There was a time, said the Mathematician,
when we had a tacit agreement that any tale told in this company should embody some point of mathematical, scientific, or philosophic interest. Of late, however, this understanding seems to have given way to a tendency for buffoonery. I am thereby emboldened to present to you a story which I call:
My friend Warlock Tombs and I were spending a quiet evening at home in our lodgings at 21c Inkerman Terrace, when our landlady, Mrs Fazakerly, ushered in a stranger. Our unexpected visitor, a tall man of middle age, spoke in an agitated manner.
Mr Tombs, I am at my wit's end; you are the only man in the country who can help me. I am .....
Tombs interrupted with an imperious gesture.
My dear sir, I already know a great deal about you. You are a former deep-sea diver who has been retired for some years. You have written a small monograph on your hobby of bee-keeping, recently published . You live in Portsmouth, whence you have just arrived on the 5.30 train.
Great Scott, Tombs, I exclaimed,
this is fantastic! How can you tell so much about someone you have only just met?
Tombs gave a shy smile, sometimes mistaken for a conceited smirk by those unappreciative of his greatness.
Elementary , my dear doctor. (I have tried in vain to dissuade Tombs from this jocular manner of address, a reference to the fact that my National Service was spent as a lance-corporal orderly in the Royal Army Medical Corps.)
As he ascended the stairs, our visitor paused every few steps, suggesting one who has been trained to rise slowly for fear of the bends. On the other hand, he does not leap wildly about, like one accustomed to wearing lead boots. Thus I deduce that he used to be a deep-sea diver, but has been retired long enough to readjust to normal footwear. The evening paper in his overcoat pocket is the second edition, which has not been on sale many minutes, yet already it is folded to the page of literary criticism. Who else would turn first to that page other than a newly published author anxious to read his first reviews? Most first books are about the author's hobby. This gentleman does not use tobacco, as one can tell from his unstained moustache, yet he brings with him a distinctly smoky smell, common in those who frequently smoke out beehives. I see by the grime on his cuffs and collar that he has recently travelled by train. His shoes, however, show that he has walked here from the railway station. The only station within walking distance is Waterloo, from which a brisk walking pace would bring him to our door in 4½ minutes. The only train to have arrived at Waterloo in the last 5 minutes is the 5.30 non-stop express from Portsmouth. Simple deduction, my dear Motson, simple deduction. (I have explained to Tombs the difference between deduction and induction, but to no avail.)
Our visitor broke in with some impatience.
If you must know, I am a clock-maker from Slough. The pauses on your staircase were occasioned by your landlady, who is over-generously built for rapid climbing. Your next-door neighbour's peculiarly obnoxious bonfire accounts for both the state of my linen and the smell which entered with me. The literary page in the newspaper also contains a crossword which has been engaging my attention, and my only hobby is collecting cast-iron coal-hole covers.
You notice, Motson, Tombs muttered to me in an aside,
that he does not deny travelling on the 5.30 from Portsmouth. He then addressed himself to our caller.
Come, come, my dear sir, do not waste my time with trivia. What is your problem?
My name is James Arbuthnot, the stranger began.
My wife and I have been married for more than 15 years without the slightest sign of discord between us until about a year ago, when we bought a house in Surrey. Janus Lodge had remained unsold for some time because of a reputation for being unlucky or even haunted. Of the previous owners,one had gone mad, one had murdered his wife, and another had committed suicide; none had remained long in the house. Being sensible level-headed people, we discounted the stories as local superstition and moved in. Since then, we have known nothing but misery. We are at loggerheads with each other the whole time. Whatever one says, the other contradicts. We agree jointly on some course of action, then when the time comes, one or other of us will deny all knowledge of the arrangement. We have almost come to blows many times, and our formerly ideal marriage is now on the point of breaking up. Whenever we leave the house and stay elsewhere, our relationship soon improves, but we have only to return home to be at each other's throats again. We do not believe in the supernatural, but there is no doubt that the house is having a pernicious influence upon our lives. If there is a natural agency behind it, Mr Tombs, you are the man to find it. Will you inspect the house and give us your views?
