Archive of Our Own

The morning consisted of introductions and banter, and it wasn't until after an expansive lunch taken together in the gardened cafeteria that Professor Hilbert at last began to expound on the nitty gritty.  He's a retired megamech crewman, so the language that he speaks is from experience and internalized understanding, rather than recitation of something from a book.

“You're all familiar, I expect, with the precursory material — Flatland, Planiverse, Euclid, Hinton, the various other texts on applied and theoretical particle physics and so forth. You also know of some pop-culture misuses of the word 'dimension,' I imagine,” to which everyone first nodded with ease, and then rolled their eyes in varying degrees of mental pain and cringing.

“Well, the funny thing is that although for the longest time one might understandably have held in low opinion the use of the term 'the fourth dimension' as some idiotically particular turn of phrase, when mathematically no one direction should show preference in geometry, and so 'a fourth dimension' might reasonably have been a better choice, the real world does so love to surprise us.

“We know now, of course, that while there is as yet no real-world parallel for the pop-cultural conception of 'the fourth dimension' as some other place as one might encounter in a comic book or half-baked garage religion, it's certainly a case of 'the  ' rather than 'a,' but we'll take this in easy steps, just to be sure that we're all on the same page here.”

“You should already have all of the standard illustrations that would normally accompany this class, but I prefer to work with my hands,” he explained as he began sketching on the board, “these sketches will naturally append themselves to the relevant sections of your note portfolios, but you should also be able to follow directly from the summary text's preformatted manipulable illustrations, each of which of course carries its own explanatory metadata.”

Saying this, the figure of a windowpane took form, which he marked as “Planiverse.”  Its X and Y dimensions extended arbitrarily, with the Z subdued to fairly degenerate thickness.

“You'll note that there's some non-zero degree of freedom to the Z-axis,” he pointed out, “but for the moment, treat it as being a physically limiting minimum, a Planck-like thickness that affords no leeway and that obtains to all objects as mass-energy events therein — equivalent for all practical purposes to delta being greater than zero, but with delta squared equaling zero. This gives a Pointlander only two potential degrees of hyperspatial freedom of motion, even though that world possesses three physical dimensions in fact.

“Our Pointlander might discover the X-axis and wonder if there were anything like a hyper-X axis or more, paralleling this first X-axis and arranged along what we'd call a Y-axis, and if they might be able to translate themselves to one of these other X-axes, or even rotate themselves to intersect some subset of them along the Y-axis. They might go on to find any number of further axes eventually, but this example's Z-axis isn't meant to be specifically the third one in the series, only representative of some eventual barrier, a very real axis or set of axes, and impenetrable to the Pointlanders, amounting to an ordinal omega-plus-one-th axis.

“We'll come back to this, but I want you to keep it in the back of your mind as we explore those accessible axes in the meantime.”

Sketching a stick figure in a stairwell, he went on.

“Here we see a blind robot walking along stairs. This robot could certainly traipse around on the same stair-step as everyone else, of course, but this one's special: it can go to a different step. Its mad roboticist designed it with extremely extensible legs, and so it finds itself in the unique position of being able to leave and return. Sadly, the roboticist didn't give it eyes, and so it can't tell you if it's going upstairs or downstairs.”

“Our early experiments revealed the very real capability of accessing the previously hypothetical higher dimensionality present in the universe, but it was soon apparent that this extra dimension possessed at least ana and kata directions. That is, our three-world is sandwiched within the middle of that dimension that extends to each side of us; we aren't sitting on the ground floor. We figured that much out after having sent out probes with overlapping mission timelines and finding that they were able to detect only half of the other probes (all falling into one of two distinct subsets), even though all probes survived their launch and return trips (or later sent live feeds reflecting the same coin-flip, once hyper-transception capabilities were developed). Clearly, we needed to develop sensors, guidance, and control to navigate hyperdimensionally, much as the robot on the stairs would otherwise blindly go up or down at random.”

“We've learned our way around a few dimensions, and there are at least a few more to come. I wonder sometimes if instead of having to tinker with each new ordinal dimension, there isn't some way to simply access them all at a single go. Our robot on the stairs is a starting point, but doesn't convey the whole problem: in reality, it's as if each next stair-step pair weren't merely farther along in some direction, but in fact went off at right angles to all of the previous stair-steps, and the robot had to discover that new direction and learn how each new direction worked every single time. Perhaps, one might be able to access all at once.

