The Four-Dimensional Lord of Dance

I wrote two posts last year dealing with the fourth dimension:

• Views of a Tesseract
•  Arts and Crafts in Four Dimensions

The focus was mathematical, but along the way I looked at how the fourth (spacial) dimension appears in the works of authors like H. P. Lovecraft, Robert Heinlein, H. G. Wells, Jorge Luis Borges, and, most memorably, Madeleine L'Engle, whose A Wrinkle in Time is about to appear as an uninspired-looking Disney movie in March (sigh). I also talked about mathematical visionaries and mystics like Paul S. Donchian and Charles Howard Hinton, both of whom made real contributions to the field, if only in the sense that they developed and humanized what the academics were saying in their inaccessible research articles, and both of whom might be labeled as cranks or crackpots.

Since then I've done a little research on Hinton, Donchian, et al., and have found a number of other links between the idea of a fourth spacial dimension and various forms of spirituality or mysticism. For instance, the German astronomer Friedrich Zöllner (1834 – 1882) apparently tried to use the fourth dimension explain Spiritualist phenomena. In his eagerness, he was imposed upon by the medium Henry Slade in experiments that have since been debunked. Fantasy and horror authors in their turn used the claims of Spiritualism in their stories; some, like Arthur Machen and Algernon Blackwood, actually subscribed to its views. Hinton, who wrote a number of "scientific romances" himself, was a post-Christian altruist who speculated that spiritual agencies might work by means of the fourth dimension and believed in something like eternal return.

Some Christians of the late Victorian era, disconcerted by the advance of materialism, attempted to colonize the fourth dimension themselves. For other Christians, such as the liberal theologian Edwin Abbott Abbott (author of Flatland: A Romance of Many Dimensions, a book much admired by Hinton), higher spacial dimensions were merely a metaphor for gradual way in which the human mind must approach divine truths. 

[source]

Salvador Dalí appears to have used the fourth dimension in a similar way, in his famous 1954 painting Crucifixion (Corpus Hypercubus), which portrays Christ crucified on the net of a tesseract / hypercube / 8-cell (Schläfli symbol {4,3,3}) hovering over a square grid (Schläfli symbol {4,4}), illustrating the incomprehensibility of God to man.

I have in my hands a Dover edition of Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, which includes several of Hinton's scientific romances. It's edited and has an excellent introduction by Rudolph v.B. Rucker, also known as Rudy Rucker, author of the Ware Tetralogy and a modern tribute to Flatland and all-around sci-fi author of note. So no doubt I'll soon be posting about all of this yet again.

*

My post on four-dimensional arts and crafts includes an account of my building the sections and net of a 120-cell. More recently, I've printed and built the sections and net of a 24-cell, which is a regular four-dimensional polytope built from twenty-four octahedra.

The sections proceed as follows, with colors given as the craft paints I bought at Wal-Mart: (I) the octahedral cell at the "south pole" (Parchment); (II) the truncated octahedral section cut by a hyperplane through the midpoints of the edges "above" the south pole (Parchment and Real Brown); (III) the cuboctahedral equatorial section cut by a hyperplane through the set of vertices to which these edges connect (Look At Me Blue and Real Brown); (IV) the truncated octahedral section analogous to Section II but in the "northern hemisphere" (Look At Me Blue and Real Brown); and (V) the octahedral cell at the "north pole" (Coffee Latte).

The net has the "south pole" at the center and the "north pole" at the base. For reasons fully known only to my subconscious, but partly inspired by Dalí's painting above, I decided to model it after traditional depictions of the Hindu god Shiva as Nataraja or Lord of Dance, with three-fold rotational symmetry.

Shiva is the destroyer, and his dance is the cosmic dance of creation / destruction. That puts me in mind of the line from the Bhagavad Gita, uttered by Krishna, quoted by Robert Oppenheimer, and used by me in the title of a short story: "Now I am become Death, destroyer of worlds."

Blade Runner 2049

The original Blade Runner is what first got me into film noir. It evokes noir's guilty heart where other attempts at "future noir" only get the trappings right. Rick Deckard isn't what you'd call a hero: compelled by self-interest, he's hunting down escaped replicant-slaves whose only crime (so far as he's concerned) is their presence on earth. Though he tries to hide it from himself, he's fully conscious of their humanity and fear as he blows them away, as in that slow, sad scene where he shoots a terrified female replicant in the back as she flees into a department store. It's Roy who "earns" his soul at the end, who, with his crucified hand, saves Deckard's life and possibly his soul; we're never quite sure whether Deckard is even really human.

Blade Runner also has a poignant but seriously messed-up love affair, another noir element ratcheted up almost to the intensity of myth. The vast dark urban abysses (reminiscent of night scenes in Phantom Lady and others) and mountainous decayed buildings (like the Bradbury, where the classic noir D.O.A. also ends) meld seamlessly with the iconic score and the plot's moral grayness and lack of resolution.

And, as with noir, I've somehow also found Blade Runner a solace when going through bad times. I can remember one period in my life when I watched it once or twice a week. I think a lot of people would say something similar. And there's not all that much to the plot. It's more of an immersive audiovisual mood experience than a movie. So there's probably a lot of other people who think it's okay but kind of dull. They're mystified by people like me, to whom it means so much.

All of which is to say, Blade Runner 2049 has a lot to live up to. Well, I went and saw it at the $4.00 matinee this weekend, and I...think it kind of succeeds. For me, at any rate. Not quite, of course. How could it? But it continues the story, adds to the background, and deepens the world without doing any violence to the thrust of the original, all while maintaining its own narrative independence.

I have to say, I went into it with pretty low expectations. I've been less than impressed with Ridley Scott's attempts to rekindle the Alien magic, and as for Harrison Ford's reprising the roles that made him famous, well, um, yeah, same thing. So I'm very excited that Blade Runner 2049 proved so much better than I'd expected.

I don't want to say too much, because it's got a good plot with plenty to spoil. But I will say that it's another true noir, with a written-off protagonist, a bizarre love story, and an ambiguous ending. What we see of Deckard (not much, thankfully) does little to explain or humanize him. Like the original, it's elliptical and rather sad. The city is the same, down to the now-retro-futuristic Atari signs. The texture is as rich, too, though perhaps a bit contrived in places, and not quite as authentic feeling. We get to see the world outside L.A., including a protein farm and a humongous waste dump, and it's a beautifully unlovely place. The CGI is some of the best I've seen, though, for me, nothing could ever quite equal the practical effects of the original.