The next day saw Tombs and myself alighting from a trap at the gatehouse of Janus Lodge. The driver had carried us thus far from the station, but refused to proceed any further, uttering many obscure prophecies of ill-omen. I threw him a coin for his services, and he tugged at a grizzled forelock.This was a painful experience for me, and I resolved to keep my hair tucked more securely inside my hat in future.
My first impression of the Lodge, as we rounded the curve of the drive, was one of extreme ugliness. Its architect, clearly a stranger to the golden ratio, had produced a heavy, inelegant, lump of a building. Its proportions suggested that it might have been a converted Saxon church, but it was not of such antiquity, although I would not have cared to date it with any hope of accuracy.The house was quite small, with four single-storey wings extending from a two-storey centre that was, I judged, 15 feet square. Three of the wings were of the same dimensions as the upper storey, and the fourth was twice as long. When I add that the height of each storey was also 15 feet, you will begin to appreciate the clumsiness of the design.
We let ourselves in with the keys that had been provided for us, the Arbuthnots being away from home for a few days. The interior gave an impression of more spaciousness than was apparent externally. In the centre was a large hall and a staircase. Rooms leading off on all sides also communicated directly with each other, so that it was possible to pass from one to another without returning to the hall. Tombs suggested that I make a swift tour of the downstairs and count the number of rooms, but I found myself utterly unable to keep track of those I had been in and those I had missed. My task was made more difficult because many of the chambers were in darkness, being either windowless or shuttered. I reported my failure to Tombs, who was sitting on the stairs idly scratching his initials into the varnish of the banisters with the small penknife which he kept upon his fob. To my astonishment, he snapped at me,
Well, what do you expect, Motson, when you insist on attempting such a foolhardy undertaking without consulting me! with which he sprang up and set off through the rooms, leaving me to trail apologetically in his wake. The edge was taken off my contrition, however, as it gradually became plain that the great man was as confused as I by the layout of the rooms. At one point we ascended the staircase to what we supposed to be the sole upstairs room, and were puzzled to find connecting doors there also. Our astonishment was as nothing compared to that which struck us when, having passed through another room or two, we found ourselves once more in the hall without having descended the stairs.
Fearful of voicing my true thoughts, lest they betray an unseemly lack of manliness, I started to berate Tombs for his inconsiderate vandalism of the banisters. My voice choked into near hysteria, however, when, seeking to point an accusing finger at the damage he had wrought, I found that no trace of his handiwork remained. Despite my by now incoherent jabbering, Tombs apparently caught the gist of what I was trying to convey, for he seized me firmly by the elbow and propelled me towards the front door, saying grimly,
Come Motson, this is no place for Christian gentlemen to loiter!
No mention was made of these events by either of us as we made our way home. In the reassuring surroundings of our own sitting-room, Tombs resorted to his habitual comforts of tobacco and music. He puffed away at one of his aromatic hand-rolled cigarettes, the special mixture for which he purchases from an itinerant vendor of Caribbean appearance, an agent presumably for tobacco growers fearful of rivals discovering the prices at which they sell their wares, for his transactions with Tombs are always clandestine. In the semi-stupor which his cigarette had, as usual, induced, my friend reached for his saxophone and improvised several choruses of Melancholy Baby, pausing occasionally to murmur
Soaring with the Bird, man, soaring with the Bird! in a slurred voice. I stole away reverentially from this poignant scene, and sought the solace of my bed.
At breakfast the following morning Tombs seemed his usual self. He was scanning the advertisements in the personal column of the Pigeon Fanciers' Gazette, as was his custom, on the offchance that the ambassador of a foreign power might be advertising for the return of a stolen secret treaty, or a scion of a noble house be seeking a brother unjustly deported to the colonies, when suddenly he threw down the paper and exclaimed,
Good Heavens, Motson, I have been a fool! Experience has taught me that this sort of remark is not an invitation to concur, so I stifled my response and waited.
The cellar, Motson! What about the peculiarity of the cellar?
But we saw no cellar, I protested.
Exactly! he proclaimed triumphantly.