“Right now, we are in the same position as a Pointlander who had discovered how to move sideways on the line-like surface of a two-dimensional ocean, and had to develop left-right understanding and movement, only to then discover the air above and water below and repeat the whole process. If that same Pointlander were to somehow find that they lived in a three-volume and could sense and move freely in any combination at once, then the process wouldn't require so tedious and arduous a waste of effort in order to leverage practical results. As it currently stands, our metaphorical Pointlander couldn't discover simply any of the other dimensions at random: they'd necessarily stumble upon the X-axis before the Y, and the Y before the Z, and so forth.”

“That Pointlander would normally, in such a multidimensional space, be rather like a person having legs and not being restricted to having to first learn ordinal forward and backward movement before even discovering the existence of sideways in order to then learn how to move so, before going on to discovering and moving upward and downward — typically, if you have legs and can walk, then there is no innate constraint against your moving any which way that you might wish, along with leaping upward and kneeling downward without having to learn anything about combining all of these directions. This would be the cardinality of the first three dimensions: any direction chosen at random and placed orthogonal to some other direction forms an arbitrarily chosen de facto coordinate plane with a forced third direction orthogonal to those first two. None has any particular reason to have been chosen as such, nor possesses any intrinsic specialness beyond the fact of having been chosen; there isn't a 'first,' 'second,' and 'third' dimension as such, any one direction is the same as another for all practical purposes — there are simply 'three' and that's that. The first three dimensions of the universe are cardinal. Here, however, is where things went sideways, as it were. We found a fourth direction to move along, one that showed preference as regarded its treatment with respect to the other three, possessed of an ordinality in fact, in contrast to the cardinality of the others; moreover, further theoretical developments later indicated subsequential ordinal dimensions beyond that, which of course began an arms race of technological attempts to access the same.”

“I suppose that this is where we move on to at least a bit of history,” he sighed.

“In short: initial field tests involved gravitomotive fountains that showed promise in scale models. Test objects went somewhere and returned, though it turned out that it was more like a flexible plastic coin being squeezed between your fingers: it'll pop out soon enough, but the direction that it takes could be left or right — this simile would be the earlier-mentioned idiom of the probes' coin-flip. These experiments brought to light an unexpected x-force Meissner-Ochsenfield-like effect, which eventually showed itself to possess a toroidal right hand rule flow that could be used to guide the test bodies. The fields interacted stochastically with the collective particle sets, but were detectable. Once their vectors became ascertainable, it wasn't long before control mechanisms were in place to expel the bodies in a preferred direction.

“In any event...” the professor trailed off as if following a path of thought all his own.

“Our developments were fueled by the value of hyperdimensionality, one that's easily demonstrated with playing cards on Flatland.”

“Just as we vied with one another, so too do these cards do so within Flatland — only to fall instantly to the superior abilities of a newfound technology that permits a card to stand above (or below) Flatland on legs that interact with Flatland three dimensionally, or walk upright along the surface (or beneath it). This technology is slightly improved when they discover three dimensional radar, so that they can still 'see' the opponent even when standing at a three-angle askew to each other, rather than what they would call head-on — that is, in front of one of their edges, rather than in either hemisphere of the subject's current orientation. The arms race then takes a dramatic turn when someone develops flight and can hover and eventually soar across the plane at some elevation, soon to perform multi-dimensional maneuvers in dogfights.

“Naturally, things become more complicated when the Flatlandian equipment and personnel find the need to function with pentatopes, tesseracts, and glomes.”

Here he returned to his desk, taking a seat.

“Tomorrow's simulation is meant to introduce you to sensory processing techniques that will allow your brains to decode and your minds to grasp what they see in n+1 dimensions, and eventually n+2, n+3, et cetera. I could express what this is like, how it feels, how it's done, and what n+p dimensional visual fields look like, but it would lose accuracy if I were to try evocative metaphor, and lose any relatable meaning if I were to take the route of mathematical precision. Once you've learned the methods and absorbed the training, you'll be able to talk about it intelligibly with others in the same fashion, but just as unable to convey any real meaning to anyone who's then in your own current position.

