I like to call this the “phantom lightbulb illusion” [1]. Its operation can be seen at the end of the video, and described as follows. A functioning lightbulb is placed in a socket upside-down inside of a box, while an identical socket is empty directly above. When the bulb is turned on and the box is in the proper place, the concave mirror in front of the box produces a real image that lies directly above it. As long as one keeps the empty socket between the observer and the mirror, the image will remain in the proper place.

This is, in short, *how* the illusion works, but doesn’t explain *why* it works — that is, what is the underlying optics that creates the image in the proper place?

What a great opportunity for me to explain a little bit of geometrical optics! By the end of this post, I will have described the basic theory of image formation in concave mirrors and, hopefully, we’ll really understand why the “phantom lightbulb illusion” works.

Though we know today that light has both wave-like and particle-like behaviors, these are surprisingly hard to observe without a carefully prepared experimental setup. For most mundane observations of light, it is sufficient to treat light as a stream of line-like rays that radiate from a source in straight lines that only deviate when they encounter some sort of matter. This model of light behavior was sufficient to explain most observations for literally thousands of years, and is seen every day. It can be seen explicitly, for instance, when sunlight peeks through clouds, as shown below.

In geometrical optics, only two things can happen to a light ray: when it bounces off of a smooth highly reflective surface, it reflects, and when it bounces off of a smooth transparent surface, it refracts. In reflection, the angle of reflection of a light ray is equal to the angle of incidence . In refraction, the light ray changes direction as it enters the transparent medium.

Since we’re working with a mirror, we’ll only need to concern ourselves with reflection [2] to explain the “phantom lightbulb.”

For a curved mirror, the law of reflection still holds at every point. The light ray reflects around the line locally perpendicular to the surface, which is different at every location. This means that parallel light rays at different heights will bounce off of the mirror at different angles and travel in different directions. Below, we have an illustration of horizontal rays bouncing off of a spherical mirror.

The red lines indicate light rays coming in parallel to the horizontal axis of the mirror, and the dashed lines indicate the perpendicular to the mirror surface at various locations. The dashed lines run from the center of curvature of the mirror (the center of the sphere that the mirror is “cut” from) to the points where the rays hit the surface; it can be seen that the angle of incidence of the rays is always equal to the angle of reflection.

But something else interesting happens: every parallel ray hitting the mirror reflects back and passes through the same point on the mirror’s axis, labeled *f*. If we put a lightbulb really, really far away from the mirror, it would produce almost parallel rays and those parallel rays would converge at *f*, producing a bright spot of light. This point *f* is called the focal point of the mirror, and from the picture it can be seen that it is located halfway between the mirror surface and the center of curvature.

We can also imagine taking the picture above and reversing the direction of the arrows. This reverses the “angles of incidence” and “angles of reflection,” but since those angles are the same, the picture doesn’t change! In other words, any ray that passes through the focal point on its way to the mirror will be reflected parallel to the horizontal axis.

With just these two rays, we now have enough information to determine the location of any image produced by the concave mirror! Let us first imagine that we have an object (represented by an arrow) at a distance greater than *f* from the mirror. We imagine that light leaves the tip of the arrow from all directions; however, we only follow the two rays that we know how to draw.

What we find is that the light from the object gets refocused into a magnified, inverted image that lies beyond the curvature point *C*. The image is known as a *real image*, as there is actually light passing through the observed image, as the diagram suggests.

We can also consider the other circumstance, when the object lies within the focal distance, but must be a little ingenious. Unlike the previous case, no ray passes first through the focal point and then hits the mirror; the ray that reflects parallel is the one that, when extended backwards (thick dashed line), hits the focal point.

Another (seeming) complication: our two rays never intersect! They extend away from each other after hitting the mirror. However, if we draw imaginary (thin dashed) lines into the mirror itself, we see that they would appear to intersect inside the mirror, forming an upright magnified image. This is a *virtual image* — there is no light actually inside the mirror — similar to the virtual images we ordinarily see in a flat mirror.

You can test these properties of a concave lens if you have a concave makeup mirror; such mirrors work well for putting on makeup because of the magnification of the upright virtual image.

Now we come back to our illusion! Suppose that we place an object exactly under the point *C*. Drawing our rays again, we see the following.

Our object produces a real image that is inverted and is almost exactly the same size and at almost exactly the same horizontal position!

This is the optical physics behind the “phantom lightbulb” illusion — an object placed directly on the center of curvature of a concave lens will produce a real, inverted, unmagnified image at the same location.

There is some additional complexity, however, associated with the simple picture of geometrical optics given here. If you look at my ray picture of the optical illusion given above, you’ll notice that the rays do *not* pass directly through the focal point, but instead hit the axis a little to the left. It turns out that the statement “parallel rays all pass through the focal point” is only an approximation, and an approximation that gets worse for parallel rays more distant from the axis. This is a phenomenon known as spherical aberration, and is an inherent limitation of spherical mirrors, which are not the best mirrors for focusing and imaging! One can see the problem easily by considering a hemispherical mirror and drawing a ray that hits very high on the mirror’s edge.

The ray hits the mirror’s surface at a glancing angle, and isn’t reflected enough to hit the focal point! Spherical mirrors are not, in fact, perfect image-forming devices, and the rule that “every parallel ray passes through the focal point” is only an approximate one for rays sufficiently close to the center axis.

I should now confess that I cheated in drawing my earlier pictures! When I drew all the parallel rays passing right through the focus, I *assumed* that they did so and drew the picture accordingly. This approximation is good enough, however, for a basic description and understanding of geometrical optics.

And what better way to end a blog post about an illusion than to admit to a little deception of my own?

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[1] I first saw it on YouTube, where it is referred to as the illusion of the famous Chinese magician Foo Ling Yu. For hopefully obvious reasons, I don’t want to perpetuate a racially-based pun.

[2] I talk in detail about refraction in an “Optics basics” post, for those wanting to learn more.

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Now, on the heels of that discovery, the LHCb (Large Hadron Collider beauty) collaboration at CERN has announced the discovery of a new particle — an exotic hadron that has *four* quarks in it, instead of the usual three quarks or quark-antiquark pair. Such a beast lies outside our current understanding of particle physics, opening the door to even more revelations about our universe. Matthew Francis has another nice summary of the discovery at Ars Technica.