Midmorning found us once more in the hall of Janus Lodge, this time armed with a bull's-eye lantern. I tried to keep my gaze averted from the banisters, but was uneasily aware that the initials W.T. were plainly visible there. Tombs went straight to the panelling under the stairs and pulled open a low cupboard door. In the closet thus revealed were three odd galoshes, a broken lacrosse stick, an umbrella with no handle, a small wooden plaque with Home, Sweet Home inscribed in pokerwork, a dog's feeding bowl bearing the name Bonzo, two tennis rackets with no strings, forty-eight issues of the Lady's Home Journal, a single ski, a frame with cracked glass holding the photograph of a cocker spaniel, a box of broken crockery, a sprung mousetrap containing the skeletal remains of its victim, and a broom head with no handle. At Tombs' behest, I threw it all out into the hall, revealing a trapdoor in the floor. Grasping its iron ring, I pulled it up, and fearfully shone the lantern into the cavity; there was, however, nothing to be seen but a ladder leading down to a large empty chamber, about 15 feet square and surprisingly deep for a cellar, having as much headroom as the rooms above ground.
With much apprehension on my part, we descended, Tombs humouring my timidity by allowing me to go first so that he could guard my escape route. The cellar, like the rooms above, had a door in each wall. Choosing one at random, we passed through into an identical chamber. With a growing presentiment of what was to come, we rapidly crossed the floor and opened another door. My knees buckled as I saw that we were again in the main hall, but I recovered my physical capabilities well enough to pass Tombs at a good pace on my way through the front door, even though he was moving at a speed that I had previously witnessed only in urchins playing Knock-Down-Ginger.
How did you know about the cellar, Tombs? I asked that evening after we had supped. Fortunately, the great man had exhausted his supplies of tobacco and was in no mood for saxophony, so we made ourselves comfortable and he began to propound his explanation.
What do you know of the fourth dimension, Motson?
It is usually considered to be Time, is it not?
Indeed, since the publication of Mr Einstein's theory of special relativity and his views on the space-time continuum, that has been one interpretation of the fourth dimension. It is also possible to imagine a fourth dimension of space, however. Naturally, experiencing only three spatial dimensions ourselves, it is not easy to envisage the consequences of extension or movement in a fourth. The best way to appreciate the concept is to imagine beings that know only two dimensions, and consider the effect on them of objects which extend and move in three dimensions. For example, if a sphere passed through their domain, what they would experience would be only what was happening at the intersection of the sphere with the plane surface that they inhabited, namely the emergence of a dot which would first grow in size, and then diminish until it disappeared. This restricted version of the truth would appear to them edgewise, so that they would observe only a line which first grew in length and then diminished. Mr Edwin A Abbott has given an entertaining account of the strange effects of a 2-dimensional existence in his book Flatland, which offers an insight into how restricted our view of the universe must be if there exists a fourth dimension of space.
Consider, next, how we might represent a 4-d object in three dimensions. Again, it is useful to step back one dimension and ask ourselves how we represent three dimensions in two. This can be achieved in more than one way. We can, for example, draw perspective views on a 2-d surface which suggest the appearance of a 3-d object. These succeed, however, only because we have experience of the appearance of 3-d objects; they would mean nothing to a Flatlander. Another way is to project the surfaces of a 3-d object onto a 2-d plane. Maps of the globe are an obvious example, and illustrate some of the strange effects which occur. For example, the extreme east of such a map must be interpreted as being contiguous with the extreme west: a movement traced off the right hand side must be continued by entering from the left - an apparent absurdity on the two-dimensional map, which makes sense only when you remember that the movement really takes place in three dimensions. Another example of surface projection is a very common one. Consider a cardboard carton: it is bounded by 6 faces deployed in 3-d space. We can, however, unfold the carton and deploy its faces in 2-d space.
Note that we still have 6 faces, but 5 of them have been rotated in 3-d space in order to bring them into the same 2-d plane as the first. In doing this, it is necessary to duplicate 7 of the edges and 4 of the vertices. More correctly, I should say that 2 vertices are duplicated, and 2 triplicated. Thus, while the 3-d carton has 6 faces, 12 edges, and 8 vertices, the 2-d projection of it has 6 faces, 19 edges, and 14 vertices. Imagine that a fly is crawling over the surface of the 3-d carton, and that you wish to trace its path on the 2-d projection. Every time the fly crosses one of the duplicated edges, your trace of its continuous progression must jump from one point to another of the projection, without traversing what appears to be intervening space. To a Flatlander, this would seem to contravene natural law, but not to one who can comprehend a third dimension.
