From ``The Number of the Beast'', by Robert A. Heinlein.

Hilda: ``Professors, may I point out that class is in session? I want to know about time machines and I'm not sure I could understand with champagne buzzing my buttonhead.''

``Sharpie, sometimes you astound me.''

``Zebbie, sometimes I astound myself. Since my husband builds time machines, I want to know what makes them tick. Or at least which knobs to turn. He might be clawed by the Bandersnatch and I would have to pilot him home. Get on with your lecture.''

``I read you loud and clear.''

But we wasted (``wasted?'') a few moments because everybody had to kiss everybody else -- even Zebbie and my husband pounded each other on the back and kissed both cheeks Latin style. Zebbie tried to kiss me as if I were truly his mother-in-law but I haven't kissed that way since junior high. Once I was firm with him he gave in and kissed me better than he ever had before -- whew! I'm certain Deety is right but I won't risk worrying my older husband over a younger man and I'd be an idiot to risk competing with Deety's teats et cetera when all I have is fried eggs and my wonderful old goat seems so pleased with my et cetera.

Class resumed. ``Sharpie, can you explain precession in gyroscopes?''

``Well, maybe. Physics One was required but that was a long time ago. Push a gyroscope and it doesn't go the way you expect, but ninety degrees from that direction so that the push lines up with the spin. Like this --'' I pointed a forefinger like a little boy going: ``Bang! -- you're dead!''

``My thumb is the axis, my forefinger represents the push, the other fingers show the rotation.''

``Go to the head of the class. Now -- think hard! -- suppose we put a gyroscope in a frame, then impress equal forces at all three spatial coordinates at once; what would it do?''

I tried to visualize it. `I think it would either faint or drop dead.''

``A good first hypothesis. According to Jake, it disappears.''

``They do disappear, Aunt Hilda. I watched it happen several times.''

``But where do they go?''

``I can't follow Jake's math; I have to accept his transformations without proof. But it is based on the notion of six space-time coordinates, three of space, the usual three that we see -- marked x, y, and z -- and three time coordinates one marked `t' like this --'' ( t ) ``-- and one marked `tau,' Greek alphabet --'' ( τ ) ``-- and the third from the Cyrillic alphabet, `teh' --'' ( )

``Looks like an `m' with a macron over it.''

``So it does, but it's what the Russians use for `t'.''

``No, the Russians use `chai' for tea. In thick glasses with strawberry jam.''

``Stow it, Sharpie. So we have x, y, and z; t, tau, and teh, six dimensions. It is basic to the theory that all are at right angles to each other, and that any one may be swapped for any of the others by rotation -- or that a new coordinate may be found (not a seventh but replacing any of the six) by translation -- say `tau' to `tau prime' by dispacement along `x.' ''

``Zebbie, I think I fell off about four coordinates back.''

My husband suggested, ``Show her the caltrop, Zeb.''

``Good idea.'' Zeb accepted the widget from my husband, placed it in front of me. It looked like jacks I used to play with as a little girl but not enough things sticking out -- four instead of six. Three touched the table, a tripod; the fourth stuck straight up.

Zeb said, ``This is a weapon, invented centuries ago. The points should be sharp but these have been filed down.'' He flipped it, let it fall to the table. ``No matter how it falls, one prong is vertical. Scatter them in front of cavalry; the horses go down -- discouraging. They came into use again in Wars One and Two against anything with pneumatic tires -- bicycles, motorcycles, lorries, and so forth. Big enough, they disable tanks and tracked vehicles. A small sort can be whittled from thorn bushes for guerrilla warfare -- usually poisoned and quite nasty.

``But here this lethal toy is a geometrical projection, a drawing of the coordinates of a four-dimensional space-time continuum. Each spike is exactly ninety degrees from every other spike.''

``But they aren't'', I objected. ``Each angle is more than a right angle.''

``I said it was a projection. Sharpie, it's an isometric projection of four-dimensional coordinates in three-dimensional space. That distorts the angles ... and the human eye is even more limited. Cover one eye and hold still and you see only two dimensions. The illusion of depth is a construct of the brain.''

``I'm not very good at holding still--''

``No, she isn't,'' agreed my bridegroom whom I love dearly and at that instant could have choked.

``But I can close both eyes and feel three dimensions with my hands.''

``A good point. Close your eyes and pick this up and think of the prongs as the four directions of a four-dimensional space. Does the word tesseract mean anything to you?''

``My high school geometry teacher showed us how to construct them -- projections -- with modeling was and toothpicks. Fun. I found other four-dimensional figures that were easy to project. And a number of ways to project them.''

``Sharpie, you must have had an exceptional geometry teacher.''

``In an exceptional geometry class. Don't faint, Zebbie, but I was grouped with what they called `overachievers' after it became `undemocratic' to call them `gifted children' ''

``Be durned! Why do you always behave like a fritterhead?''