The details of this discovery, and its long-term implications for physics, are out of my depth these days (I haven’t been a particle physicist for a while). However, the press releases assumes a lot from the reader — do they know what quarks are, and why they only come in threes or quark-antiquark pairs? With this in mind, I thought I could write a short post giving some background for those who aren’t familiar with the details — the “Cliff Notes” of quarks, so to speak! As I tend to do, I’ll approach this from a historical perspective, though this history will be simplified for the sake of brevity.

The discovery of quarks can be thought of the result of humankind’s desire to understand what we’re made of, and our instinct that a theory of matter must be quite simple. For instance, as far back as ancient Greece, philosophers broke down all of nature into the four classical elements earth, water, air and fire. They were wrong, of course, but that desire to simplify nature, specifically matter, has been a large part of science ever since.

Most everyone is familiar with one of the greatest accomplishments in this effort: the periodic table, introduced by Dmitri Mendeleev in 1869, which arranges chemical elements in order of atomic number and chemical properties. From the structure of the periodic table, Mendeleev was able to predict as yet undiscovered elements from gaps in the existing table.

When I say “chemical elements,” of course, I’m referring to different types of *atoms*. The regular behavior of the periodic table hinted that atoms are built of more fundamental particles, though it took quite some time to figure out exactly what. The first piece of the new puzzle was the discovery* of the negatively-charged electron by J.J. Thompson in 1896, which was immediately recognized to be a part of the atom. The next important piece was the discovery of the positively-charged atomic nucleus by Ernest Rutherford in 1911, based on experiments done by his assistants in 1909. This observation gave us the most familiar picture of the atom seen today, with electrons orbiting a tiny nucleus**.

But what is the nucleus made of? In 1917, Rutherford managed to show that the hydrogen nucleus (the smallest one, with a positive charge equal in magnitude to the negative charge of the electron) is present in other, heavier elements. This was the first evidence that all nuclei are built from these single positive charges, now called protons.

Something else was missing, however, as the mass of the heavier nuclei is greater than could be explained by just protons alone. The mystery was solved in 1932, when James Chadwick discovered the neutron, an almost “twin” of the proton with roughly the same mass but no charge.

So ordinary matter, in the form of atoms, consists of a small nucleus of protons and neutrons, with electrons “orbiting.”

Even in 1932, however, this was not the entire story: in 1928, Paul Dirac predicted theoretically the existence of antimatter, negative twins of ordinary matter with opposite electric charge that annihilate with ordinary matter. The anti-electron (positron) was discovered in 1932 by Carl D. Anderson. In 1933, one seemingly final piece of the puzzle of matter was proposed, the neutrino, a (nearly) massless, chargeless particle that is closely related to the electron.

So, in the early 1930s, there was a seemingly quite complete picture of matter: electrons, protons, neutrons, neutrinos, and their antiparticle compliments. However, things would quickly change: in 1936, a new negatively-charged particle was discovered, the muon, that would later turn out to be in essence a heavier version of the electron. Seemingly serving no real purpose in nature, the muon was a new puzzle. Muons are created all the time by collisions of high-energy particles in the upper atmosphere, and they decay into an electron and neutrinos. The existence of the muon was baffling, famously leading physicist I.I. Rabi to remark, “Who ordered that?”

Things got worse — or better, depending on one’s view of scientific mysteries! To discover the nucleus, Rutherford smashed alpha particles (helium nuclei) against gold atoms. To probe inside the nucleus required higher-energy collisions of atoms and particles, and this set the stage for a century of particle physics experiments that are still going on today. However, as collisions of higher and higher energies were generated, new heavy particles were created that were distinct from the protons, neutrons and electrons already known! These particles were broadly grouped into baryons (including protons and neutrons) and mesons (something new), and suddenly the “fundamental” set of particles had grown to a set of dozens.

Clearly something else was missing, and it took until 1964 for the answer to come. In that year, physicists Murray Gell-Mann and George Zweig independently proposed that all of the “fundamental” particles — baryons and mesons — were composed of a much smaller set of even *more* fundamental particles, labeled “quarks.”

Over the course of several decades, six distinct quarks — and their antiquarks — were predicted and/or discovered experimentally, the last — and heaviest — being the top quark, discovered in 1995. A picture of all known fundamental particles is illustrated below.

A few things to note here. The “leptons” include the electron (e), the muon (μ), and the electron and muon neutrinos (the ν’s), as well as an even heavier cousin of the electron, the tau (τ). The “gauge bosons” are particles that “mediate,” or create, the fundamental forces of nature, including the photon γ, or “particle of light.” Also on the list is the Higgs boson, a particle that, loosely speaking, “gives” mass to other massive particles. The Higgs was originally hypothesized in 1964, but it was not until 2012 that it was officially discovered at the Large Hadron Collider. The Higgs is the heaviest fundamental particle known, and so it was the last to be discovered, requiring collisions at energies that only the LHC could provide.

You may note that even this list seems like a really large number of supposedly “fundamental” particles! This somewhat paradoxical situation is the motivation for string theory, which speculates that all of the particles in the standard model are really just different excitation/vibrations of a single fundamental string-like particle. There is no way to experimentally test this theory for the foreseeable future, however.

Putting all of that aside, however, let’s focus on the quarks! The six quarks, in order of increasing heaviness, are listed below.

- up quark: mass 2.3 MeV, charge +2/3e
- down quark: mass 4.8 MeV, charge -1/3e
- strange quark: mass 95 MeV, charge -1/3e
- charm quark: mass 1.275 GeV, charge +2/3e
- bottom quark: mass 4.18 GeV, charge -1/3e
- top quark: mass 173 Gev, charge +2/3e

The masses are given in units of energy; MeV = million electron-volts, and GeV = billion electron-volts. The charges of the quarks are given as fractions of the fundamental charge of the electron, e. This in itself raises an odd question, though: how can the quarks have a smaller charge than the *fundamental* unit of charge?

The answer is that quarks are never seen alone: they only appear in trios or in quark-antiquark pairs. The strong nuclear force that holds quarks together never allows them to exist in isolation. Smash two protons together with a tremendous collision energy, and that energy will result in additional quark-antiquark pairs being created, not in the separation of individual quarks. The output of such a high-energy collision is a huge zoo of particles traveling in all directions, as illustrated below.

