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“I’m here, Grandma,” Duncan called into the antique intercom which had been presented to her by some anonymous admirer long ago. (It was still clearly marked “Made in Hong Kong,” and had been dated circa 1995. Shameful to relate, there had been one attempt to steal it, though since theft was virtually unknown on Titan, this was probably only a childish prank or an anti-Makenzie gesture.)

There was, as usual, no reply, but the door unlatched at once and Duncan walked through into the tiny foyer. Grandma’s electro cycle occupied the place from which it had not moved for years. Duncan checked the battery and kicked the tires, as he always did with great conscientiousness. No need for any pumping or charging this time; if the old lady suddenly felt the impulse to descend upon the city, there was nothing to prevent her.

The kitchen, which was a unit lifted intact from a small orbital passenger shuttle, was a little tidier than usual. Presumably one of the voluntary helpers had just made her weekly visit. Nevertheless, the usual sickly sour smell of slow culinary disintegration and inadequate recycling was heavy in the air, and Duncan held his breath as he hurried through into the living room. He never accepted more than a cup of coffee from Grandma, and feared accidental poisoning if he ever sampled the products of her robot reconstituter. But Ellen

seemed to thrive on it; over the years she must have established some kind of symbiosis with her kitchen. It still lived up to the manufacturer’s “failsafe” guarantee, even though it did produce the most peculiar odors. Doubtless Grandma never noticed them.

Duncan wondered what she would do when the final disaster occurred.

The main living room was as crowded as ever. Against one wall. were the shelves of carefully labeled rocks-a complete mineralogy of Titan and the other examined moons of Saturn, as well as samples from each of the rings.

As long as Duncan could remember, there had been just one section empty, as if, even now. Grandma was still waiting for Kleinman to return.

The opposite wall was more sparsely occupied with communications and information equipment, and racks of micro modules which, if completely saturated, could have held more knowledge than all the libraries of Earth up to the twenty-first century. The rest of the room was a compact little workshop, most of the floor space being occupied by the machines that had fascinated Duncan throughout his childhood, and that he would associate with Grandma Ellen as long as he lived.

There were petrological microscopes, polishing and cutting tools, ultrasonic cleaners, laser knives, and all the shining paraphernalia of gemologist and jeweler. Duncan had learned to use most of them, over the years, though he had never acquired more than a fraction of his grandmother’s skill and almost wholly lacked her artistic talents. What he did share, to a much greater extent, were her mathematical interests, exemplified by the small computer and associated holographic display.

The computer, like the kitchen, was long overdue for retirement. But it was completely autonomous, so Grandma did not have to rely in any way upon the immensely larger storage facilities in the city. Although her computer had a memory scarcely larger than that of a human brain, it was sufficient for her rather modest purposes. Her interest in minerals had led her, inevitably, to crystallography, then to group

theory, and then to the harmless obsession that had . dominated so much of her lonely existence. Twenty years ago, in this same room, she had infected Duncan with it. In his case, the disease was no longer virulent, having run its course in a few months; but he knew, with amused tolerance, that he would suffer occasional relapses throughout his life. How incredible that five perfectly identical squares could create a universe that neither man nor computer would ever be able to explore fully…. Nothing in the familiar room had changed since his last visit, three. weeks ago. He could even imagine that Grandma had not moved; she was still sitting at her worktable, sorting rocks and crystals, while behind her the read-out screen intermittently flashed solutions of some problem the computer was analyzing. She was, as usual, wearing a long gown that made her look like a Roman matron, though Duncan was quite sure that no Roman matron’s dress ever appeared quite so disheveled or, to be perfectly frank, so overdue for the laundry. While Duncan had known her, Ellen’s care of her equipment had never extended to her personal appearance.

She did not rise, but tilted her head slightly so that he could deliver his usual affectionate kiss. As he did so, he noticed that the external world, at least, had been touched by change.

The view from Grandma’s picture window was famous-but by reputation only, since few indeed had been privileged to see it with their own eyes. Her home was partly countersunk into a ledge overlooking the dried-up bed of