I began to fear that Tombs had lost track of the subject matter, so I took the liberty of prompting him.
That is all very interesting, Tombs, but what has it to do with Janus Lodge?
You will soon appreciate its relevance, Motson, when I tell you that a four-dimensional hypercube can likewise be projected onto 3-d space. In its 4-d form it is bounded by 8 cubes, which between them have 24 faces, every face being shared by two adjacent cubes. In its 3-d projection, 17 of the faces have to be duplicated, resulting in a figure with 41 faces. The 3-d projection of a hypercube may be visualised as a cube with a similar cube stuck to each of its 6 faces, and an eighth stuck to one of the ends. Is that a familiar shape, Motson? It should be: it is precisely the shape of Janus Lodge once the cellar is taken into account.
But surely, Tombs, you are not suggesting that any house built in this shape would exhibit the same strange properties as Janus Lodge?
Indeed not, Motson. I believe that the Lodge was actually built in four-dimensional space by someone with extraordinary powers. What we saw from the outside was the 3-d projection of the house, but once we entered, we were moving in four dimensions. In the real 4-d Janus Lodge, each of the 8 rooms is abutted by 6 similar rooms. This is why we were able to move between rooms in a manner apparently impossible in 3-d space. In effect, Motson, we were the fly crawling through the cubes which bounded the hypercube. Every time we passed through one of the duplicated surfaces, we appeared to be transported between separated points without passing through the intervening space, but only because in the real 4-d Janus Lodge the points were truly adjacent.
But Tombs, how does that account for the disappearance and reappearance of your penknife scratchings, or for the disagreements between the Arbuthnots and, for that matter, between ourselves?
It accounts for those effects, my dear doctor, because the fourth dimension is not only a spatial dimension, it is also the time dimension. In moving around the house in a certain manner, we were moving in time. Oh, not in the dramatic way that Mr Wells would have us imagine - perhaps only by a second or two for each circuit of the house. Remember, Motson, that when two people live together amicably, they may both have changes of intention and emotion from time to time, but discord is avoided because they adapt and adjust to each other's moods. If they become unsynchronised, however, the harmony is destroyed. Imagine a piano duet in which one part is played just one beat behind the other - result, utter discord.
But Tombs, surely the fourth dimension is either spatial or temporal - it cannot be both at once.
But it can, Motson, and it is. Let me then put the matter as clearly as I can. Once again, it will be easier to grasp if we suppose that there are beings who are cognisant with only two dimensions, and explain how the third dimension of space might represent time to them. Have you ever noticed that when you drive a screw into a threaded hole, no movement is apparent if you restrict your field of view to the immediate vicinity of the hole? Although the screw is turning and moving forward into the hole, the cross-section at the plane where it enters remains unchanged. Now imagine Flatlanders inhabiting that cross-section. Their world would appear to remain unchanged spatially. As the screw turns, they would be fed a new cross-section every instant, but such changes as accompanied this progress would appear to them as events taking place in time, rather than a shift in space. In the third dimension of which they are totally unaware, there would exist a length of screw still to come - their future - and a length that has already passed through - their past. Now imagine that one of them discovers a means to travel in the third dimension. He becomes able to escape from the plane in which his fellows are trapped, and to move along the threads of the screw in either direction. If he moves outwards, he will be travelling into the future; if inwards, into the past.
This analogy enables us to avoid the elementary mistakes made by most of the writers who have chosen to fantasise upon this subject. For one thing, it becomes clear that one does not travel forward into the future, or backwards into the past. On the contrary, to visit (or revisit) the past, one must hurry forward in the direction that time has gone, so as to catch up with it. Likewise, one must travel backwards, against the tide of time, to experience the future before it is borne naturally into one's plane of normal existence. Similarly, writers who have not thought clearly about the matter suppose that if one visits the past, there might be some danger in interfering, in case one's natural present is affected, or even destroyed. Thus they ask, What would happen if you killed your own father before he begat you? The answer, of course, is: absolutely nothing untoward. The effect would be the same as rewriting history. You could alter all the books to say that Napoleon won the Battle of Waterloo, but it would not make any difference to the present state of Europe, because those events have already happened and are not going to be re-run. The length of screw that has already passed through will not return to the same plane of reference again, so one may interfere with what is on it with impunity.