There are some interesting allusions. The protagonist, a replicant Blade Runner, is called K, seemingly a reference to Kafka's bewildered protagonists. One of the short "prequel" films that came out in advance (Blade Runner 2048: Nowhere to Run) centers on a copy of Graham Greene's The Power and the Glory, a novel about a hunted priest – a broken "whisky" priest – in the aftermath of the Mexican Revolution. The parallel is obvious, and, given the film's preoccupation with the nature of the soul, fitting. Other religious and mythological allusions abound.

In the end, I think it's movie that I'll have to rewatch once or twice to decide what to make of it, but it's passed the first test so far.

Warbles and Bots: Out of the Hive

SHE WAS A CYBORG: PART-ORGANIC WHALE, PART-MECHANIZED SHIP

What happens to humanity when it "adjusts" to hive living? 

Some writers write to earn a living. Others write because they've got to get something out of their system. When it comes to science fiction, I've learned to be leery of successful commercial writers. Not because they're bad, necessarily, but because I'm almost sure to be bored by them. My favorite authors tend to be the ones who wrote themselves out fairly quickly, expending their visions in an explosive spurt, leaving behind rich mines for more conventional writers to exploit. Their work might be naive or outre, but to me that's just part of their charm.

It was a different time. A time when sci-fi
still brought terror, wonder, and arresting
strangeness to the reader. A time when a
publishing company could proudly decorate
its books with Magritte-esque parades of
schlubby naked nebishes, and people would,
apparently, still buy them.

I'd say that Thomas J. Bassler, M.D., who wrote under the pen name T. J. Bass, sits squarely in that category. I was recently reminded of his work by Fletcher Vredenburgh's review of Half Past Human over at Black Gate. I'd never that novel before, but I had enjoyed The Godwhale a long time ago.

Bass's career spanned only six years, lasting from 1968, when he sold his first story, until 1974, when The Godwhale was published. He wrote only the two novels I've mentioned. Both take place in a far-future earth in which the vast majority of humankind's teeming trillions live in a global subterranean society called the Hive. I can't really compete with Fletcher's review, which I think captures the spirit of the books perfectly, and I urge the reader to go take a look at that if interested. Instead, as usual, I'll just natter on about what strikes me about them.

Part of what makes Bass unique / bizarre is his habit of describing everything, from sex to warfare, in medical terms.

When he flashed his helmet light down, vertigo gripped his cardio-esophageal junction. (Half Past Human, p. 96)

Perhaps this is just a reflection of his day job, but it has the effect of making his characters seem like rats in a gigantic, not-very-ethical experiment. Overall, Bass writes with the air of a physician conducting the examination of a patient, gently squeezing pustules and prodding necrotic tissues with a cool detachment. His outlook isn't grim or moralistic, and the weird, disgusting scenes he describes are often quietly amusing as well. In general, Bass isn't so much sounding an alarm as saying, more or less cheerfully, "Hey! Let's see what happens if we extrapolate this trend in our society!" Which, in a way, ends up being much more disturbing than the hysteria of other entries in eco-science fiction.

It's hard to select a favorite passage. Many stand out in one way or another. So let's just look at the scene quoted above, in which a Hive inhabitant goes "ratting" in the dark, dusty world of 'tween walls, hoping to score some extra "flavored calories."

He dust-waded along the top of a large pipe. It was hollow. Voices and and shuffling vibrated. It was a crawlway. The larger rats became more numerous – and bolder. They remained stubbornly in his path until he nudged them with his toe. They wouldn't be too tasty. The sweet stink of the nests hit him. Moist and dripping, the huge cool sphere of the membrane filters loomed ahead. The city's sweat condensed and trickled down the sphere's outer sphere – providing drops of drinking water for the rodents. [...]

Selecting a large nest he thrust in his hand. Expecting mother-with-food, the soft young rats swarmed onto the glove. He pulled out three handfulls and squeezed them through the sphincter of the anoxic bag. Their squirming and squeaking ceased. (Half Past Human, p. 97)

The hunter goes on to have his catch pressed into moist wafers. He shares them with a friend, who savors "the salty fluids, tangy viscera, and iron-rich muscle and blood" (HPH, p. 99). They go on to discuss his friend's devotion to Dabbing ("'Dirt, adobe, and bamboo – DAB'"), a stress-reducing quasi-religious practice.

"The most important thing [...] DAB protects you from is suicide. That is the number one killer. Inappropriate Activity – old I.A. Without DAB your ectodermal debris sensitizes you. All your skin scales, hair and skin oils get into the house dust and feed the mite, Dermatophagoides. The mite acquires ectodermal protein antigens. As you live with the mite and breath [sic] in dust – mite fragments – you build up antibodies against them. Antibodies against your own ectodermal antigens. When the titre gets high enough the antibody cross reacts with your own neuroectoderm – your brain. Hence the logarithmic correlation between crowding and I.A. Between house dust sensitivity and suicide. Humans who nest with rugs, drapes and stuffed furniture have the highest suicide rate. Humans who live with dirt, adobe and bamboo the lowest." (Half Past Human, p. 97)

The "erotic" scenes retain a certain tenderness, despite being described in the same oddly technical terms.

"I am going to enjoy living with a man who is good with his hands," she said. Taking his wrists she moved his trembling hands over her tunic. Her soft erogenous zones radiated warmly. His autonomic synapses struggled with the increasing excitement. Passion flared somewhat erratically, and then, abruptly, faded. While he stood there, the heat in his loins faded away – leaving fatigue. [...]

"You have just recently polarized," she consoled. "Your meld reflexes need time to synchronize. We will work on it, and it will improve." (Half Past Human, p. 18)

Half Past Human takes place in and above the Hive's vast subterranean network of shafts, tubes, and cubicles, following the intersecting lives of large class of characters: artificially prepubescent nebishes, polarized males and females, paleolithic warriors and wizards, ageless wanderers, earnest mecks, and, of course, G.I.T.A.R. and Olga. The Godwhale, on the other hand, takes place mostly in the ocean. The main character, if it has one, is Rorqual Maru, a lone cyborg whale-ship plankton harvester bent on aiding humanity.

