The first incursion was so sudden, and so unlike those that followed, it was a long time before anyone even referred to it as such. It was a brisk autumn day that November twelfth; I should know, because I was there, back in the beginning of this tribulation. I myself was taking a morning walk to arouse my mind for the day's philosophical labors. It was to be a day that I would have been named head of the Department of Physics at my university- but that is only idle speculation. For on a day in which the leaves should have reflected orange-brown hues and fallen lazily to the soil beneath, instead they charred black in a fraction of a second then rent themselves into emancipated elementary particles without a pause.
Initially, a violet ripple spread from low in the sky, leaving a translucent indigo veil starting roughly thirty feet in the air and terminating somewhere underground. Its width equaled that of a six-lane highway, but on it rode not cars, but visions of another world: the clouds behind that gossamer film failed to match those around them, and desert sand alighted on what once was sparse grass dotted with bushes. Yet what instantly snatched my attention was the dull argent sphere near the center of that disjunction, bedecked in jutting rectangular boxes of similar color and numerous characters and glyphs.
My pleasant ambling transformed into a furious sprint. Only my thought of nearby sanctuary sustained my mind from screaming terror, for though not a soul was near to tell, I knew. That device was a distinctly human creation, one my field of study invented: a thermonuclear explosive.
Retrospection fails concerning much of what followed. A man faced with absolute destruction has little immediate need for long-term memory; imminent obliteration works wonders for focus. My recollection picks back up after the shutting and locking (by four separate mechanisms) of a strikingly ponderous steel door. I can scarcely imagine how my aged and admittedly poorly kept up body managed now, but at the time, I regained brief youth. At that point my senses returned: I was alone inside the university subterranean vault. We kept it around as storage space, but I now adopted it for its intended purpose ordained in the 1940s, as a bomb shelter. I was now secure; my faith in engineers surpasses that of almost anyone’s, having seen what they are capable of, so my heart needed no additional assurance.
As was my expectation, the blast wave soon enough rolled over my head. After what was probably an hour, I donned one of the radiation suits hidden in a disused corner, behind stacked cardboard boxes of laboratory equipment. Taking with me an archaic radio set with the most look of the set with me, I unlatched the enormous vault door, this time letting creak clamorously open of its own accord. I rechecked the seals on my suit and stepped over the threshold.
The hundreds of movies and books plus the hundred thousand television broadcasts on the subject are far more than enough to describe to anyone what lay outside: nothing. The same publications detail my hailing of and subsequent rescue by military helicopter. And still there would have been twice as many, had the incident not happened again. Repeatedly.
Through media dramatization, the occurrences were titled "warstorms". I never cared for the name, thinking it melodramatic, something more suited to  Hollywood hack work. Not to say that my own creativity produced any better; and I cannot call it entirely inappropriate, when purple waves ripple through the sky like onrushing cumulus and an endless variety of ingenious modes of destruction spew forth as though every man, woman, and child on Earth has been struck by sudden homicidal madness. Each one posed a different problem: sometimes gray-suited soldiers marched out with rifle barrels ablaze, other times tight formations of black-winged jets issued in blurs with bleached smoke streaming behind them. One peculiar event dominated speculation both idle and deadly serious for years: an entire squadron consisting of unknown engines of war, each supported by five limbs never quite touching the earth below them invaded the Ruhr of Germany. They bore fearful flame-belching cannons and a not-quite-familiar coat of arms; the German industry has not yet recovered. Devoid of any specialized tactics or intelligence, the arrayed military might of the entire world fought to their utmost against a foe with no pattern, no movements, and no motive. Valiantly, to be sure: worldwide enlistment numbers reached all-time peaks in every civilized nation, even as men dropped on the battlefield like so many china plates from a jarred cupboard. My own grandson, whom I had hoped would enter academia as I had, instead chose to enlist, joining the army with consuming ardor; he must have thought he could save the world single-handedly. Militias as well remained constantly at the ready to drive back whatever might arrive in an hour or arrive in a year, while mobile command centers which awakened dim memories of the SAC deployed troops at the barest evidence of incursion.
We did win some battles. I recall the rout of the invading force in South Africa last May, and the surprising naval battle off the coast of Japan two years ago (before that, no one knew that incursions could develop over bodies of water) which ended in the capture of an enemy vessel; the Japanese still haven't revealed anything, but I feel quite certain that they've been examining every last bolt and panel for something they can use to turn the tide. The fact is, every incursion eventually found itself repelled; but the high cost in men and metal of most prevent me from describing them as victories.
The flow of this conflict met a serious, but extremely welcome detour with the fruition of my team's research. Amongst numerous masks and media blackouts, the government had diverted my person to head the Four Sides Project, if one interprets "head" very loosely. Their research was in my field, but everyone there knew the primary reason I was privy to the project's secrets was to keep me out of the public eye. I was an elderly burden to the quick thinking young men and women who otherwise constituted the effort, one largely pushed aside and ignored. I suppose that they hoped to avoid the possibility of a comment I made being seized by the news networks as a weapon to incite public panic; and in the process incite public television viewing. Empowered to do little else, I took it upon myself to go over the hasty and often erroneous figurings of my brilliant yet arrogant subordinates. Our myriad flaws notwithstanding, the countless hours came to a resounding conclusion, in the form of mankind's first planar transposition generator.