“For the moment, an unsatisfactory balance between the two is to imagine the difference in views afforded to the Flatlandian megamechs. Within the visual field of their world, they see a one dimensional strip of visual data ringing around them, with some depth perception to indicate a full two dimensionality to their world. Upon launching their megamechs into some crazy third dimension of hyperspace, the early pilots would be able to see other megamechs only if they're at the same distance from Flatland, hence occupying the exact same literal geometric plane, assuming that nothing caused their physical orientation to rotate — but more on that when we get to the mechanics of sabotage — and they'd be playing a game of hide and seek on elevators, popping out on one floor or another to see if anyone's there. Later, with technical refinements, they could cause their megamechs to take a negative pitch so that their flight deck pointed downward toward Flatland and present themselves with a one-dimensional swath of their world and any intervening megamechs below them; adjusting their roll would of course then rotate their cross-sectional view clockwise or counterclockwise relative to the plane below them, and their yaw would translate their point of focus; technical improvements might permit their sensors to do the trick for them, or permit their megamechs to bend or even be articulated in a two-orientational manner, perhaps going on to snaking three-dimensionally or resembling a koosh ball of plane segments.”

As he expanded on these items, an augmented reality display appeared before them, with an option to view it at classroom scale, as a desktop manipulable, or both.

“—now, take a moment to think about this,” he interjected, looking up from his notes, “It sounds from our perspective as if the Flatlanders in three-space might then be afforded a view of their world that's outside of their normal view, but still one that's at least in keeping with what they'd normally see when looking at anything: a one-dimensional strip of whatever they're looking at. That's true to an extent, but misleading. It would be for them about as 'familiar' as it would be for us to be presented with a picture that took a two-dimensional cross-section of everything in whichever direction we were looking — not the surface features that we'd normally see, mind you, but what we would see if we could see only whatever lay along a specific flat disc with its origin at a given distance away and oriented at whatever four-angle relative to us, no more nor less: concrete, internal organs, blank air with nothing behind it, disconnected bits of glowing filament encircled by a ring of glass. Not what you'd call familiar, or easily interpreted, or much use at all.”

“Moving right along, though,” he said, returning to his notes, “Their language would have to develop both absolute and relative terms for position, translation, orientation, and rotation within and through three dimensions — after all, they would hardly be expected to move sideways, upward, and onward as three separate motions, rather than one continuous action across the Euclidean n-hypotenuse or some curving path. Eventually, if they were to develop cybernetic feeds and appropriate mental models, their brains could process a two-dimensional visual field, taking in the entire visible scope before them at once, showing them the interior structure of buildings without need for deep-penetrating x-radar or infrabands, or presenting opportunities for surgical strikes or extractions of far greater precision than afforded by bunker-busters and normal two-dimensional line of sight.”

He looked up for moment, considering a few of the quieter, rougher-looking individuals.  This didn't escape them.

After a moment, he found where he had left off.

“This would then immediately require address with some form of third-dimension sheets of armor — on both the upper and the lower faces of such structures and people — preventing such indiscreet probing. They're already beringed with two-dimensional armor annuli, so this amounts to adding sheets of plating to each face of their megamechs, both front and back — positive Z and negative Z — in order to present an englobing armor, a two-dimensional boundary manifold that curves around them in three dimensions.

In a seeming tangent, he continued, “You'll note that the text has a simple formula standing alone: the integral of e to the x is some function u to the n,” and indeed, there it was, clear as day: ∫e^x = f(u)ⁿ, at which a low chuckle made the rounds, “That's not there simply for humor. It speaks to the fact that a boundary manifold of some dimension n-minus-one shows what is intuitively two faces with respect to the n-body, and that this intuition is deceptive: just as one can choose some arbitrary u to the n, so too is the obverse-reverse facing issue an accident of circumstance, as shown when one considers things from the perspective of n-plus-one-or-more dimensions and the rotation of the boundary manifold as its own axis. The equation remains true if in changing base u, one also chooses an appropriate change of exponent.

