “turn everything grey.” The bipolar don’t get the extreme highs and lows, which is good, but they often don’t experience any variation from the middle.

I think the alternatives are either experiencing all life has to offer, good and bad, or experiencing little or nothing at all.

**§**

And, as Stan Lee taught us in Spiderman: *“With great power comes great responsibility.”*

In fact, that’s a message to *all* humans, because we *all* have great power for good or ill.

*Stay great, my friends!*

∇

]]>