This is where things get interesting, and even beautiful! The theory of strong nuclear interactions — quantum chromodynamics — indicates that quarks in fact come in three “colors” — call them red, green, and blue — and their corresponding “anti-colors.” The nuclear force only allows “colorless” combinations of quarks to exist, either in three quarks of different colors — one red, one green, one blue — or in a quark/anti-quark pair of opposing colors, for instance red and anti-red. The implication of this is that quarks can only be combined in ways that produce a non-fractional total charge! For instance, a proton consists of two up quarks and one down quark, which has a total charge:

+2/3e (up, red) + 2/3e (up, green) -1/3e (down, blue) = e.

A neutron consists of two down quarks and one up quark, with a total charge:

-1/3e (down, red) – 1/3e(down, green) +2/3e (up, blue) = 0.

A simple meson, the pi+ meson, consists of an up quark and an anti-down quark:

+2/3e (up, red) +1/3e (anti-down, anti-red) = e.

Additional rules apply for the more exotic quarks. It turns out that “strangeness,” “charmness,” “bottomness” and “topness” are all conserved quantities, as well. That is, charm quarks can only be produced in charm/anti-charm pairs, where the total “charm” is zero, with similar considerations for the other heavy quarks.

This simple model of quantum chromodynamics (which of course in practice is much more complicated) allows the construction of all the mundane and exotic baryons and mesons. With quarks, then, we have a wonderful and complex hierarchy of matter, from the smallest (quarks) to the largest aggregates, such as a beaker of water.

At long last, we can come back to the new discovery at CERN! As we’ve alluded throughout this post, new discoveries come at increasingly more advanced accelerators that can (a) smash things together harder and (b) do so at a faster rate, to build up enough statistics to see exceedingly rare events. In particular the LHCb collaboration sifted through 180 trillion proton-proton collisions at the LHC, investigated 25,000 promising meson decays among thetse, and found unambiguous evidence that a new particle was being formed from a collection of *four* quarks. This result confirms earlier tantalizing evidence of this particle that was seen in 2008 by the Belle collaboration.

The new object, dubbed Z(4430) because of its mass of 4.43 GeV, does not last long before decaying into lighter particles. However, the collaboration was able to determine the quark composition of this new transient particle: a charm quark, a charm antiquark, a down quark, and an up antiquark.

This new particle does not break any of the rules we have set out previously. The total amount of “charm” is zero, thanks to the present of the charm quark and the charm antiquark, so charm is conserved. Also, the total charge is -e, an integer amount of fundamental charges. The total color can still be zero, since we could have a red down quark, an antired up antiquark, a blue charm quark, and an antiblue anticharm quark.

However, the theory of quantum chromodynamics does not predict that this combination of quarks will form a distinct particle, which means it is new physics of some sort. As Matthew Francis notes at Ars Technica, this could be some sort of unusual fusion of two mesons (down/anti-up plus charm/anti-charm), or it could be a new type of particle entirely — a tetraquark!

These results are a bit of a relief, I imagine, for the particle physics community. After the discovery of the Higgs boson, there was at least some worry that there might not be anything else to discover, at least in the short-term! The discovery of this unusual particle resonance indicates that there is new physics to be found at the LHC, and this work will give experimentalists and theorists something to ponder and study for a while.

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* By “discovery,” I should really say “proof that the electron is a distinct particle and not an atom or wave.” So-called cathode rays had been observed years earlier, but their precise nature was unclear.

** This picture of an atom is oversimplified, as the electron has wavelike properties and is more aptly described as being a sort of blur around the nucleus.

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It is not difficult to deduce from the title that *The House of the Wolf* is about werewolves! In fact, that realization made me reluctant to pick up the book, which has been sitting on my shelf for at least six months, simply because I (personally) find werewolf stories a little tedious. (One notable exception: Valancourt’s collection of very early werewolf stories that I’ve blogged about before.)

As I should have known, however, from my previous experiences with Copper’s writing, *The House of the Wolf* is quite fun! A Gothic novel similar to his earlier *Necropolis* (1980) and his later *The Black Death* (1991), it is a mixture of horror and mystery that is slow to start but quite stunning by the end.

Set in the Victorian era near the end of the 1800s, the tale begins as Professor John Coleridge arrives in the remote Hungarian village of Lugos. He, and a select group of folklorists, have come at the invitation of Count Homolsky, the lord and owner of Castle Homolsky, also known locally as “The House of the Wolf.”

The castle’s name turns out to be immediately apt; as Coleridge arrives, he witnesses the funeral procession of a villager, the latest victim of a black wolf that has been terrorizing the countryside. Though tragic, the event seems unconnected to the impending folklore meeting, and Coleridge busies himself with meeting his colleagues, as well as the Count’s mother, wife, and lovely daughter Nadia.

Things change quickly, however, when Nadia takes Coleridge into her confidence. Her sleep has been disturbed by the sound of an animal outside her door and — impossibly — the rattling of her doorknob, as if that animal were trying to gain entrance. Coleridge promises to investigate, and soon the investigation becomes a matter of life and death, as people both inside and outside the castle are found savaged by an unknown creature. As the folklore meeting progresses, the participants and residents of the castle find themselves in deadly danger, as well as trapped by blizzard conditions outside.

The survivors soon suspect that the killer is among them, and also that this killer is something more and less than human. Will they be able to uncover the murderer before he or she strikes again?

*The House of the Wolf* is one of those books I read at a leisurely pace, matching the pace of the novel itself. Copper fills his Gothic tales with lots of detail, and I found that one or two chapters a night was the best rate to digest the prose. It seems a little *too* detailed at first, however, and I found the book slow-going until about the halfway point, when the danger becomes very apparent and the supernatural implications unavoidable.

The best part about *The House of the Wolf*, however, is its ending. Though it is a supernatural novel, it is also a mystery, and the resolution of that mystery was genuinely surprising and unexpected to me. It made the slog through the earlier sections of the book worthwhile!

In comparison to his other Gothic stories, I would rank *The House of the Wolf* above Necropolis but below *The Black Death*. Amusingly, there is a shout-out to *Necropolis* about midway through ‘*Wolf*, demonstrating that the stories take place in the same continuity!

In short, *The House of the Wolf* is an enjoyable supernatural tale with a quite good atmosphere. It doesn’t really break any new ground where werewolf stories are concerned, but it manages to fold the werewolf legend into a murder mystery in a satisfying way.

Sadly, the novel has been out of print since its 1983 release. Hopefully it (and the wonderful cover and interior illustrations) will be made available again sometime in the future.