Good Heavens, Tombs! What is to be done about the Arbuthnots and Janus Lodge, then? Incidentally, the name of the place gains added significance from your explanation of its properties.
It does indeed, doctor, it does indeed. I shall advise our client to move out instantly and to have the house dismantled as soon as possible. What motives its creator had we shall never know, but it clearly is of no beneficial use, and has already caused a great deal of anguish and danger.
A year or so later, a photograph of a familiar face caught my eye as I perused the evening paper, and I read the accompanying story with interest.
Mr James Arbuthnot astonished the City today by buying a controlling interest in the Bolivian Mining Company. Shares had fallen to rock-bottom following the success of the Xenophobe Party in the presidential elections. Mr Arbuthnot's purchase came just hours before a coup d'etat by General Montmorency Mendez, who immediately announced the abolition of income tax for all foreign businesses.
Mr Arbuthnot has had a meteoric rise in financial circles recently. Earlier this year he bought controlling interests in eight US insurance companies when share prices fell in the wake of the prediction by the Seismological Institute of major earthquakes in San Francisco, Chicago, Dallas, and Las Vegas. The Director of the Institute was subsequently found to be insane. During last June's torrential rainstorms it emerged that Mr Arbuthnot had recently become sole proprietor of Rainwear Conglomerates and Umbrellas International. He sold these holdings and re-invested in Worldwide Refrigeration Inc just prior to August's unprecedented heatwave.
Mr Arbuthnot is known for his shrewd dealings on the currency exchanges. He is a keen follower of the turf, and is reported to have as good an eye for a horse as for an investment. He was the centre of sensational scenes in Monte Carlo last summer when he backed six successive winning numbers at the roulette table. Officials of the Principality refuse to comment on reports that Mr Arbuthnot now owns Monaco.
I passed the paper to Tombs, and he read the item with a smile.
It is gratifying, is it not, my dear doctor, when our efforts are rewarded with such success? Our client obviously took my advice to heart, and had Janus Lodge destroyed. Had he not done so, there is no doubt that its baneful influence would have had a deleterious effect upon his mind, and would have prevented the success which freedom from its machinations has enabled him to achieve.
It is perspicacity of this order which makes my friend what he is.
I must confess, said the Author,
that I find it quite impossible to imagine a four-dimensional object, and am baffled as to how Tombs could so confidently describe the shape of one.
No apology is needed for being unable to visualise a 4-d object, said the Mathematician generously.
Nobody can, because we have no visual or tactile experience of more than three dimensions. But as to describing a 4-d shape, that is simply a matter of extrapolating from our knowledge of three dimensions.
At the Author's request, he continued,
Suppose that there were no dimensions. The only shape possible would be a point. Now suppose that a single dimension of space becomes available. The point can move in that dimension and trace out a straight line. The line is bounded by 2 points, at the start and finish positions of the tracing point. The line itself can be regarded as an infinite number of points occupying the space between those two positions.
Now suppose that a second dimension becomes available. The line can now move at right angles to itself in that dimension and sweep out a square area. The square is bounded by 4 straight lines, namely the sweeping line in its start position, the sweeping line in its finish position, and the 2 lines traced out by its bounding points. The square has 4 points or vertices, namely the 2 that bound the start line and the 2 that bound the finish line. The square may be regarded as an infinite number of lines occupying the space between the start and finish positions of the sweeping line.
Now suppose that a third dimension becomes available. The square can move at right angles to itself in that dimension and sweep out a cube. The cube is bounded by 6 planar surfaces, namely the square in its start position, the square in its finish position, and the 4 squares swept out by its bounding lines as it moved. The cube has 12 edges, namely the 4 lines that bound the start square, the 4 lines that bound the finish square, and the 4 lines traced by the 4 vertices of the square as it moved. The cube has 8 vertices, namely the 4 vertices of the square in its start position, and the 4 vertices of the square in its finish position. The cube may be regarded as an infinite number of squares occupying the space between the start square and the finish square.