“Put simply: if one considers a disc-shaped soap bubble bounded by a metal ring, then the two dimensional inhabitants might be reasonably justified in considering themselves to be within that ring. Now set that ring into the two-sphere of a three-ball and tell me if the inhabitants of the cap are within the ring or without — and the same question for inhabitants on the other side of that ring.”

He let them consider this for a moment before returning to the question of relative dimension.

“Bumping that preceding analogy down one dimension, that picture now becomes the story of Linelanders in their second-dimensional hyperspace, wherein their access to second-dimensionality necessitates line segment armor to be affixed to each second-dimensional edge of their one-dimensional assets — both positive Y and negative Y. You might anticipate their next problem: having encountered this weird second-dimensional vulnerability and armored themselves against it, they then go on to discover yet a third physical dimension to hyperspace... and a concomitant need to fully sheath Linelandian people and such with insulating manifolds of two dimensional surfaces that would be closed, or mostly so, in three dimensions; one imagines that some allowance might reasonably be made for points of access within Lineland itself. Intuition, however, would inform the Linelandians initially that they need only emplace one-dimensional armor 'above' and 'below' Flatlandianly, and now to their 'left' and 'right' Spacelandianly, without regard for rotation: that just as the Flatlandian line segments of armor would guard against attack in one plane, so too would a similar arrangement guard against it in another plane — or that perhaps they'd need some third sandwich to account for a third possible plane of attack along the cross-section — and these are true of the orthogonal pairs of axes, but completely misses the question of rotation and thereby attack from some angle relative to any pair of planes.

“With every stepwise increase, even partial ones, comes exactly this same need. Hyperdimensional emplacements can provide overwatch and suppressive fire and countermeasures, but cannot negate the assets' vulnerability to hyperdimensional vectors.”

Grabbing a bacon sandwich and a pint of milk from the snacks that they'd brought back from lunch, he went the impromptu route.

“Hyperspace is devoid of life and materiality, but not empty. There are our own probes, ships, stations, et cetera out there. There are those of other groups — science establishments, artists and philosophers, researchers into maintenance practices and surgery, hobbyists, private groups, high-end and low-end housing, transportation, universities, industry, waste storage and retrieval, militaries, corporate concerns, the whole nine yards. There's also at least one body of unknown origin, several minutes away.”

This raised a stir in the class, the murmurs mostly focusing on the obvious of an “unknown origin,” but with a clear undercurrent about “minutes.”

“In the early days, we had little control of the probes' motion, and had to gauge 'depth' in minutes — how long they'd been gone, as an estimate of how 'far.' Over time, this convention became cemented into the language of hyperdimensional theory and so remains present today, even now that we have the ability to move at different speeds and to gauge hyperdistance, or spissitude, from the origin — 'origin' not being meant to refer to any universally agreed upon origin point nor to some mission-specific point of origin as such, mind you, but just a shorthand euphemism for 'the real-world' — both visually and through our equipment. Although absolute spissitude is best measured in units of distance, it's more naturally measured in units of time — and this isn't because of light speed as the default, though it's something like that. Instead, it's that there's a natural rate of movement associated with anything in motion there. Either you're sitting still, or you're moving at a set pace; you can move faster or slower than that, but either one takes effort to maintain anything other than zero or hyper-C — 'hyperspatial-C,' or 'C-H,' denoted in double-capitals with the 'H' in subscript, as indicated in your texts — which has significant impact on computing, communication, and transportation, incidentally.”

He paused to let everyone verify the same: C_H.

“Conveniently for us, hyper-C is actually the same as our own C, so the convention offers little to object to, as long as the subscript isn't forgotten by the writer or the reader.

“If an object reaches a certain state, it sits still in hyperspace, otherwise it will either return to its original settled state in three-space like those early probes or so much dust settling to the ground, or else take off at hyper-C; in this sense, the 'C' of hyper-C isn't so much a question of the constant speed of massless phenomena in vacuo, but the absolute and only speed of any phenomenon whatsoever above three-space and not in an accelerated frame of reference — other than that massless phenomena still maintain what we would call 'C' relative to us, and that normal speeds obtain within the body and immediate vicinity of a given ship or station. A static object's spissitude doesn't seem to have a minimum distance — that is, there's no apparent Planck metric to hyperspace — but there are some factors that tend to form preferred hyperquantum resonances, analogous of Fermi levels and paramagnetism. In any case, you can see why there are strictures on leaving and returning being confined to hyperports, especially those below the Kármán line — you wouldn't want to drop down to three-space only to find yourself suddenly trying to occupy the same space as some other physical object, and air displacement would present issues if it weren't for evacuated drop-points.