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This makes my third appearance, I believe, after an appearance on WCCB TV last year to talk invisibility and cloaking and a short spot on News 14 Carolina (now Time Warner Cable News) that I can’t even find any more.

I love doing these appearances because it gives me an opportunity to see “behind the scenes” at how news crews work. Larry Sprinkle was a lot of fun to work with and the cameraman was awesome — no problem with any of my crazy stuff! :)

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Twitter is a great place to waste time, but it is also a great place to get inspired with really ridiculous ideas. After I pointed out that a sequel to the movie Prometheus is in the works, PZ Myers of Pharyngula responded with this tweet:

THERE IS NO GOD. Or his name is Loki. RT @drskyskull: "Prometheus 2 is (probably) arriving on March 4th, 2016" io9.com/prometheus-2-i…—

PZ Myers (@pzmyers) March 24, 2014

My response was this:

@pzmyers Or his name is "Asshole Ra." #ImOpenMinded—

skullsinthestars (@drskyskull) March 24, 2014

And a new meme was born! Ra, for those unfamiliar, is the ancient Egyptian deity of the midday sun, a major god of Egypt from somewhere about 2500 BC onwards. He was typically represented with a falcon head, and sun disk on top of it, as pictured below.

The ancient Egyptian god Ra is a powerful, benevolent deity, responsible for the creation of life and the one who protects and sustains it. On the other hand, Asshole Ra is petty, annoying, and pretty much, well, an asshole.

Taking classic Egyptian artwork of Ra from around the web, I produced my own conception of Asshole Ra. Go below to see his glory…

My first image of Asshole Ra shows his appreciation of fine arts.

Asshole Ra likes to troll his believers something fierce.

Asshole Ra is not a deity you want to go on a roadtrip with.

Asshole Ra does not use his great powers with great responsibility.

Humor is not one of Asshole Ra’s strong suits.

Asshole Ra is picky about the offerings he receives.

Asshole Ra is also very picky about his beverages.

It’s really not worthwhile to be in a relationship with Asshole Ra.

Asshole Ra also has serious entitlement issues.

Did I mention that humor isn’t one of Asshole Ra’s strong suits?

Really, it’s best not to rely on Asshole Ra for anything.

I may present further adventures of Asshole Ra in the future — stay tuned!

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Images of Ra were taken from ancient Egyptian artwork that is available from a variety of sources on the web, including Wikipedia. I used a number of images of ancient Egyptian stelae from www.hethert.org, which has a nice compilation. The photograph of the Double Temple of Haroeris and Sobek at Ombos came from the very nice Amentet Neferet blog.

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This is a tragedy because mathematics is one of the best fields to find incredibly beautiful objects, experience mind-blowing concepts that challenge ones imagination — or sometimes both simultaneously.

A wonderful example of this is the geometric object constructed by the Italian mathematician Giuseppe Peano and described in his 1890 paper “Sur une courbe, qui remplit route une aire plane.” This beautiful and mind-boggling object is illustrated below.

Uh… wait a minute…

*shuffling of papers as I check my notes*

No, that is correct! The object above is obviously a square, which at first glance would seem to be the most boring geometric figure possible. What Peano did, however, was demonstrate a new way to fill the square, inventing a mathematical construction that allows the square to be completely filled in with a single, continuous line! This “Peano curve” was the first example of what is now known as a *space-filling curve*, which has surprising and insightful connections to modern mathematics.

Giuseppe Peano (1858-1932) was a brilliant and creative mathematician who had a major influence on a number of fields of mathematics, including set theory, logic, and axiomatic mathematics. We have recently come across his work on this blog in the context of the science of falling cats, which inspired Peano to dabble in the physics of the Earth’s motion — though his geophysics research was not particularly groundbreaking*.

In the early 1890s, however, Peano had come across the stunningly novel work of Georg Cantor on set theory, in particular the description of the different sizes of infinity. We have discussed this in a series of posts on this blog before**: in short, Cantor noted that the set of real numbers between 0 and 1, dubbed the *continuum*, is an infinitely larger set than the set of counting (natural) numbers 1,2,3,… . The “size” of the continuum is labeled , while the size of the natural numbers is labeled : they are “numbers” of a sort, though outside the realm of ordinary numbers.

This was a difficult enough concept for many to accept; even stranger, Cantor proved that the size of the continuum — the set of points between 0 and 1 — is just as large as the set of all point in a square! That is, if one considers every distinct point on the number line, they can be lined up in a one-to-one correspondence with every distinct point in a square.

This is, obviously, a very non-intuitive result! We know that a square is a two-dimensional object, while a line is a one-dimensional object; it does not seem that a line could contain the same number of points. Nevertheless, Cantor demonstrated that this is true, using very simple logic (described in my infinity post series, part II).

In the wake of Cantor’s revelation, however, another question arose: is it possible to *continuously* map a line into a square? In other words: is it possible to draw (mathematically, at least) a single continuous curve that fills a square entirely? To be continuous, the square must be filled by a mathematical pen that is never lifted from the mathematical paper, kind of like a “connect the dots” kids puzzle whose outcome is a filled black square.

Peano was the first one to introduce such a curve, now known as a *space-filling curve*. As one might expect, such a curve is quite unusual, and won’t quite look like anything encountered before. In fact, the complete curve is impossible to visualize, since it literally fills the square and, in the process, takes an infinite number of twists and turns along the way. However, we can get a feel for its behavior through an iterative process that generates curves of increasing complexity that approach the true space-filling curve in the limit of infinite iterations. The result of the first four iterations of the Peano curve are shown below.

The starting point of the Peano curve is basically a sideways, backwards “S.” In the next iteration, that basic “S” is preserved – the path meanders up, then down, then up again — but takes additional detours along the way. In the next iteration, detours of detours are taken, and so on. Even by the fourth iteration, one can see that the curve is starting to fill in the entire square; with an infinite number of iterations, it completely fills it in.

Curiously enough, the first person to give a graphical representation of a space-filling curve was not Peano, but the great German mathematician David Hilbert. Peano’s paper (an English translation by me can be found at this link) gives a detailed mathematical description of the curve, and a proof that it does in fact completely fill a square, but does not give any illustrations. Curiously, Peano apparently knew how to do so, as it is reported that he installed a tiled representation of the curve on the terrace of a villa he purchased in 1891***. On reading Peano’s paper, though, Hilbert immediately realized that one could generate the curve by an iterative procedure such as pictured above.