But the story begins much earlier in time, following the adventures of the hapless hemihuman Larry Dever, who starts his career by getting cut in half in a singularly stupid accident. Thanks to medical advances, he doesn't die, but his quality of life isn't what it was.

Larry turned on his refresher and grasped a ceiling rung of his horizontal ladder. The mannequin walked away slowly, pulling flexible tubing out of his various surgical stoma. Sucking  sounds. Drops of urine and feces soiled the meck's breastplates with yellow and granular brown. Larry progressed across the monkey bars to the hot shower, where he emptied his visceral sacs down the drain. (The Godwhale, p. 23)

Life isn't so pleasant for Larry without his lower half, despite his talking prosthesis.

"This is great! It feels like I am really running. It's the lactate you're putting in my Blood Scrubber. Now if you can just give me back my sex life."

Mannequin shared and updated with distant Library. "That too can be arranged; a mechanical penis for me and midbrain electrodes for you. Meck sex can be pleasant with a wired reticular system."

Larry grinned, assuming that he was the object of a very funny robot joke. (The Godwhale, p. 16)

Larry eventually enters suspended animation, hoping for a future "cure" that doesn't involve raising and harvesting a clone of himself. He awakens instead in the decidedly unpleasant world of the Hive. He escapes, coming into contact with Rorqual Maru, a spunky little meck named Trilobite, and a race of humans adapted to life underwater, teaming up with his latter-day genetic progeny: a gargoyle Tweenwaller named Big Har, and a bio-engineered superman known as A.R.N.O.L.D. (Augmented Renal Nucleus Of Larry Dever) who becomes King of the ocean and sometimes thinks of himself as a chicken.

"BACK OFF!!" shouted ARNOLD, riled to the point of hearing "cluck, cluck" in his subconscious. (The Godwhale, p. 167)

I read The Godwhale back when I was in college. Then I went to grad school and purged my shelves of such oddities. But now it's back. Rereading it has been a bit strange, because I'm realizing that I must have internalized more of it than I'd thought. Taken together, Half Past Human and The Godwhale amount to the most bizarre, disgusting, wonderful stuff I've read in quite a while.

A hundred miles up-sump the sewer conduits sang with pneumatic belches of dead city gases: incoles, skatoles, methane, ozone, and carbon monoxide. [...] 

Their mold-flecked dinghy drifted sideways, its bow wedged into a raft of nondescript, floating debris. Hemihuman Larry hunkered down, swatting flies. The blackness and echoes told them nothing. Their progress was marked by aerial mycelia which swept across the boat's wet ribs and snagged in their hair. Persistent swarms of sucking botflies hovered over them. Their throbbing backs sponged-out with bots and warbles – the cutaneous abscesses that contained the vigorous fly larvae. 

"The damned itching is getting worse," complained Larry. "A new crop must be maturing." He wiped his hand across his scaly, lumpy back, breaking open pus pockets and catching the wriggling, bristly maggots as they emerged. "Damn!" He rubbed at the pasty crusts of pupa cases, wings, legs, and dermal scales. (The Godwhale, p. 78)

They're both a bit hard to follow, partly because they bristle (sorry, bad word choice) with technical terms, but partly also because the action jumps around a lot, leaping across years in fits and starts, hopping from character to character without any kind of clear direction. But, that comes with the territory. And they certainly don't shy away from questions of human sexuality, bioethics, philosophy, and religion, posing quandaries without providing solutions. What is humanity? What is the individual's meaning and worth? You could say that the books are a prolonged and poignant evocation of the anxiety of anonymous, post-religious Man and the terrifying, faceless masses that surround him.

The little I can find about T. J. Bass himself raises more questions than it answers. Who was he, really? What was he trying to do with these novels? Did he accomplish it? How did his writing fit in with the rest of his life? He died in 2011, having written nothing else but a diet and exercise book, and that was in 1979. That's a long time ago! Did he try and fail to get published again? Or did he just move on with his life? For that matter, can you imagine a scientist, mathematician, engineer, or physician in our own time, with little or no experience in fiction, penning an eccentric novel laden with precise technical knowledge and actually getting it published? Such a thing would be relegated to an online discussion board somewhere, to be skimmed by a few people and then forgotten.

Times have changed.

Views of a Tesseract

…the breadth, and length, and depth, and height…

– Ephesians 3:18

And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length and the breadth and the height of it are equal. And he measured the wall thereof, an hundred and forty and four cubits, according to the measure of a man, that is, of the angel. And the building of the wall of it was of jasper: and the city was pure gold, like unto clear glass.

– Revelation 21:16-18

Un homme qui y consacrerait son existence arriverait peut-être à se peindre la quatrième dimension. [A man who devoted his life to it could perhaps succeed in picturing to himself the fourth dimension.]

– Henri Poincaré

This spring I have scaled the awful, sanity-threatening Unknown Kadaths of the fourth dimension in a desperate, god-provoking quest to visualize the six regular polytopes.

What is a polytope, you ask? The word polytope is the general term in the sequence whose first terms are the line segment (dimension one), the polygon (dimension two), and the polyhedron (dimension three). A regular polytope is a polytope which is "completely symmetric."

Theatetus, a contemporary of Plato, proved that there are exactly five regular polyhedra: the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron. They are called the Platonic solids because Plato identified each of the first four with a material element (fire, air, water, earth), and the fifth with "the delineation of the universe" [Timaeus]. Their construction is the crowning achievement of Euclid's Elements, written in about 300 BC. But the world had to wait more than two thousand years for the "discovery" of their analogues in the fourth dimension.

Fourth-dimensional geometry, thought it might seem mysterious to the uninitiated, is defined axiomatically, just like Euclid's three-dimensional geometry, and has an intuitive basis. It was first described by Ludwig Schläfli, a Swiss mathematician, in the 1850s, but his work remained relatively inaccessible and unknown. Then, between 1880 and 1900, the geometry of higher dimensions was rediscovered in nine different publications written independently of each other. The time, it seems, was ripe. It was the dawn of a new era.

Not that era. [source]

This phenomenon of numerous researchers all suddenly reaching the same conclusion at the same time, though surprising when it happens, isn't all that uncommon in the history of math, science, and technology. What's striking is the way four-dimensional geometry fired the popular imagination, which seems in some cases to have outstripped academia.