“As for this station of unknown origin, the UnSub, or 'Unknown Subject'... what little is known of it is almost entirely classified way above my pay grade, much less yours, but I'll tell you what I can.

“The UnSub sits about half an hour ana of reality — a little over three hundred million miles out, or five hundred million kilometers; right about three point six astronomical units away, to be precise; a bit more than twice the distance of the sun to Mars, or halfway between Mars and Jupiter, around where the asteroid belt peters out — in the eighth dimension. In our own familiar three dimensions, no one dimension possesses a unique frame of reference, preference over any other orthogonal dimension, nor even identifiable dimensionality without reference to some arbitrarily chosen personal relative orientation to and metric translational distance from a set of four separate coordinate points (five, if you wish to consider tensors for changes over time). This isn't true of hyperspace. 'There,' there is a very truly 'fourth' dimension; this isn't some poorly chosen colloquialism for cardinality that should be rephrased for interchangeability of any dimension with any other, nor a slightly less nonsensical choice from it having been discovered later than the other three making it de facto the fourth. It's based upon a physically meaningful ordinality that asserts itself only above three physical dimensions — and not merely any random subset of three, but only and exactly those specific three that we occupy right here in the real world. Had we been any other civilization in the world, at any other time in history, that specific 'fourth' dimension would necessarily still have been the next accessible one after our own three, as would the fifth one only after that particular fourth one, and so forth. Some mathematicians have pondered the significance of the UnSub sitting there in the eighth dimension; the most popular guess is that it has to do with the then-decreasing ratio of surface area to contents.”

At this, he selected a formula and an equation from the notes, along with a table of figures, and slid it all to the AR model: A:V, then some scribbles showing the gamma function, and derivatives and antiderivatives of spheres and balls in n-dimensions¹.

“We haven't actually gone as far as the UnSub physically, as yet. It's known to be present only though sensory pings, so far, and we can't determine its structural dimensionality. We have no idea who put it there — aliens, God, a lost civilization, time travelers, some incursion from another reality, a secret organization with next-level D-tech. Who knows? Maybe it's just an anomaly, or some naturally occurring event that we just don't know enough yet to have expected? Everyone wants to know what's there, to get the first crack at it; who knows but that maybe when we can reach it, we'll have already obtained whatever insights could be obtained from it? I'd say that it's either dead or dormant, but that's just my own two cents.

“Now, I mentioned structural dimensionality. Picture our Pointlander in two-space with two-radar that expands in circular rings, not spherical. With that, they could tell that some ping presented a two-dimensional profile, but not its contents without x-ray or something. They could tell the precise internal structure of a linear form without needing x-ray, and from that determine whether it were a one-structure of Lineland versus a linear arrangement of Pointlandian objects, and if its internal structure made no sense then they might deduce that it had some two-dimensionality to it but at some z-rotation to their own plane of orientation — or even some body of two or more dimensions that happened to be presenting just a one-edge along their plane of orientation. They couldn't look at a two-profile and determine it to be just two-dimensional and not some cross section of a three or more dimensional body. Same here: we know only that the UnSub is of at least eight dimensions in profile, but necessarily cannot determine more detail than that at this remove.

“Getting back to the point, though: in a certain sense, it doesn't really matter any. A single hyperdimension contains all of the space that we could ever need, and additional dimensions — presumably infinite in extent though they might each be — don't increase the amount of infinitude already present with that first additional hyperdimension. Consider Linelanders in two-space: you could pile an infinite number of one-dimensional structures atop one another like a sandwich, and still have increased the pile's two-thickness not a whit; a pile of n-dimensional objects in n+p dimensions doesn't simply have an infinite amount of space to work with — aleph null — but instead at least beth one space, as far as we can tell, and Schwarzschild densities aside, of course. Given that, what would it matter if they went on to one more hyperdimension or one million of them? I mean, that might help decongest their flight paths a bit, sure, or offer some similar shortcuts architecturally, but that's about it. Again though, that's just my own two cents.”