Hilbert demonstrated the process by creating a new, (relatively) simpler space-filling curve, which he published in an 1891 paper, “Ueber die stetige Abbildung einer Linie auf ein Flächenstück.” Hilbert’s original drawing of the creation of the curve is shown below.

These pictures give a simple idea of how the process works. Hilbert noted that, in his curve, the line could be divided into four segments, and each segment of the line corresponds to 1/4 of the entire square. Then, each of the four line segments can be divided into four smaller segments, and each of those segments corresponds to 1/4 of each of the smaller squares, and so on. This process results in an increasingly convoluted path with each iteration.

Using my own code, the first 6 iterations of Hilbert’s curve are shown below.

The key to generating the Hilbert curve is to notice that the structure looks similar, albeit smaller, at each level. In fact, the 2nd form of the Hilbert curve above consists of four shrunken copies of the original curve, appropriately joined together. Step 1 is shown below.

Step 2 is simply shrinking the original square and moving it to the upper left corner.

Note the red line that has been added to connect the two pieces! The third step involves shrinking the original square and moving it to the upper right corner.

Finally, we perform a final series of transformations to get the final piece.

We have the next iteration of the Hilbert curve! To get the next iteration, we take this current iteration and perform the same four operations on it, and so on, and so on! A Peano curve is generated in a similar manner, the main difference being that the square is divided into nine pieces, and each iteration produces nine shrunken, rotated and mirrored copies of the previous iteration.

This process, however, indicates that a Hilbert curve and, similarly, a Peano curve, have the same structure on different size scales. In other words: if you zoom in on a part of the curve closer and closer, it looks pretty much the same. For some readers out there, this will sound suspiciously familiar: it sounds like a fractal!

For those unfamiliar, a fractal may be loosely defined as a geometrical object that is self-similar at different zoom levels. The concept was introduced by Polish mathematician Benoit Mandelbrot in the 1970s, and it can describe many real-world objects that look similar near and far away: clouds, coastlines, mountaintops.

A classic and very familiar example of a fractal is a Sierpinski triangle, as shown below.

This triangle can be created in a number of different ways, but one way is directly connected to the space-filling curve construction. Start with a full triangle (or a single point) in the middle of the unit square. Shrink this triangle or point in half, and move three shrunken copies to the corners of the original triangle. Keep doing this process, and the Sierpinski triangle forms.

One of the most unusual properties of many fractal is that they possess a *fractional dimension*! For instance, although the Sierpinski triangle is spread out over the unit square, it is filled with “holes.” In a loose sense, it takes up more space than a traditional one-dimensional curve, but takes up less space than a traditional two-dimensional surface. In fact, it can be proven that the Sierpinski triangle has an effective fractional dimension of 1.585, between 1 and 2!

The space-filling curves of Hilbert and Peano may also be considered self-similar fractals. However, because they fill the entire square, their fractal dimension is simply 2! They are, in a sense, the weirdest sort of fractals because they are a complicated, self-similar structure that results in a very simple geometric object.

The choice of Sierpinski’s triangle to illustrate the relationship between fractals and space-filling curves is no accident: its “discoverer,” Wacław Franciszek Sierpiński (1882-1969), developed his own space-filling curve in 1912, after becoming obsessed with set theory a few years earlier. Wikipedia provides a charming summary of how this obsession began:

In 1907 Sierpiński first became interested in set theory when he came across a theorem which stated that points in the plane could be specified with a single coordinate. He wrote to Tadeusz Banachiewicz (then at Göttingen), asking how such a result was possible. He received the one-word reply ‘Cantor’. Sierpiński began to study set theory and, in 1909, he gave the first ever lecture course devoted entirely to the subject.

Sierpiński’s space-filling curve, not surprisingly, ends up filling a triangular region (though it can be adopted to fill a square quite readily). The first few iterations of this curve are shown below.

What about higher dimensions? Remarkably, Cantor not only demonstrated that it is possible to draw a curve that fills an area, it is also possible to draw a curve that fills a *space of any number of dimensions*. In three-dimensions, a Hilbert curve is readily constructed, as illustrated below.

Even more mind-boggling, it is possible to draw a curve that will fill a countably-infinite-dimensional region of space, i.e. a region of dimensions! Obviously, however, we cannot draw a nice picture to illustrate this effect.

There is at least one subtlety in the construction of space-filling curves worth noting. Even before Peano derived his curve, it had been shown by the German mathematician Eugen Netto (1848-1919) that it is not possible to draw a continuous curve that *exactly* fills the square. The implication of Netto’s theorem is that the Hilbert and Peano curves in fact *over*-cover the unit square: in the limit of infinite iterations, these space-filling curves run over themselves countless times.

So where does all of this information leave us? The work of Peano (and Cantor before him, and Sierpinski and Mandelbrot after him) demonstrates that the relationship between the dimensionality of an object and its structure is non necessarily a simple one. It is possible to construct a continuous “curve” that is one-dimensional, two-dimensional, fractional-dimensional, or *infinite*-dimensional. Peano’s work also demonstrates that beautiful mathematics can be found in even the simplest of squares!

*P.S. The title of this post was inspired by this classic series.*

*Update: Interestingly, the Sierpinski triangle can also be constructed as a continuous line, known as the arrowhead curve. This is an explicit demonstration that a curve can be constructed that forms a structure with a fractional dimension.*

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For this post, I learned a lot about space-filling curves from the wonderful mathematical book by Hans Sagan, *Space-Filling Curves* (Springer Science & Business Media, New York, 1994).

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* I’m so sorry for that.

** See part I, part II, part III, and part IV.

*** As described in Hubert Kennedy’s *Life and Works of Giuseppe Peano* (Springer, 1980).

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John Blackburn (1923-1993) was a prolific author of books containing a unique blend of thriller, science fiction and horror, producing some 25 novels over the course of his career (I talk about a number of them on this blog). His work was also extremely popular in his time, but was sadly almost completely forgotten after his death — until Valancourt started reprinting his books over the past year.

*The Cyclops Goblet* marks an interesting departure from Blackburn’s usual fare. It can be considered a classic “caper” story, in which a series of crooks attempt to pull off a spectacular heist — and try to double-cross each other in the process!