Last year I blogged about Flatland: A Romance of Many Dimensions, a strange geometrical fantasy written by the English schoolmaster Edwin A. Abbott (1838-1926) and published in 1884. In it, Abbott gives what must be the first popular description of the tesseract, or four-dimensional hypercube, by way of analogy.

In One Dimension, did not a moving Point produce a Line with TWO terminal points?

In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points?

In Three Dimensions, did not a moving Square produce – did not this eye of mine behold it – that blessed Being, a Cube, with EIGHT terminal points?

And in Four Dimensions shall not a moving Cube – alas, for Analogy, and alas for the Progress of Truth, if it be not so – shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points?

Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this – if I might quote my Lord's own words – "strictly according to Analogy"?

Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"?

How much exactly did Abbott know of contemporary research? One imagines he must have encountered something, but I can't seem to find anything definite. A matter for further research, I suppose.

Inspired by Abbott, a high school teacher and amateur mathematician by the name of Charles Howard Hinton (1853-1907) wrote a number of "scientific romances" exploring higher dimensions. It was Hinton who coined the term tesseract, and his book A New Era of Thought, published in 1888, provides a detailed account of the hypercube's structure. It also offers a mystical interpretation of the fourth dimension, following to some extent in Abbott's footsteps, but with considerably greater gravity and self-importance.

We have been subject to a limitation of the most absurd character. Let us open our eyes and see the facts.

Now, it requires some training to open the eyes. For many years I worked at the subject without the slightest success. All was mere formalism. But by adopting the simplest means, and by a more thorough knowledge of space, the whole flashed clear.

Space shapes can only be symbolical of four-dimensional shapes; and if we do not deal with space shapes directly, but only treat them by symbols on the plane – as in analytical geometry – we are trying to get a perception of higher space through symbols of symbols, and the task is hopeless. But a direct study of space leads us to the knowledge of higher space. And with the knowledge of higher space there come into our ken boundless possibilities. All those things may be real, whereof saints and philosophers have dreamed.

Hinton was read by Jorge Luis Borges, and his book is mentioned "Tlön, Uqbar, Orbis Tertius."

Through his father, described by some as a religious crank, Hinton came to know the family of the late George Boole, the father of algebraic logic, whose untimely death had left his wife, Mary Everest* Boole, with their five daughters to raise. Mrs. Boole's interests ranged from mathematics to mysticism to politics; she wrote a number of pedagogical works, organized controversial discussion groups, and hobnobbed with the denizens of the fringes. Among these were the polygamy advocate James Hinton and his son Howard.

Howard married the eldest daughter, Mary Ellen Boole, in 1880, and they had four children together. A few later, he married a second woman under an assumed name, had two children with her, was convicted of bigamy, spent a few days in jail, lost his job, and moved to the United States with his (first) wife to become a university professor. He died unexpectedly in 1907, and Mary Ellen committed suicide the next year.

H. S. M. Coxeter's Regular Polytopes, published (in its second edition) in 1963, remains the main authority on its subject. I've entertained myself by constructing the various solids he describes in it.

More importantly for us, each chapter concludes with historical notes. There Coxeter discusses Alicia Boole Stott (1860-1940), another of George Boole's daughters, with whom he was personally acquainted in her later years. Curiously, though he mentions both Hinton and his book (in deprecatory terms), he says nothing about the family connection or about the fact that Stott assisted in finishing and publishing A New Era in Human Thought when Hinton left the country.

When Alice was about thirteen the five girls were reunited with their mother (whose books reveal her as one of the pioneers of modern pedagogy) in a poor, dark, dirty, and uncomfortable lodging in London. There was no possibility of education in the ordinary sense, but Mrs. Boole's friendship with James Hinton attracted to the house a continual stream of social crusaders and cranks. It was during those years that Hinton's son Howard brought a lot of small wooden cubes, and set the youngest three girls the task of memorizing the arbitrary list of Latin words by which he named them, and piling them into shapes. To Ethel, and possibly Lucy too, this was a meaningless bore; but it inspired Alice (at the age of about eighteen) to an extraordinarily intimate grasp of four-dimensional geometry. Howard Hinton wrote several books on higher space, including a considerable amount of mystical interpretation. His disciple did not care to follow him along these other lines of thought, but soon surpassed him in geometrical knowledge.

In 1890, she married an actuary and "led a life of drudgery" [Coxeter] as a wife and mother with a small income. But she continued to explore the fourth dimension as a kind of hobby, building cardboard models of three-dimensional "slices" of four-dimensional figures. Somehow her husband came across the work of the Dutch mathematician Pieter Hendrik Schoute, whose published diagrams mirrored her models. She contacted Schoute and the two began a long and fruitful collaboration. As Coxeter puts it,

Mrs. Stott's power of geometrical visualization supplemented Schoute's more orthodox methods, so they were an ideal team.

It was she who coined the term polytope.

Among other "enthusiasts" (as opposed to academicians) who contributed to four-dimensional geometry, Coxeter mentions Paul S. Donchian, an Armenian American.

His great-grandfather was a jeweller at the court of the Sultan of Turkey, and many of his other ancestors were oriental jewellers and handicraftsmen. He was born in Hartford, Connecticut, in 1895. His mathematical training ended with high school geometry and algebra, but he was always interested in scientific subjects. He inherited the rug business established by his father, and operated it for forty years. At about the age of thirty he suddenly began to experience a number of startling and challenging dreams of the previsionary type soon to be described by Dunne in 'An Experiment with Time'. In an attempt to solve the problems thus presented, he determined to make a thorough analysis of the geometry of hyper-space.

Donchian built delicate three-dimensional models of four-dimensional polytopes which were displayed at expositions in Chicago and Pittsburgh, several pictures of which appear in Coxeter's book.

I built a wire-solder model of the hypercube many years ago, using what I suppose are the same principles, though I didn't know it at the time. It remains in good shape, but it's in my parents' possession, and I don't have a picture of it handy.

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These days I'm working on a set of 3D printer files reproducing Stott's model of the 120-cell, a polytope composed of 120 dodecahedral cells. From her 1900 paper "On Certain Series of Sections  of the Regular Four-dimensional Hypersolids," I've created the virtual constructions from which I'll derive the vertex coordinates.