Now suppose that a fourth dimension becomes available. The cube can move at right angles to itself in that dimension and sweep out a hypercube. The hypercube is bounded by 8 cubes, namely the cube in its start position, the cube in its finish position, and the 6 cubes swept out by its 6 bounding surfaces as it moved. The hypercube has 24 planar surfaces, namely the 6 bounding surfaces of the cube in its start position, the 6 bounding surfaces of the cube in its finish position, and the 12 areas swept out by its 12 edges as it moved. The hypercube has 32 edges, namely the 12 edges of the cube in its start position, the 12 edges of the cube in its finish position, and the 8 edges swept out by its 8 vertices as it moved. The hypercube has 16 vertices, namely the 8 vertices of the cube in its start position and the 8 vertices of the cube in its finish position. The hypercube can be regarded as an infinite number of cubes occupying the space between the the start cube and the finish cube.
You see how easy it is to build up a description of the four-dimensional shape without having to visualise it, simply by analogy with our three-dimensional experience. By the same process one can describe a five-dimensional rectangular object: it would have 32 vertices, 60 edges, 80 planar surfaces, 40 cubic aspects, and 10 bounding hypercubes. If you construct a table, the computations required to describe rectangular objects in any number of dimensions call for simple arithmetic only. The hardest part is finding names for the shapes.
Of course, said the Physicist,
that all applies only to shapes formed by extrusion. It is also possible to create shapes by rotation. Consider the uni-dimensional line, for example. When a second dimension becomes available, instead of moving the line bodily in the new dimension at right angles to itself, why not anchor one end and rotate the other? You would then create a 2-dimensional circle. The stationary end of the sweeping line becomes subsumed in the interior of the circle, and the moving end traces out a line whose finish point is coincident with its start point, so that the line is endless. Thus the circle has no vertices and only one bounding line, its circumference. The circle is a two-dimensional area; its circumference, although it is only a line and has no area, also requires two dimensions because it is curved.
"When a third dimension becomes available, you can rotate the circle about one of its diagonals to create a sphere. The original circle becomes subsumed into the interior of the sphere; its circumference makes a complete revolution in the third dimension until it meets with and merges with itself. In so doing, it sweeps out a surface which has no boundary line. That surface is the boundary of the sphere. The sphere is a three-dimensional volume; its surface, although it has only area and not volume, also requires three dimensions because it is curved."
Now comes the mind-extending part. If a fourth dimension is available, you can rotate the sphere about the circle of its cross-section to create a hypersphere. The original sphere becomes subsumed in the interior of the hypersphere; its surface makes a complete revolution in the fourth dimension until it meets with and merges with itself. In so doing, it sweeps out a volume which has no boundary surface. That volume is the boundary of the hypersphere. The hypersphere is four-dimensional; its boundary, although it has volume, requires four dimensions because it is curved.
The Author was clearly unhappy with this explanation.
How can you rotate something about its cross-section?
The Physicist appeared surprised.
Naturally, rotation in the fourth dimension takes place about a two-dimensional axis.
Naturally? asked the Author.
Well, of course. Look here: to produce the two-dimensional circle, you rotated a one-dimensional line about a zero-dimensional point, did you not? The Author nodded.
And to produce a three-dimensional sphere, you rotated a two-dimensional circle about a one-dimensional line, did you not? The Author nodded again.
So to produce a four-dimensional hypersphere, you will have to rotate a three-dimensional sphere about a two-dimensional axis, will you not?
Seeing the look on the Author's face, he continued,
If you are more at home with the concept of curvature than with rotation, you can look at it this way. Start with a uni-dimensional straight line. Now curve it in 2-d space, joining the ends together so as to enclose a 2-d circle. Now curve the area of the circle in 3-d space, joining the edge together so as to enclose a 3-d sphere. Now curve the volume of the sphere in 4-d space, joining the surface together so as to enclose a 4-d hypersphere. If you find that last step difficult, remember this: when you curved the line, every part of its length went to form the boundary of the circle, and when you curved the circle, every part of its area went to form the surface of the sphere. So when you curve the sphere, every part of its volume must go to form the boundary of the hypersphere. Is that clear?
The Author looked at him dourly for a moment, then intoned in a peculiar cadence, "
On the whole, I'd rather be in Philadelphia discussing the Ontological Proof, upon which the assembly dispersed.