An illustration popped into the classroom as he said this, showing a black section of string-like length that was labeled Lineland, a second line segment that had its midsection sliced into multiple blue strings separating at one point of the string and taking separate paths to either side before reconnecting at another like the shadow of a whisk and labeled Lineland with DIM-2 slices, and a third that showed the same as the second with added red and green strings that leapt over and under the extra string sections at random points and labeled Lineland with DIM-3 transit.  The illustration as a whole was captioned as Linear space of Lineland technologically re-formed for inhabitants' use and convenience.

“We've broken the sixth barrier, as it were, with some spillover much like a sonic boom or Cherenkov radiation that presents some unique challenges, but dimension seven still stands as an unscalable obstacle. It's thought that passing seven and one quarter embedding dimensions represents a watershed, past which things should become increasingly easy to bootstrap and leapfrog. It's also postulated that the actual watershed dimension itself — seven point two five six nine... — is specifically unattainable, and not solely because of its irrational fractality acting as a knife-edge to balance against divergent force derivatives.”

“That brings us around to the question of cosmo-topology.

“I mentioned incursions for a reason. I didn't necessarily mean multi-world, though that's always a possibility. I meant more like space aliens.”

This time the board drew its own sketches without his having to get up.

The classroom rippled with variations of puzzled frowns.

“Picture number one, on the left, shows Lineland set metrically flat as a circumference of a sphere. That sphere is necessarily a plane that's curved in three dimensions, closed without boundaries, but is itself set within a metrically flat space of three or more dimensions. If you look to the right, you'll see a similar Line-in-sphere land.

“Imagine that they develop megamechs in Lineland two, explore their sphere, and discover three-space, only to eventually run into sphere one.  Maybe we don't face that now, maybe we never will, but it's something to keep in mind.

Once more, the board re-drew itself.

“In the second case, even if hyperspace shows a negative Gaussian at some point, we could easily enough — in principle — find ourselves in one Lineland-like flat universe out of a bunch, and all piled up hyperspatially like so many Pringles.”

Again, the familiar re-drawing of the board.

“Alternatively, it could take a course like the preceding two, but in some less expected direction, if you'll pardon the pun.”

The board now showed two doughnuts, fashioned in such a way that the body of each went through the hole of the other, like a pair of links in a chain.

“Genus one, interlinked, not a horned sphere, though that's not ruled out yet either. It might not be obvious, but aside from a simple variation on the earlier slides, this could play a vital part in our next big advancement in universal wave function modeling. See your notes on topological quantum fluids, if you're curious, and the side-notes on quasicrystals, Bose-Einstein condensates, and phase transition.”

“OK, so you might recall those degenerate Z-dimensions that I alluded to in the windowpane analogy, the ones that I said to keep in mind for later? They're not known to be a real constraint, only a postulated possibility beyond our potential eventual technological limit. If such exists, then the hows and whys thereof would fall within the realm of ontology and perhaps teleology, rather than strictly answerable geometries.

“Their basic nature might yet be accessible to a degree though, in that a megamech could arguably tilt its relative plane of reference through a limited range of arc in such Z-dimensions. This might have little effect of value, but wouldn't be without some use for banking, evading, and parallax. In such a case, they couldn't rotate freely through a complete one hundred eighty degrees, hence onward through three hundred sixty degrees, but would still not be limited entirely to a relative sandwich of translatable space. As well, there could very well exist exploitable aspects that a megamech might yet use in some unforeseen fashion — for example, our two-dimensional Flatlandian megamechs might find themselves in a world of only a millimeters' width in their third dimension, though more reasonably a Planck length, and hence be unable to simply spin about their X- or Y-axis at will, but nonetheless find some way to spindle themselves finely around a very small radius — Planck over two, as it were —” he chuckled, “in order to become one dimensional for practical purposes, and then rolling that thin string down toward a practical zero dimensionality.

“Of course, if such were even possible,” he paused sententiously for the throttle jockeys, and those minded of logistics and intel, “they would then be able to operate within their clearly finite Z-space as if it were of infinite relative-breadth for any practical purposes.”