The following image represents a series of slices slices cut by hyperplanes parallel to a dodecahedral cell, starting with the cell itself (at the center of the image) and ending with the "equatorial" slice midway up the polytope (at the outside of the image). In my file the layers are numbered from VIII to XIV, in accord with the partial nets illustrated in her paper shown above.

And here is part of the "net" from which the 120-cell can be "folded." The "equatorial" layer of dodecahedra (not shown) fits in the interstices, with one for each edge of the dodecahedral cell forming the "base." A second set identical to the one shown then "caps" the 120-cell above the equator.

However, I find that I'm not the first to attempt reconstructing Stott's fascinating models. Well, I'll do the 600-cell as well, and that will be impressive. Here is my projection of the 600-cell to the plane.

I hope to recreate it in string art, the use of which in teaching children was pioneered by Mary Everest Boole.

Here are some of my printed polyhedra, which I built myself in Blender: we have a compound of five tetrahedra, a compound of five cubes, a compound of five tetrahedra (edges only), a great dodecahedron, and four rhombic dodecahedra, but no hypersolids yet. (Chessboard chosen advisedly: see below.)

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The geometry of the fourth dimension has made appearances in a number of imaginative works. Aside from Flatland, the earliest instance is probably The Time Machine by H. G. Wells, published in 1898. Unfortunately, he makes the rather common mistake of conflating temporal extension with a fourth spacial dimension.

"Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing.

"Scientific people," proceeded the Time Traveller, after the pause required for the proper assimilation of this, "know very well that Time is only a kind of Space. Here is a popular scientific diagram, a weather record. This line I trace with my finger shows the movement of the barometer. Yesterday it was so high, yesterday night it fell, then this morning it rose again, and so gently upward to here. Surely the mercury did not trace this line in any of the dimensions of Space generally recognized? But certainly it traced such a line, and that line, therefore, we must conclude was along the Time-Dimension."

H. P. Lovecraft gives a much better account of the fourth dimension in "The Dreams in the Witch House," published in 1933 and described by its Weird Tales tagline as "a story of mathematics, witchcraft and Walpurgis Night, in which the horror creeps and grows." Whatever you think of Lovecraft as a writer, one thing you can say is this: he knows when to be explicit and when to be vague and ominous. It serves him well here.

Toward the end of March he began to pick up in his mathematics, though the other studies bothered him increasingly. He was getting an intuitive knack for solving Riemannian equations, and astonished Professor Upham by his comprehension of fourth-dimensional and other problems which had floored all the rest of the class. One afternoon there was a discussion of possible freakish curvatures in space, and of theoretical points of approach or even contact between our part of the cosmos and various other regions as distant as the farthest stars or the transgalactic gulfs themselves – or even as fabulously remote as the tentatively conceivable cosmic units beyond the whole Einsteinian space-time continuum. Gilman's handling of this theme filled everyone with admiration, even though some of his hypothetical illustrations caused an increase in the always plentiful gossip about his nervous and solitary eccentricity. What made the students shake their heads was his sober theory that a man might – given mathematical knowledge admittedly beyond all likelihood of human acquirement – step deliberately from the earth to any other celestial body which might lie at one of an infinity of specific points in the cosmic pattern.

Reminds me of my own college days! Ha ha, actually, it doesn't. I spent an entire year of my life working a problem of 10- and 26-dimensional geometry, got stuck on a minus sign for most of its duration, and finally had to give up and start a new problem. My dissertation advisor may very well have wondered about my nervous and solitary eccentricity, and my fellow students may have shaken their heads at my theories, but not for the reasons Gilman found himself the source of such disturbance…

[source]

The net of a tesseract figures in Robert A. Heinlein's 1941 story "And He Built a Crooked House," in which an architect builds a house in the shape of the three-dimensional "net" of a tesseract (from which the polytope can be "folded" much as a cube is folded from a two-dimensional cruciform net); an earthquake causes it to collapse into an actual tesseract from which other worlds can be reached. The story was anthologized in Fantasia Mathematica in 1958.

Madeleine L'Engle's A Wrinkle in Time, which contains the most well-known tesseract (and verbs the word as tesser), was published five years later, in 1963. I wonder if L'Engle got her idea (which is rather garbled) from the Heinlein story?

Meg sighed. "Just explain it to me."

"Okay," Charles said. "What is the first dimension?"

"Well, a line."

"Okay.  And the second dimension?"

"Well, you'd square the line. A flat square would be in the second dimension."

"And the third?"

"Well, you'd square the second dimension. Then the square wouldn't be flat any more. It would have a bottom, and sides, and a top."

"And the fourth?"

"Well, I guess if you want to put it into mathematical terms, you'd square the square. But you can't take a pencil and draw it the way you can the first three. I know it's got something to do with Einstein and time. I guess maybe you could call the fourth dimension Time."

"That's right," Charles said. "Good girl. Okay, then, for the fifth dimension you'd square the fourth, wouldn't you?"

"I guess so."

"Well the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points."

Terrible! Just imagine an inhabitant of Flatland speaking like that: "The third dimension is Time. The fourth dimension's a cube. You add that to the three dimensions and you can travel through the plane without having to go the long way around. In other words, to put it into linear terms, a straight line is not the shortest distance between two points." Ugh! A novel is not a math textbook, it is true, but, for me, it's harder to overlook such nonsense than scientific speculation. There's nothing like 1 + 1 = 3 to break the suspension of disbelief. (Not that it's a bad book mind you.)

#

[source]

Higher-dimensional geometry appears in art as well. Cubism is an oft-cited example, but geometry figures more directly in Salvador Dali's Crucifixion (Corpus Hypercubus), which depicts Christ crucified on the net of a tesseract. Just as the net permits us to approach what lies beyond our comprehension, in God, so does the Incarnation provides a "picture" of God comprehensible to humankind. That's how the picture usually seems to be interpreted.

Well, somehow this has turned into one of those posts of mine in which I draw connections between whatever unrelated topics I happen to be interested in. Here it's higher-dimensional geometry, science fiction and fantasy, the early twentieth century, and art. I do still want to describe the catalog of regular polytopes, but that will have to wait for a subsequent post.

* The mountain was named after her uncle. Quite a dynasty!