*The Cyclops Goblet* features as protagonist the scoundrel Bill Easter: a former gangster, bodyguard, and current and perpetual con-man. As the novel begins, he is flat-broke in London with his common-law wife Peggy Tey, and the two of them are struggling to pay the rent when they are contacted by Colonel Wellington Booth, the head of a fascistic British Nationalist movement. Booth tasks them to help steal a priceless Renaissance art collection, the Danemere Treasure, which is currently locked in a perfectly impregnable vault. Or is it impregnable? And is it locked in the vault? As Bill and Peggy join the plot, they learn that the treasure is reportedly cursed, and supposed to bring death and misery upon any who owns it. What follows are twists and turns, crosses and double-crosses, as multiple parties work together and against each other in order to seize the treasure for themselves. In the end, they will all find that the curse is not mere legend, and a truly deadly secret is tied to the fate of the artifacts.

This is a really good Blackburn novel and, as I have said, it has a rather different tone than many of his books. There is a very dark humor laced into the story, but it is at heart still a tale of horror. It is the third book featuring the anti-hero Bill Easter, following *Deep Among the Dead Men* and *Mister Brown’s Bodies*, though it is not necessary to read the earlier books to enjoy this one. (I myself read *The Cyclops Goblet* before the others.)

In my introduction, I give a bit of background on John Blackburn and his works, and summarize the earlier stories of Bill Easter to get readers up to speed on his adventures, which do play a significant role in this novel. I also note several possible inspirations for the story and settings of *The Cyclops Goblet*: Blackburn often featured current events and unusual locations in his books, in addition to his own life experiences.

In short, I can recommend *The Cyclops Goblet* as a very enjoyable horror-thriller-comedy! Its unusual characters and bizarre story make it worthwhile to explore.

I expect to write at least one more introduction to a Blackburn novel in the near future, possibly for one of his best works! Keep an eye on this blog for details.

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* “Masterful,” of course, in my own humble opinion, which is also masterful, so you know it’s true. (See how this works?)

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It really is a great collection, and I feel proud to have made it in once again, especially in light of the company I keep there, which includes Deborah Blum, Jennifer Ouellette, Maggie Koerth-Barker, Christie Wilcox, Melanie Tannenbaum, Jason Goldman, Krystal D’Costa, Blake Stacey and many more! The full list of authors and articles can be read at this blog post, and the collection can be ordered through The Creativist at this link.

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The more I research, the more it becomes clear that cats caused all sorts of mischief in the scientific community in the late 1800s! The source of this mischief is the feline ability to turn themselves over in freefall and land on their feet, even when released at rest with no rotational motion. As I have noted in a previous post, this ability is, at a glance, seemingly at odds with the conservation of angular momentum — though in reality it is not! In a rigid body, the angular momentum of the object is directly proportional to its rotational speed. In a flexible body such as a cat, however, different sections can rotate in different ways, producing a net overall rotation even if the cat’s total angular momentum remains zero.

The debate, and confusion, was sparked in 1894 when Étienne-Jules Marey presented a sequence of photographs to the Paris Academy showing a cat flipping over at rest. As was later reported in the New York Herald, Marey’s observations were met with hilarious incredulity at the meeting:

When M. Marey laid the results of his investigations before the Academy of Sciences, a lively discussion resulted. The difficulty was to explain how the cat could turn itself round without a fulcrum to assist it in the operation. One member declared that M. Marey had presented them with a scientific paradox in direct contradiction with the most elementary mechanical principles.

Fortunately for the dignity of the scientific community, researchers quickly realized that Marey was correct: non-rigid bodies can flip over, even starting from rest, while conserving angular momentum. This led to a century-long investigation into how, exactly, a cat achieves this feat (you can read about the history in another blog post of mine).

Other researchers, however, found immediate inspiration in the cat’s newly-appreciated ability and its implications for physics. Inspired by Marey’s work, mathematician Giuseppe Peano in fact argued that the cat’s flipping talent provided a lesson and a solution for a problem in the most unlikely of places: geophysics!

Giuseppe Peano (1858-1932) is not well-known to the general public, but he was a formidable voice and researcher in mathematics, publishing over 200 books and papers during his lifetime. He is perhaps best known for the so-called Peano axioms, a set of axioms describing the natural numbers that he formulated in 1889. Peano was also one of the founders, along with Georg Cantor, of mathematical set theory, a subject also discussed extensively on this blog recently [1]. One of Peano’s most intriguing results is related to Cantor’s demonstration that there are the same number of points in a one-dimensional line and a two-dimensional plane; in 1890, Peano introduced what is now called the Peano curve, an iterative process to construct a continuous line that, in a limit of infinite steps, fills the plane entirely.

This curve is quite fascinating, and I will blog more about it in a future post, but for now let’s focus on Peano’s interest [2] in cats! By the 1890s, Peano had become a very distinguished and well-known academic. In 1890, he became a full professor in the University of Turin, and in that same year he founded his own journal, *Rivista di Matematica* (“Journal of Mathematics”), to promote new and exciting results in the field. Only a year later, he started work on the *Formulario Mathematico*, an attempt to compile all of the fundamental theorems of mathematics into a single “encyclopedia.”

I mention all these achievements to emphasize that Peano was extremely active and ambitious, and there seemingly wasn’t any mathematical issue he wasn’t aware of or afraid of investigating. When Marey’s observations, and the controversy surrounding them, came to his attention, Peano devised his own explanation, that he published in the January 1895 issue of *Rivista di Matematica* [3]:

But the explanation of the cat’s motion appears to me quite simple. When the animal is left to itself, it describes with its tail a circle in the plane perpendicular to the axis of its body. Consequently, by the principle of the conservation of angular momentum, the rest of its body must rotate in the sense opposite to the tail. When it has rotated as much as it wishes, it halts its tail and with this simultaneously stops its rotary motion, saving in this way itself and the principle of angular momentum.

In short: Peano suggests that a cat flips itself over by a rotation of its tail! By rotating its tail over multiple circles in, say, a counter-clockwise manner, the cat’s torso should rotate in the clockwise direction, albeit at a slower rate, keeping the total angular momentum at zero. Bringing back my cylindrical cat from an earlier post, we have something like below.

This model is physically possible, but incorrect! As we have discussed in my previous cat post, the bulk of a cat’s rotation comes from a torso twist, not a tail rotation. Also, a cat would need to rotate its tail in many complete circles in order to turn its much heavier body completely over, and this is not borne out by the photographic evidence. Finally, it is known that even bob tail cats cat turn over without a problem! This isn’t to say that cats don’t ever use their tails to aid in rotation, but that it is not the most important, or even essential, part of their flip.