“In fact,” he went on, “it would even present the counterintuitive, but deductively irrefutable, result that such objects of a given interior size would then — under the spindled and rolled zero-dimensionality condition — while not actually being caused to become any bigger at all on the inside than they had been to start with, nonetheless de facto be much smaller on the apparent outside than they still would be within, practically infinitely so, and all without having to resort to inflating or deflating spacetime: no specific Ricci curvature or Weyl tensor as such, though there are some effects to consider regarding the resulting energy density as seen from the outside perspective. As such, fermionic bodies could arguably occupy essentially bosonic states, though there would still be a hyperspatial Schwartzschild radius to bear in mind, and presumably no light speed boost deriving from the merely effective state. One certainly wonders if the same could be brought about in reverse for fermionic states of bosons, or even perfectly extractable information from within an event horizon — to say nothing of the infinite regression of such a megamech containing within it a number of megamechs of effective codimension n... that is of effective codimension three in our case, or effective codimension two for the Flatlanders, or effective relative dimension negative-n in all cases, with each such epsilon-megamech in turn carrying their own respective complement of Cantor dust megamechs of effective codimension epsilon, and so on. It could even be that our universe is ultimately just such an epsilon-megamech.”

Everyone laughed with this, until they realized that he wasn't smiling.  This sobered them as the implications sank in.

“Of course,” he noted, “such considerations wouldn't necessarily negate M-theoretic leakage or path lensing and metric displacement detection.”  This he said as he looked across the room over the rim of his glasses in order to underscore the ramifications thereof.

“Another possible application of such deformation is that they fold back upon themselves as previously described, and flip themselves over while yet remaining entirely within the volume afforded to them even though they themselves are several inches across in each of their non-zero axes — though this last begs chiral annihilation were they to come into contact with any normal matter afterward without first having reversed this process. Obviously, this could present a problem with self-interaction, whether by direct self-intersection or through quantum tunneling or other means.”  At this, he glanced at several students who'd shown interest in energy and weapons applications.

“In any event of folding or spindling though, this raises the question of locative self-intersection with or without effective self-interaction. 'Without' would be alike to a more-nearly literal bosonic state, permitting megamechs to tie themselves up into contorted Gordian knots of Calabi-Yau manifolds — metaphorically, since you can't make loops in fewer than three dimensions, and can't keep them stuck in place above three — with leading edges passing right through any and all faces in their paths; 'with' would permit one to cut a tongue from one portion of a Flatlandian megamech and a matching groove in another, and glue them together — temporarily or permanently, where 'temporarily' would leave open the possibility of rotating pairings — to obtain what would amount to wormhole portals, though the one-dimensional domain wall line segment groove-discontinuities might form impenetrable barriers from the other side and their pointed ends would slice through anything embedded within that continuum (aside from other such domain walls, and themselves simply translating along any tongues encountered) and present a personnel hazard (though if the tongues could cross through each other to their respective grooves — somewhat like a pair of escalators, for the more gear-headed engineers among you — then any Flatlanders would be left with only the two monopolar defects to contend with, since each tongue would then cover the other's domain wall); some variable state, permitting choice between the two states might even permit such portals' tongues to pass ephemerally through intervening spindle layers before reaching their target layer's groove.”

“Negative dimensionality is still very much a hotly-contested point of conjecture...” he trailed off and eyeballed the more-theoretical R&D-track students meaningfully.

“Fractal dimensionality,” he continued, after pausing to collect his thoughts, “is also a divisive issue.

“Although fractality has geometric meaning in several ways, it has yet to bear fruit even in theory with regard to megamechs. Perhaps their component parts could operate when wholly separate, or even atomized; perhaps the armor could be five dimensional, with six-dimensional radar and one-dimensional tachyonic circuits; maybe hyperspace could be treated as if it were a Menger sponge, begging the question of accessing that not-necessarily-continuous space between the spaces. In less rarified realms, it might be as mundane as a bosonic scaling transform, the megamechs literally performing coordinate conjugation through themselves as they would otherwise merely appear to be to a Flatlander seeing the projected two-shadow of an n-body that's rotating in three-space. Whatever the reality of it, if any, the topic will be discussed in detail, and you will be graded on it in the final exam.