Vast Active Living Intelligence System

I've never been much of a science fiction reader. My attempts at writing it have been pretty unsuccessful, too, at least to my mind. Sometimes this embarrasses me. When a "real-life" acquaintance finds out I write, and they ask, "Oh! What sorts of things?" I stammer a bit, and say, "Oh, you know, sci-fi fantasy stuff." But that's not quite honest, is it? It's just that, if I come right out and say, I write fantasy, people don't seem to know what I'm talking about. Maybe it's just where I live. It sounds like I'm confessing to writing, you know, fantasies, as in the "fantasy suites" down at the Ramada Inn. But sci-fi they understand, and it gets the basic idea across.

Me, I like the prose laid on thick, as in Conrad or Melville. That's probably obvious to readers of my books. But you don't find many gothic edifices or thickets of purple prose in the science fiction field. Descriptions are terse and often rather vague. Everything is done by suggestion. My eye just glides right over it. I'm not particularly intrigued by fictional advances in science or technology, either. Maybe I spent too many years studying spin geometry and quantum field theory.

Leaving aside romancers of the cosmic future like Herbert and Wolfe, whom I personally consider to write a species of fantasy rather than science fiction, what I find when I look at the authors I most enjoy (A. E. van Vogt, Alfred Bester, Philip K. Dick) is a preoccupation with philosophy, religion, and psychology.

What brings on this little soliloquy is the fact that I've just read (or listened to, at any rate) Philip K. Dick's VALIS yet another time. I find myself reading Dick whenever I go through a particularly severe bout of depression and isolation. His books tend to be about people who are depressed and isolated, of course, but I find them strangely consoling for that very reason. He's a humane writer with a sympathy that never becomes sentimental.

Though possibly his most idiosyncratic work, VALIS is one that I've returned to again and again. It's Dick's hilarious yet deeply sad semi-autobiographical account of one seriously fucked-up (as Dick would put it) person's (or is it persons'?) slightly brain-fried search for God (or whatever) in paranoid post-Nixon 1970s California. Unlike a lot of Dick's fiction, drugs play only a peripheral role. Religious experience has taken their place. I remember my mother once telling me that, to her, it seemed like the Jesus People of the seventies had merely traded one drug for another (that is, dionysian ecstasy). Maybe that sums up Dick's experience.

The milieu described in a number of Dick's drugs-and-religion books bears a disturbing resemblance to a certain phase of my own life, when I spent all my time with card-carrying prophets, traveling evangelists, drug addicts, professional bead-necklace-making drifters, hippies living in station wagons, cell group leaders, paranoid schizophrenics, Hare Krishnas, recovering vampires, and undercover missionaries. I was scared out of my mind, autistic without knowing it, and, for a while, on the verge of homelessness and utter ruin. But that was my world, and the only thing I could do was try to find meaning in it. My general situation was, like Horselover Fat's, fucked up. Actually, it was just after I'd divorced myself from that world that I first started reading Philip K. Dick, starting with And Now Wait for Last Year. What I'm trying to say is, Dick's whacked-out religious novels hit close to home.

You might call VALIS an exploration of lowercase-g gnosticism, that is, salvation through special knowledge (or "information," as Dick puts it). The Exegesis runs all through it, with frequent citations from the New Testament and the pre-Socratic philosophers. What's strange is that it seems to call into question the very concept of saving knowledge, reaffirming John of the Cross's path of negation. A message is only as reliable as the messenger. Sensory impressions, interior locutions, and emotional experiences can always be questioned. Maybe they come from a divine source, or maybe someone's just playing with microwave transmissions. Maybe you're speaking to an incarnation of the pleroma, or maybe it's just a little girl rigged with wires and speakers. Looking in from the outside of Dick's universe, it seems to me that a god whose communication of himself takes place solely on the plane of "information" will always turn out to be a resident of some star system or other, that is, a being more powerful but commensurate with human beings.

It strikes me as ironic that Fat's friend David, a Catholic (and, apparently, a stand-in for Tim Powers), always tries to bring C. S. Lewis into his theodicy. Lewis was himself a gnostic, in that his descriptions of faith, in both his fiction and his apologetic works, amount to a kind of secret knowledge. The heaven described in The Last Battle is a thoroughly gnostic heaven; both Dick and Lewis routinely cite Plato. I wonder how much Lewis Dick had actually read, if any. The David character resists, but ultimately accepts the import of Fat's revelations and the significance of the Eric Lampton film. But then it all just fizzles out, gets explained away (if one can call it that) as the operation of mundane technology and delusion...

I think I'll read The Divine Invasion sometime soon. No doubt I'll have more to say then.

Some other posts on Dick-related material that you might enjoy:

• Dick's Broken-Backed Novels
• Can-D and Other Matters
• The Man Who Fell to Earth

The Incredible Shrinking Man

I recently watched (for the second time, I think) The Incredible Shrinking Man (1957). The screenplay is by Richard Matheson, the writer to whom we owe the novel upon which it is based, as well as I Am Legend (1954), Hell House (1971), and What Dreams May Come (1978). Amazingly, he also wrote the screenplays for that best of Poe adaptations, The Pit and the Pendulum (1961), as well as two other Roger Corman productions, namely, the memorable but somewhat inferior House of Usher (1960), and the ridiculous but entertaining The Raven (1963); he wrote the famous Twilight Zone episode "Nightmare at 20,000 Feet"; the Star Trek episode "The Enemy Within"; The Legend of Hell House (1973), which I talked about earlier this year; the made-for-TV movie Trilogy of Terror (1975), whose little doll continues to pursue me through my worst nightmares; and numerous other things.

Anyway, The Incredible Shrinking Man. It's one profound movie. An ordinary man starts shrinking, more or less inexplicably, becoming a tiny bit smaller each day. He undergoes humiliation, alienation, and, finally, cosmic isolation. His resentment becomes tyrannical petulance, then terror, then utter loneliness. His white-collar insecurities gradually crumble before Pascal's fear of the eternal silence of the infinite spaces. In the last act, the basement-hell to which he finds himself consigned stretches before him like an alien landscape in which he must fight like an insect to survive. More so even than the protagonist of I Am Legend, he finds himself marooned in an inimical new universe. He triumphs over the spider, only to go on shrinking smaller, and smaller still.