The cat’s ability, however, served to inspire Peano in tackling a significant and hot topic in geophysics at the time, related to the overall motion of the Earth itself!

By Peano’s time, astronomers had already long known that the direction of the Earth’s axis of rotation is not fixed. Analogous to the motion of a spinning top or gyroscope, the axis traces out a circular path, known as a precession, with a period of 26,000 years, and wobbles slightly about that circle, or undergoes a nutation, with a period of 18.6 years. This precession and nutation are driven by the interaction of the Earth with the gravitational forces of the Sun and Moon. (A nice video of gyroscope precession and nutation can be seen on YouTube.)

However, another form of nutation was predicted in 1765 by the great mathematician Leonhard Euler, who suggested that the spheroidal (slightly non-spherical) shape of the Earth allows for a free nutation: an additional small wobble of the Earth’s axis with respect to the solid Earth that is self-contained and not induced by external forces. With some serious mathematical gymnastics, Euler predicted that this free nutation should have a period of 305 days.

The amplitude of this small wobble — which is equivalent to a variation in lattitude — is very small, and it was not until 1891 that astronomer Seth Carlo Chandler presented the first definitive measurements of what became known as the Chandler wobble. Chandler observed a variation of axial position of about 30 feet (9 meters) with a period of 433 days.

The discrepancy between Chandler’s measurements and Euler’s prediction was initially somewhat baffling: in fact, numerous astronomers had previously tried and failed to find the Euler nutation simply because they were too focused on Euler’s estimate. In a February 1891 meeting of the Royal Astronomical Society [4], a simple explanation of the difference was given:

Astronomers had hesitated to accept the 427-day period, even in face of the very strong evidence of the 1860-1880 observations, owing to the difficulty in accounting for it theoretically. It had been pointed out by Euler that, treating the Earth as a rigid body, the period of rotation of the pole must be 306 days. Professor Newcomb, however, happily pointed out that a qualified rigidity (either actual viscosity or the composite character due to the ocean) afforded an explanation of this longer period; and after this suggestion Mr. Chandler’s 427-day period was well and even warmly received.

In short: Euler had assumed the Earth to be a perfectly rigid body; however, the flow of material in the planet’s interior, as well as the motion of the crust and the oceans, result in significant differences from Euler’s model [5].

It is one thing to explain an unexpected phenomenon, and another thing entirely to provide a rigorous theory to back up the explanation. When Peano encountered the problem of the falling cat in 1894, he immediately saw in it a kindred spirit to the nutating Earth, and began working on mathematics to explain the latter. Both problems involve an object changing its orientation in space entirely in the absence of external forces, and both problems can be qualitatively explained by internal motions of the object in question.

There is something terribly ironic about Peano’s inspiration: where physicists are normally known for oversimplifying problems — there is the famous joke about approximating a cow by a sphere — Peano went in the other direction, envisioning a sphere as a cat!

In an 1895 paper titled “Sopra lo spostamento del polo sulla terra” (“Concerning the pole shift of the Earth”), Peano presented his own mathematical theory of the phenomenon, giving due acknowledgement to the cat:

At the end of last year at the Academy of Sciences in Paris it was proven by experiment that certain animals, such as cats, can, as they fall, through internal actions, change their orientation. It is immediately explained by mechanics the possibility of this motion. In a short article published in the

Rivista di Matematica(in the beginning of January 1895), I discussed briefly the question, I tried to describe the cyclic motions by which the cat actually rightens, and added other examples.This naturally brings us to the question: Can the globe change its orientation in space, through internal actions alone like every other living being? From the mechanical aspect the question is the same. But Prof. Volterra has the merit of proposing it first. He made it the object of some notes presented in this Academy, and the first of which is on the 1st of February 3.

“Prof. Volterra” referred to in the passage quoted is Vita Volterra (1860-1940), another Italian mathematician and physicist of significant influence. Superficially, it probably seems quite decent of Peano to have acknowledged Volterra’s work and its primacy in the scientific literature. In reality, however, Volterra took Peano’s entire presentation as a huge scientific slap in the face. First of all: Volterra and Peano were both working together at the University of Turin at the time, and would literally see each other every day, but Peano never mentioned his own work on the Earth’s wobble, though he well knew that Volterra was working on it as well. This snub would have seemed at best uncollegial, and at worst outright deceptive. Second: reading between the lines of Peano’s description above, it strongly implies that Volterra’s own work was inspired by Peano’s cat physics discussions! Note that Peano very clearly emphasizes that his cat article appeared in *January*, and that Volterra’s first article on the Earth appeared in *February*. Volterra had in fact been working on the problem for a significant period of time, and it so happened that the publication only appeared in February.

Reading accounts of the May 1895 meeting of Turin’s Royal Academy of Sciences [6], one can almost picture the smoke that must have been pouring from Volterra’s ears as he listened to Peano speak. In fact, as soon as Peano finished, Volterra leapt to his feet and argued not only that his work had come first but that Peano’s paper was based on flawed incomplete data. Volterra then asked for permission to go home and get an additional paper to present to the Academy to back up his claims; he was given this permission.

What followed was a year-long, very public feud between the two scientist-mathematicians. At the heart of it, there was more than arguments of authorship: the two men had very different, even conflicting, approaches to research, and the wobble of the Earth became the perfect battleground for their views. Peano was very much a pure mathematician, attempting to develop cutting-edge mathematical formalisms for solving problems. In the case of the wobble, he championed the use of “geometric calculus” and argued its superiority over the classical method of solution by Volterra, who was more of a physicist and was employing conventional calculus.

Over the course of the next year, the two mathematical giants would fire shots at each other through various papers, using subtle and personal jabs at each other as well as criticisms of each other’s work. In a paper on 23 June, for instance, Peano fails to acknowledge Volterra’s work on the Earth’s wobble at all! Later, Peano implicitly responded to Volterra with a paper of exactly the same title as Volterra’s. Volterra was much more direct, and criticized Peano in print of “forgetting” to cite his work, and condemning him outright for misrepresentations. Peano, in turn, continued to mention the damn cat in his papers, continually hinting that Volterra had drawn inspiration from it!