“Now, I mentioned radar earlier without going into detail about it. The example just now used ships that might be traveling in up to five-space, but that have six-space radar capability. I'm sure that this sounds as if it could be of use, but just how so might not be quite as clear.

“Imagine a Linelandian ship in two-space. If they were to encounter an oncoming fleet with only their standard two-radar, then they could see the lead ships well enough, since there's no heat distortion or airy index of refraction or diffractive dust haze or anything in their second-dimensional hyperspace, but only a general cloud of ping-back beyond that due to the fleet's generally blobby signature. Were they to use three-radar however, then the picture would resolve itself in much greater detail, since the signal wouldn't encounter a wall of noisy shapes en route, nor would there be any radar shadow to obscure anything. It would instead return a clear image of just how many ships were present, where they were, their exact two-footprints, and any radar-reflective mass concentrations within each ship. This last might not be necessary, and would presumably fail to identify personnel at all, but would serve to highlight likely engine nacelles and computer locations and weapon bank distribution — key target points.”

“Not only that, but remember that this is a Linelandian ship, hence crewed with Linelanders whose natural senses are geared for one dimension. Here they'd not only need to be capable of processing the two-dimensional coordinate data of Flatland, but moreover would now be being given a field of two-dimensional view from a third-dimensional perspective. That's two full dimensions above their norm: back home, they see a single point ahead of them and another right behind; after training for two-space, they can understand a whole one-dimensional boundary circle of visual data around them, or at least some arc segment thereof, and their minds will grasp the two-dimensional depths and parallaxes of that visual field; now they're being fed three-radar and presented with a two-dimensional picture head-on. There might not be any depth perspective needed three-dimensionally, but it's still a very different result from the previous strip of the nearest through farthest objects.”

The professor paused again to take a few sips of water before continuing.

“Both the Flatlanders' down-pitch of the megamech's bridge or sensors that I began with, and the x-radar example that I made reference to later, approach the visual processing issue from the perspective of an n-dimensional being viewing their world from some non-zero n+p dimensional altitude without any special hyperdimensional visual cortex or any other special workaround. In the former, they still see the world as a one-dimensional strip for their frame of reference, and in the latter they receive a full two-dimensional image as viewed from the edge as if it were layers of one-dimensional strips. You're three-dimensional people with two-dimensional fields of vision, and so can appreciate how neither of these would do justice to your taking in an image of the Mona Lisa with a single glance.

“Imagine looking at the Mona Lisa through the eyes of a Flatlander. You might get the 'surface  '-edge along one side of the canvas, then probe to the next 'depth  ' sideways with your adjustable x-ray vision, and look deeper and deeper sideways at the vertical strips. This would amount to multiple one-dimensional strips that you might process in some way, but wouldn't add up to a single integrated two-dimensional field — and this same issue is simply compounded when viewed by a Linelander, who might think of x-raying from one point to the next along a single line of vision, then look upward at the next point and repeat the x-ray, and continuing this way until they'd reached the last point at the top of the edge. That's the difference: integration. Just as three plus three is hardly the same as three times three, much less three to the third power or any other hyperoperator, information processed within a given dimensional context — even when added layer upon layer — doesn't equal the whole as integrated from the simultaneous perspective of some higher-dimensional position. X-raying the Mona Lisa edge-on simply isn't the same as standing in the Louvre and taking her in face-on, no pun intended.

“The purpose of the simulators is to begin mapping your visual cortices' processes to the algorithm required to integrate a three-field, rather than to see a pile of multiple two-dimensional cross-sections. We can't genetically engineer you to possess four-dimensional eye-glomes instead of, or in addition to, three-dimensional eyeballs, or hypersurgically graft something like that to you, at least not yet — but we can feed three-field visual data to you that wouldn't make much sense now, but would be properly interpretable to you once trained. Initially, it'll be disconnected; not gibberish, but hardly forming a whole. You might guess that it would eventually feel like seeing through things, but that would be like the Mona Lisa metaphor: you won't see layers of onions within other layers, you'll see the entire onion at a glance.”

He took off his glasses at this, looking around the room.

“That's what it means to crew a megamech.”

O ~~~ O