But despair turns to acceptance and finally hope. The film ends in epiphany, as the protagonist wanders out through the grating, looking up at the stars, soliloquizing, as the viewer receives flashes of distant galaxies:

I was continuing to shrink, to become... what? The infinitesimal? What was I? Still a human being? Or was I the man of the future? If there were other bursts of radiation, other clouds drifting across seas and continents, would other beings follow me into this vast new world? So close – the infinitesimal and the infinite. But suddenly, I knew they were really the two ends of the same concept. The unbelievably small and the unbelievably vast eventually meet – like the closing of a gigantic circle. I looked up, as if somehow I would grasp the heavens. The universe, worlds beyond number, God's silver tapestry spread across the night. And in that moment, I knew the answer to the riddle of the infinite. I had thought in terms of man's own limited dimension. I had presumed upon nature. That existence begins and ends in man's conception, not nature's. And I felt my body dwindling, melting, becoming nothing. My fears melted away. And in their place came acceptance. All this vast majesty of creation, it had to mean something. And then I meant something, too. Yes, smaller than the smallest, I meant something, too. To God, there is no zero. I still exist!

Profound words, but merely quoting them fails to convey the feeling they give me when I watch the movie. It really brings tears to my eyes, because it captures so well what has tormented me all my life. That is, the terror of the senseless cosmos.

What people miss when they talk about how insignificant man is in the universe is the fact that man is also gigantic. The truth is, man hangs suspended between two extremes, the infinite and the infinitesimal, a fact that the mathematician-philosopher Blaise Pascal, mentioned above, wrote about at some length:

Let man [...] contemplate the whole of nature in her full and grand majesty, and turn his vision from the low objects which surround him. Let him gaze on that brilliant light, set like an eternal lamp to illumine the universe; let the earth appear to him a point in comparison with the vast circle described by the sun; and let him wonder at the fact that this vast circle is itself but a very fine point in comparison with that described by the stars in their revolution round the firmament. But if our view be arrested there, let our imagination pass beyond; it will sooner exhaust the power of conception than nature that of supplying material for conception. The whole visible world is only an imperceptible atom in the ample bosom of nature. No idea approaches it. We may enlarge our conceptions beyond all imaginable space; we only produce atoms in comparison with the reality of things... 

Returning to himself, let man consider what he is in comparison with all existence; let him regard himself as lost in this remote corner of nature... What is a man in the Infinite? 

But to show him another prodigy equally astonishing, let him examine the most delicate things he knows. Let a mite be given him, with its minute body and parts incomparably more minute, limbs with their joints, veins in the limbs, blood in the veins, humours in the blood, drops in the humours, vapours in the drops. Dividing these last things again, let him exhaust his powers of conception, and let the last object at which he can arrive be now that of our discourse. Perhaps he will think that here is the smallest point in nature. I will let him see therein a new abyss. I will paint for him not only the visible universe, but all that he can conceive of nature's immensity in the womb of this abridged atom. [...] Let him lose himself in wonders as amazing in their littleness as the others in their vastness. For who will not be astounded at the fact that our body, which a little while ago was imperceptible in the universe, itself imperceptible in the bosom of the whole, is now a colossus, a world, or rather a whole, in respect of the nothingness which we cannot reach? He who regards himself in this light will be afraid of himself, and observing himself sustained in the body given him by nature between those two abysses of the Infinite and Nothing, will tremble at the sight of these marvels; and I think that, as his curiosity changes into admiration, he will be more disposed to contemplate them in silence than to examine them with presumption. 

For in fact what is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything. Since he is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret, he is equally incapable of seeing the Nothing from which he was made, and the Infinite in which he is swallowed up. [Pensées 72]

So you see, the universe is considerably more terrifying than your rather unimaginative New Atheist types generally realize. The Incredible Shrinking Man is a profound exploration of that terror, and a moving attempt to answer it. But it's also a gripping film apart from its philosophical implications, with wonderful sets and meticulous attention to detail in its depiction of the protagonist's rapidly growing world. And the battle with the giant spider toward the end is very cool.

Truly one of the great science fiction films of all time.

On Reptoids from Alpha Draconis

In keeping with my time-honored tradition of writing a blog post whenever I encounter a network of ideas connected in any way, shape, or form with speculative fiction, but also in keeping with my equally time-honored tradition of not commenting on current affairs, I hereby commence a post about

THE REPTOIDS AMONG US

without explaining what, precisely, has motivated me to pursue this particular avenue of discovery.

I will tell you what set me upon the path, however.

One day at the office, whilst devouring the salami sandwich, granola bar, and green apple that constitute my lunch, I happened to watch a Mental Floss YouTube video about weird conspiracy theories. Among the factoids presented therein is the assertion that 12 million Americans believe that shape-shifting lizard people secretly hold the reins of world government. This is according to Public Policy Polling.

Now, if a survey company called me to ask whether I believe shape-shifting lizard people are in control of the government, I would respond with an enthusiastic YES, YES, I DO STRONGLY BELIEVE THAT. And surely I'm not alone in my love of trolling the strangers who call my house to ask stupid questions. So, perhaps the number is a little misleading. Still, there must be some truly devout lizard-people believers, right? A little googling (not for the faint of heart in this case) would seem to confirm that, yes, there are some believers out there.

The leading proponent is one David Icke, British ex-footballer and conspiracy theorist. He asserts that reptilians (or reptoids, as I prefer to call them) from Alpha Draconis have been selectively breeding the human race since its inception, creating the Babylonian Brotherhood, a hybrid race of Illuminati who now control the world. He suggests that the aliens first came to earth in search of monatomic gold, which enhances the capacity of their nervous systems ten thousandfold.

Hm. A race of superintelligent extraterrestrials who inhabited earth long before man's coming and now treat the human race as a pet project and secretly orchestrate events from behind the scenes? Remind you of anything? Here we have the connection with my blog's purported focus. It has been suggested that the Reptoid Hypothesis owes a large debt to the writings of H. P. Lovecraft, Robert E. Howard, and Clark Ashton Smith; they, in their turn, took much of their material from nineteenth-century theosophical writings on Atlantis and Lemuria.