Finally, in early 1896, Volterra seems to have tired of the dispute. He wrote a final letter to the President of the Accademia dei Lincei (“Italian Academy of Science”), arguing yet again that Peano had no standing in their quarrel [6]:

Having thus shown to be empty and unfounded any of the points of criticism made of me by Peano, and that his assertions are neither original nor exact, he himself having recognized them as such, for my part I hold this polemic definitively closed.

As I have already noted, both mathematicians would, over the course of their careers, make far more significant contributions to mathematics and physics than the wobble, so I consider their duel a “draw.”

But what is the source of the Chandler wobble? Though it was appreciated that the Earth can wobble, it was less clear what keeps the wobble going or, in fact, occasionally changes the nature of it dramatically. It was only in the year 2000, in fact, that a paper appeared in Geophysical Research Letters that used computer simulations to suggest that 2/3 of the wobble comes from fluctuating pressure on the ocean bottom, and another 1/3 comes from atmospheric fluctuations.

It is remarkable, though, that some of the earliest discussion of this wobble was sparked by a falling cat — and that the cat led to an angry argument among scientists!

*Postscript: I almost wish I could say that this is the last cat physics blog post, but there still may be yet one more coming down the line in the near future!*

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[1] My four-part series on the weirdness of infinity in set theory: part 1, part 2, part 3, part 4.

[2] This discussion is based partly on the book by Hubert C. Kennedy, *Peano: Life and Works of Giuseppe Peano* (D. Reidel, Holland, 1980), and partly on Peano’s original papers.

[3] Here I use Kennedy’s translation of the paper, which is much better than my Google translate version.

[4] Monthly Notices of the Royal Astronomical Society, Vol. LIII (1893), p. 295.

[5] An aside: astronomer Simon Newcomb was also an early author of science fiction, as I have discussed on this blog before.

[6] The feud is discussed in Judith R. Goodstein, *The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician 1860-1940* (American Mathematical Society, 2007), Chapter 9.

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Case in point: about a month ago, while my wife was out of town, I broke out my DVD copy of *The Lost Skeleton of Cadavra* (2001), a delightful spoof of low-budget science fiction from the 50s, and *The Lost Skeleton Returns Again* (2009), its very silly sequel. Both movies, and others, were written, directed, and starred in by the all-around auteur Larry Blamire; on a whim, I checked to see if he was on Twitter, and to my delight, he is, and to my further delight, he graciously acknowledged my existence!

While following him on Twitter, I recently learned that Blamire has also written a compilation of horror stories, *Tales of the Callamo Mountains* (2008):

*Tales of the Callamo Mountains* compiles 13 previously unreleased stories of horror, focused in and around the fictional Callamo Mountain Range. Set in the turbulent time following the Civil War, the tales feature settlers, marshals, laborers, soldiers, cowboys and others who, traveling in the remote and untamed West, find themselves up against nature and forces far more diabolical.

These stories are *good*. I was immediately hooked once I starting reading the first of them, and could hardly put the book down until I had finished it a couple of days later. A couple of the tales are absolutely brilliant, in my opinion.

The stories fall into the relatively rare sub-genre known as “Western horror,” mixing classic Western stories and characters with supernatural horror. Possibly the first author of this genre was none other than Robert E. Howard, creator of Conan (and inspiration for this blog). In the 1930s, he wrote a slew of “Weird West” stories, such as The Horror from the Mound (1932), in which a Texas cowpunch breaks into an old burial mound and unearths something nasty. Since then, there has been a steady stream of Western horror stories in film, television, literature, and comic books, notably in some of the stories of my favorite author Laird Barron (see, for example, here).

For me, such tales are at their most powerful when they serve as a metaphor for an indifferent and unforgiving natural world. Settlers and frontiersmen traveling the Wild West had only themselves to rely on, and a single minor misstep could be fatal. (Those who grew up in the 1980s learned this lesson well from the video game Oregon Trail.)

The 13 stories of *Tales of the Callamo Mountains *fit this mold perfectly. All of the stories are good; I summarize some of my favorites below.

- The Line Shack. Two cattle drovers set up to spend the winter in the shadow of the Callamo Mountains, to watch the herd until the mountain passes clear. To keep his overly talkative colleague quiet, Tom Sidrow suggests that he investigate the dense tangle of dead trees and bramble in the back of their shack. But
*something*is back there, and disturbing it leads to deadly consequences. - The Unexpected Stop. When a late night passenger stagecoach makes an unexpected stop in a remote section of the mountains, it precipitates a nightmare of paranoia and terror.
- On Tuesday I’ll be in China. The monotony of farming is broken when an unusual flying machine passes over Enoch in the fields one day. When the machine returns, at the same time several days later, he vows to investigate further. As Enoch gets closer and closer to the machine and its occupants, he finds that it may be more — or less — than it seems.
- Tub Seven at Engel-Reis. John Box and his crew have been assigned to rehabilitate the old encrusted amalgamating pans at an abandoned mill. The process is hard work, but uneventful, until the men get to Tub Seven, overflowing with an unknown brew of toxic chemicals. Something unnatural and unspeakable has been hibernating at the bottom of Tub Seven, and it does not take kindly to being disturbed.
- The Last Thing One Sees in the Woods. Red Henry is undertaking a routine mapping assignment from the army in the desolate wilderness when he comes across an object that has no business being out there. Inside, there is something that has no business existing at all.

Blamire does a wonderful job of clashing the mundane existence of frontier folk with inconceivable supernatural horrors. His characters are reminiscent of those of Laird Barron; as I have noted previously, the working-class nature of the protagonists makes their fates even more horrifying, as they are clearly in over their heads in most cases.

Blamire absolutely shines in a lot of little details which provide incredible punch to his stories. In* The Line Shack*, look out for the terrible irony of a man wielding a powerful gun in a situation where he can’t use it. In *The Unexpected Stop*, look for what may be the best use of darkness as a palpable force and menacing character I have ever seen.

Speaking of characters, the Callamo Mountains are the only recurring character in the stories, and they are a formidable presence throughout. Many horror authors, such as the excellent W.H. Pugmire, have developed their own sinister fictional locations to serve as the backdrop for their tales, and the Callamo Mountains work well in this respect. The mountains even overtly exhibit their own personality in several stories, and it isn’t a pleasant one!

To summarize: *Tales of the Callamo Mountains* is a great collection of western horror stories, and well worth reading. It’s been a few years since the collection came out, but I hope Larry Blamire has a few more tales in him to share sometime!

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