``Why don't you ever look beneath the surface, young man! I laugh because I dare not cry. This is a crazy world and the only way to enjoy it is to treat it as a joke. That doesn't mean I don't read and can't think. I read everything from Giblett to Hoyle, from Sartre to Pauling. I read in the tub, I read on the john, I read in bed, I read when I eat alone, and I would read in my sleep if I could keep my eyes open. Deety, this is proof that Zebbie has never been in my bed; the books downstairs are display; the stuff I read is staked in my bedroom.''

``Deety, did you think I had been sleeping with Sharpie?''

``No, Zebadiah.''

``And you never will! Deety told me what a sex maniac you are! You lay your lecherous hands on me and I'll scream for Jacob and he'll beat you to a pulp.''

``Don't count on it, dear one.'' my husband said mildly. ``Zeb is bigger and younger and stronger than I ... and if I found it needful to try, Deety would cry and beat me to a pulp. Son, I should have warned you; my daughter is vicious at karate. The killer instinct.''

``Thanks. Forewarned, forearmed. I'll use a kitchen chair in one hand, a revolver in the second, and a whip in the other, just as I used to do in handling the big cats for Ringling, Barnum, and Bailey.''

``That's three hands,'' said Deety.

``I'm four-dimensional, darling. Professor, we can speed up this seminar; we've been underrating our overachiever. Hilda is a brain.''

``Zebbie, can we kiss and make up?''

``Class is in session.''

``Zebadiah, there is always time for that. Right, Pop?''

``Kiss her, Son, or she'll sulk.''

``I don't sulk, I bite.''

``I think you're cute too,'' Zebbie answered, grabbed me by both shoulders, dragged me over the table, and kissed me hard. Our teeth grated and my nipples went sprung! Sometimes I wish I weren't so noble.

He dropped me abruptly and said, ``Attention, class. The two prongs of the caltrop painted blue represent our three-dimensional space of experience. The third prong painted yellow is the t-time we are used to. The red fourth prong simulates both Tau-time and Teh0time, the unexplored time dimensions necessary to Jake's theory. Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three -- you seem to be smart at such projections.''

I closed my eyes and thought hard. ``Zebbie, I don't think it can be done. Maybe Escher could have done it.''

``It can be done, my dearest,'' answered my dearest, ``but it is unsatisfactory. Even with a display computer with capacity to subtract one or dimensions at a time. A superhypertesseract -- a to the sixth power -- has too many lines and corners and planes and solids and hypersolids for the eye to grasp. Cause the computer to subtract dimensions and what you have left is what you already knew. I fear it is an innate incapacity of visual conception in the human brain.''

``I think Pop is right,'' agreed Deety. ``I worked hard on that program. I don't think the late great Dr. Marvin Minsky could have done it better in flat projection. Holovision? I don't know. I would like to try if I ever get my hands on a computer with holovideo display and the capacity to add, subtract, and rotate six coordinates.''

``But why six dimensions?'' I asked. ``Why not five? Or even four, since you speak of rotating them interchangeably?''

``Jake?'' said Zeb.

My darling looked fussed. ``It bothered me that a space-time continuum seemed to require three space dimensions but only one time dimension. Granted that the universe is what it is, nevertheless nature is filled with symmetries. Even after the destruction of the parity principle, scientists kept finding new ones. Philosophers stay wedded to symmetry -- but I don't count philosophers.''

``Of course not,'' agreed Zeb. ``No philosopher allows his opinions to be swayed by facts -- he would be kicked out of his guild. Theologians, the lot of them.''

``Let me see,' I said. ``If I understood earlier, each dimension can be swapped for any other.''

``By a ninety-degree rotation, yes.''

``Wouldn't that be the combinations taken four at a time out of set of six? How many is that?''

``Fifteen,'' Zebbie answered.

``Goodness! Fifteen whole universes? And we use only one?''

``No, no, my darling! That would be ninety-degree rotations of one Euclidean universe. But our universe, or universes, has been known to be non-Euclidean at least since 1919. Or 1886 if you prefer. I stipulate that cosmology is an imperfect discipline, nevertheless, for considerations that I cannot state in nonmathematical terms, I was forced to assume a curved space of positive radius -- that is to say, a closed space. That makes the universes possibly accessible to use either by rotation or by translation of this number.'' My husband rapidly wrote three sixes.

``Six sixty-six,'' I said wonderingly. ``The Number of the Beast.''

``Eh? Oh! The Revelation of Saint John the Divine. But I scrawled it sloppily. You took it that I wrote this: `666.' '' But what I intended to write was this:'' ''6<sup>66</sup>'' ``Six raised to its sixth power, and the result in turn raised to its sixth power. That number is this:'' 1.03144 X 10²⁸ ``--or written in full:'' 10,314,424,798,490,535,546,171,949,056 ``--or more than ten million sextillion universes in our group.''

What can one say to that? Jacob went on. ``Those universes are our nextdoor neighbors, one rotation or one translation away. But if one includes combinations of rotation and translation-- think of a hyperplane slicing through superhypercontinua not all at the point of here-now -- the total becomes indenumerable. Not infinity -- infinity has no meaning. Uncountable. Not subject to manipulation by mathematics thus far invented. Accessible to continua craft but no known way to count them.''