To understand, first hear our operational concept. Painstaking review of the available incursion footage yielded precious few data; however, field teams supplemented them with readings from the wine-tinted edges of several event regions. Relying on theoretical extrapolations little better than wild guesses, my think tank inferred this model: with time assumed as the fourth, incursions were visible manifestations of fifth-dimensional movement. Truly, the mechanics of such an action are, like may subjects in physics, only properly described by mathematics, but for the layman, think of one brick in a wall pushed out and replaced by another. Our universe is like that wall of bricks, with each brick a part of space; in fact, space is not nearly so discrete, but one must ignore that for the sake of the analogy. What our team found was that our wall was bordered by other walls, forming an entire multiverse composed of innumerable layers.
From this paradigm, we deduced the incursions to be an exotic form of attack from a plane of existence not entirely unlike our own, perhaps a "wall" or so off. Also, from the empirical evidence, that plane must be somewhat more advanced technologically, and intent on conquest. Therefore, the PTG: an unearthly hand to push our brick back into place, and beyond if necessary. It had an unwholesome look to it when complete, an off-white pyramid laid on its side and run through with angular trenches like gashes dripping with crimson the alphanumeric code of its assemblers. A monstrosity to make a pharaoh weep to be sure, but to us, the shining instrument of our salvation.
Operating a PTG took a link to a particle accelerator in the beginning, before refinement of the relevant principles freed us from the need for synthetic particles. Flipped on, a phantom region proceeding from large end of the pyramid created a familiar amethyst shimmer precisely two hundred and seventeen point seven meters away, disregarding any conventional requirement for line of sight. The dimensions of the translocation zone depended entirely on the device's power supply. Men and tanks and planes and ships stood at the ready, advancing into enemy territory once the skin of the universe was breached, guns firing even before they passed those auroral gates. The devices only manipulated space from one dimension, so they marched into lands corresponding to the ones they had just left. Early attempts produced many portals into desolate uninhabited regions, but dead reckoning from those coordinates supplied us with ever increasing accuracy. Ignorance of from whence our attackers hailed forced us to "push" in both directions. The first direction was so much wasteland, but with fierce inhabitants, while the second  resembled our beloved Earth more, with greater but less prepared forces. We set upon one when all forays into the other showed blasted ruins. We could keep one of our counter-incursions in play for a couple of hours at most, before strained energy reserves and battered forces necessitated withdrawal. Now, we could make progress, we thought.
My grandson sent letters back home when he could; with the near constant hostilities, these letters weren't often received, and all the more cherished. In one such missive, he detailed a most disconcerting experience:
"I saw one of the Enemy today. His helmet had been knocked off when he fell. I had shot him during my squad's advance, and we were still on the move so I didn't get to see him very well, but I don't think that I'll be able to fight like I have. He had a human face! We were told that the Enemy was like us, but somehow I always saw the black visors down and imagined that we were fighting monsters. I know that wars always have men kill one another, but somehow this feels different. Wrong. My tour of duty is almost up. I hope to come home soon. I pray that my war buddies never have to see what I have."
Later messages took on a decidedly happier cast, as my grandson slowly came to terms with the war. I felt good for him, and was gladdened by his martial success: his platoon earned many decorations from it's valor. In truth, it seemed that we, humanity as a race, were learning how to deal with our foe. Our technological level grew in great strides, not in small part from retro-engineering captured equipment, so that we were no longer terribly outclassed. The count of our triumphs rose, and with it worldwide spirits, even in the face of unprecedented death.
Yet a curious thing happened. Regardless of all the casualties we inflicted on the other side, incursions grew only more frequent. Interpreting this phenomenon as the fierce struggle of a dying beast, our generals ordered ever grander assaults, our politicians proposed ever broader alliances, and our engineers dreamt up ever more innovative engines of domination. Notably puissant was the true hovercraft: French-developed, an arachnoid vehicles whose anti-gravity field generators (based on my teams PTG technology) produced enough lift and shock absorption to allow it to function as a main battle tank. They allowed their country to secede from the European State and rebrand their nation with a unique new flag, asserting newfound tactical supremacy. Each nation drove its resources to the limit, and in a way, it was good.
In the meantime, I persisted in rechecking my team's figures, no doubt a service to my country. One evening, I tripled my efforts. I couldn't stop, because I needed absolute verification. You see, one section of the basic equations never ceased to bother me. I couldn't help but think something was the matter with it. No sleep slowed my progress that night, and by five-thirty the next morning, my doubts found their credence. I rushed to the rest of my team, presenting my finding with near-manic swiftness and verve. It was not the best strategy; their disdain for me, bolstered by their pride over their immense accomplishments refused to let the light of my findings in. I was curtly dismissed, and unceasingly shunned afterwards. My own grandson died on the battlefield before I could reach him, tell him that all his bravery was in vain!
So I alone know the unexpected reality. The PTG had never altered the fifth dimension: only the fourth.
As I write this, the President is on the television announcing a plan to use PTGs in conjunction with hydrogen bombs. They can never say I didn't see it coming.