The idea that disguised aliens secretly control everything of course puts me in mind of John Carpenter's They Live (1988). I love eighties sci-fi actioners, and They Live is one of the most enjoyable. The screenplay was written by Carpenter under the pseudonym Frank Armitage, an allusion to Dr. Henry Armitage of "The Dunwich Horror." Like most Carpenter assays, They Live has a B-movie feel, but it's also hilarious, sharply satirical, and eerily contemporary. It takes place in a not-so-distant future when the middle class is being squeezed out of existence, the underemployed live in shantytowns on vacant lots, racial discord and police brutality are on the rise, and a media-induced malaise seems to have overcome the human race. One day, a down-and-outer (played by pro wrestler Rowdy Roddy Piper) discovers a pair of cheap-looking sunglasses that let him see a whole new reality. Turns out the rich and powerful are all aliens or people who have sold out to the aliens. This leads to one of the greatest and cheesiest one-liners in action sci-fi history: "I have come here to chew bubblegum and kick ass...and I'm all out of bubblegum."

Now, it's one of my little pleasures in life to post extremely weird links and opaque commentary on Facebook to see if my aunts, cousins, in-laws, coworkers, and high school friends will think I've gone off my rocker. (I used to go in for politics, but eventually you learn what everybody is going to say, so why bother?) So I recently posted the above picture of George W. Bush, Tony Blair, and Queen Elizabeth II as reptoids. Imagine my surprise when, immediately afterward, the "people also posted" thing popped up with three stories about...Donald Trump.

Let me pause here to assure whomever might be reading this that I, for one, welcome our new reptilian overlords.

Now, what's interesting to me about They Live (which is awesome and you should watch soon if you haven't yet) is that it appeared shortly before David Icke began his career as New Age prophet. Was it an influence on his ideas? Icke is also supposed to have been influenced by 1940s non-fiction writer Mark Doreal, who took some of his ideas from Robert E. Howard. So apparently there's been a strange cross-fertilization between weird horror and cranky fringe conspiracy theories. Each influences the other; both grow together.

For all that, though, there's a distinct psychological line between the weird horror fan and the wild-eyed conspiracy theorist. For the former, there's a big element of play, of fun. The latter is in deadly earnest. I would even assert that the fan is the unlikeliest person to go down that rabbit hole. They're inoculated by their sense of humor.

But what does make someone start thinking that the celebrities and politicians they see on TV and in the news are all reptoids in disguise? That, I'm afraid, is a question above my pay grade, so I'll just conclude with a word from our sponsor:

ADDENDUM: D'oh! While writing this, I forgot to mention another ridiculous eighties movie making use of the same idea: the inimitable The Adventures of Buckaroo Banzai Across the 8th Dimension (1984), which pits humans and Black Lectroids against the evil Red Lectroids of Planet 10. Less pointed, much weirder, with an awesome cast. I'll have to start reviewing fantasy and sci-fi schlock from the eighties sometime soon.

D.O.A.

I can breathe and I can move, but I'm not alive because I took that poison, and nothing can save me.

I watch a lot of film noir. To some people, the term means trenchcoated detectives, laconic voiceovers, femmes fatales, and cigarette smoke. I think of it more as a directorial style, a style that uses harshly geometric, minimalistic (= dark) photography and unpityingly logical plots taking place beyond the boundaries of "normal" society to create an atmosphere of emotional detachment and alienation. Yes, chain-smoking dicks and dames figure largely in such films, but that's because the world they inhabit lends itself so readily to the style. Some of my favorite films noir don't involve detectives at all.

That includes D.O.A., a film that wasn't appreciated at the time of its release (1950) and isn't very well known today. It's in the public domain, and you can watch it at the Internet Archive, which no doubt has to do with its obscurity. Edmund O'Brien plays Frank Bigelow, an accountant and notary public who, months prior to the opening, notarized a seemingly innocuous transaction, a chance happening that sends him down a black rabbit-hole of madness and murder. Most films noir have a certain sense of predestination, but this one takes it a step further: it's narrated by a dead man.

Grim humor and dark irony punctuate Bigelow's nightmare urban odyssey. In confronting his nonsensical, undignified position (the human position), he seizes control of his destiny, loses all fear, and achieves a kind of justice, only to keel over in the film's final minutes. One of the last lines is the homicide captain's dispassionate "Call the morgue!"

D.O.A. has some corny elements toward the beginning, as many films noir do, but rapidly descends into darkness, madness, sweat, and grit. I appreciate it a little more every time I watch it. It also has an almost science-fictional feel, with a search for stolen iridium and a "luminous toxin" that glows in the dark. And it must have one of the best openings of any movie ever:

Bigelow: I want to report a murder.
Captain: Sit down. Where was this murder committed?
Bigelow: San Francisco, last night.
Captain: Who was murdered?
Bigelow: I was. 

Anyway, because this is my blog, and I can do whatever I like, here's some of my favorite stills:
 

The opening sequence. Bigelow marches through a dark police station.

"I want to report a murder."

Pamela Britten plays Paula, Bigelow's sweet but slightly clingy
secretary-girlfriend back home.

The madness begins: a San Francisco jive club.

"He's flipped. The music's drivin' him crazy." Meanwhile, the faceless
killer prepares the luminous toxin.

"You've got it, all right." A doctor confirms his diagnosis with a glowing
test tube. Bigelow has one day to live.

"I don't think you fully understand, Bigelow. You've been murdered."
Bigelow races down a crowded street. A neat sequence filmed without
permit, and without the bystanders' knowing what was happening.

Irony.

Now on the trail of his own killer, he journeys to Los Angeles, where he
encounters tart, unprofessional secretaries...

...and gun-toting models.

Soon he's getting shot at...

...and chasing phantom gunmen through abandoned warehouses.

Someone doesn't like him sticking his nose in.

But he's utterly fearless now. "He's not afraid," says Majak. "You can tell
from a man's eyes when he is afraid. Look at his eyes."

Chester, one of the creepiest henchmen I've seen. "That's the way I wanna
see you go, Bigelow...niiice and slooow."

But it's Chester who goes, in a drugstore shoot-out.

Beautiful on-site night photography.

The final interview with a not-so-grieving widow.

Surrounded by people but utterly alone.

Crawling on hands and knees to gain a few hours of life.

A few hours to go to the final confrontation at...

...the Bradbury Building, Los Angeles. Is this actually the exterior of the
Bradbury, though? I don't think so.

This, however, is clearly the interior. Recognize it? It's where the climax
of Blade Runner takes place. A lot of movies have been filmed there.

The faceless killer.

The final shot. "Better make it 'Dead on arrival.'"